From 4d37f3e2d2db14a96a7bc4488ab483c81d3bbb69 Mon Sep 17 00:00:00 2001
From: Aleks Kissinger <aleks0@gmail.com>
Date: Sun, 29 Dec 2024 22:35:22 +0000
Subject: [PATCH] started writing code for errors and corrections

---
 demos/PauliWebs.ipynb | 1278 ++++++++++++++++++++++++++++++++++-------
 pyzx/pauliweb.py      |   88 ++-
 2 files changed, 1101 insertions(+), 265 deletions(-)

diff --git a/demos/PauliWebs.ipynb b/demos/PauliWebs.ipynb
index fb959b81..ba6738c8 100644
--- a/demos/PauliWebs.ipynb
+++ b/demos/PauliWebs.ipynb
@@ -18,21 +18,23 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "This notebook demonstrates using PyZX's methods for automatically computing and drawing (bounded) Pauli webs. A Pauli web (a.k.a. _correlation surface_) is a coloring of the edges of ZX-diagram with the following properties for every Z spider `v`:\n",
+    "This notebook demonstrates using PyZX's methods for automatically computing and drawing bounded Pauli webs. A Pauli web (a.k.a. _correlation surface_) is a labelling of the edges of ZX-diagram from the set $\\{I, X, Y, Z\\}$ with certain properties. To explain these properties, it is useful to make some definitions.\n",
     "\n",
-    "- _all-or-nothing_: either no incident edges or all incident edges of `v` are labelled X or Y\n",
-    "- _parity_: an even number of incident edges of `v` are labelled Y or Z\n",
+    "To account for Hadmard edges, we label \"half-edges\" of a diagram. For a pair of connected vertices `v`, `w`, we represent the half-edge closest to `v` as `(v,w)` and the half-edge closest to `w` as `(w,v)`.\n",
     "\n",
-    "All X spiders satisfy analogous conditions, swapping the role of X and Z. A _bounded Pauli web_ is a Pauli web with a chosen set of spiders, called the _boundary_ where the parity condition is allowed to be violated.\n",
+    "We say a spider `v` is _stabilised_ by a Pauli web if `v` (considered as a state) is a +1 eigenstate of the Pauli string labelling its adjacent half-edges and we say `v` is _anti-stabilised_ by a Pauli web if it is a -1 eigenstate of its adjacent half-edges.\n",
     "\n",
-    "We say a spider `anti-commutes` with a Pauli web if it is touching an edge of a different colour.\n",
+    "A Pauli web is said to be _closed_ if:\n",
+    "1. every spider `v` in the ZX-diagram is stabilised or anti-stabilised\n",
+    "2. for every half-edge `(v,w)` labelled by a Pauli $P$, the other half-edge `(w,v)` is labelled by $P$ for a simple edge and $HPH$ for a Hadamard edge\n",
+    "\n",
+    "Otherwise, we say a Pauli web is _open_. The spiders and edges violating conditions 1 and 2 above are called the _boundary_ of a Pauli web.\n",
     "\n",
     "The function `compute_pauli_webs` automitically associates each non-input vertex `v` in the diagram to an integer `order[v]` giving a time-ordering, and one or two bounded Pauli webs, with the following properties:\n",
-    "1. the boundary of the web consists of only `v` itself and inputs\n",
+    "1. the boundary of the web consists of only `v` itself, earlier non-Clifford spiders `w` (`order[v] < order[w]>`), and inputs\n",
     "2. Z spiders have a web `zweb[v]` with a Z-colored edge incident to `v`\n",
     "3. X spiders have a web `xweb[v]` with a X-colored edge incident to `v`\n",
     "4. output vertices have two webs `zweb[v]` and `xweb[v]` corresponding to both colors at `v`\n",
-    "5. for non-Pauli spiders `v, w`, if `v` anti-commutes with the Pauli web of `w` then `order[v] < order[w]`\n",
     "\n",
     "Under the hood, this is using PyZX's gflow-finding algorithm to compute a focussed Pauli flow and translate this data into Pauli webs."
    ]
@@ -52,7 +54,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-tZ7Nj7Ld\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-9dj8FiGJ\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -416,7 +418,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-tZ7Nj7Ld',\n",
+       "showGraph('#graph-output-9dj8FiGJ',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 200, 200, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -449,7 +451,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-awKePwp0\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-SnzbMlXA\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -813,7 +815,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-awKePwp0',\n",
+       "showGraph('#graph-output-SnzbMlXA',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 1, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 1, \"t\": \"X\"}, {\"source\": 3, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}]}'), 200, 200, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -828,7 +830,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-Dm0snofX\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-NaMJywMd\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -1192,7 +1194,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-Dm0snofX',\n",
+       "showGraph('#graph-output-NaMJywMd',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 1, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 1, \"t\": \"X\"}, {\"source\": 5, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 5, \"t\": \"X\"}]}'), 200, 200, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -1207,7 +1209,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-ZPwbKnLM\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-Hokb0H2r\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -1571,7 +1573,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-ZPwbKnLM',\n",
+       "showGraph('#graph-output-Hokb0H2r',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 0, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 0, \"t\": \"Z\"}]}'), 200, 200, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -1586,7 +1588,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-vDPyzMv9\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-ZeSLq1Sp\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -1950,7 +1952,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-vDPyzMv9',\n",
+       "showGraph('#graph-output-ZeSLq1Sp',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 2, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 0, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 0, \"t\": \"Z\"}]}'), 200, 200, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -1979,7 +1981,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-S0RKmySn\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-LjtoNhmm\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -2343,7 +2345,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-S0RKmySn',\n",
+       "showGraph('#graph-output-LjtoNhmm',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 5, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 5, \"t\": \"X\"}]}'), 200, 200, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -2358,7 +2360,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-RRErRC6R\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-9VAy6xYd\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -2722,7 +2724,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-RRErRC6R',\n",
+       "showGraph('#graph-output-9VAy6xYd',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 1, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 1, \"t\": \"X\"}, {\"source\": 5, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 5, \"t\": \"X\"}]}'), 200, 200, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -2737,7 +2739,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-oC3I9e6C\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-kxSr7u8g\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -3101,7 +3103,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-oC3I9e6C',\n",
+       "showGraph('#graph-output-kxSr7u8g',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 0, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 0, \"t\": \"Z\"}]}'), 200, 200, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -3116,7 +3118,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-gjSkoX4A\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-75Qf0jAA\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -3480,7 +3482,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-gjSkoX4A',\n",
+       "showGraph('#graph-output-75Qf0jAA',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 2, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}]}'), 200, 200, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -3513,7 +3515,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-1Qq9pyHA\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-v3dI7tCV\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -3877,7 +3879,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-1Qq9pyHA',\n",
+       "showGraph('#graph-output-v3dI7tCV',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -3915,7 +3917,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-teMJhLnd\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-wJg9KGzD\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -4279,7 +4281,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-teMJhLnd',\n",
+       "showGraph('#graph-output-wJg9KGzD',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 4, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 8, \"t\": \"X\"}, {\"source\": 6, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 6, \"t\": \"X\"}, {\"source\": 8, \"target\": 4, \"t\": \"X\"}, {\"source\": 8, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 8, \"t\": \"X\"}, {\"source\": 8, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 8, \"t\": \"X\"}, {\"source\": 12, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 12, \"t\": \"X\"}, {\"source\": 5, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 5, \"t\": \"X\"}, {\"source\": 9, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 9, \"t\": \"X\"}, {\"source\": 15, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 15, \"t\": \"X\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -4294,7 +4296,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-eFeBvnbi\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-DWEtXQph\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -4658,7 +4660,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-eFeBvnbi',\n",
+       "showGraph('#graph-output-DWEtXQph',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 3, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 5, \"t\": \"Z\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 7, \"t\": \"Z\"}, {\"source\": 13, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 14, \"t\": \"Z\"}, {\"source\": 14, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 13, \"t\": \"Z\"}, {\"source\": 8, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 8, \"t\": \"Z\"}, {\"source\": 10, \"target\": 14, \"t\": \"Z\"}, {\"source\": 14, \"target\": 10, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -4673,7 +4675,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-MZqx5g64\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-sYKVf5Ub\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -5037,7 +5039,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-MZqx5g64',\n",
+       "showGraph('#graph-output-sYKVf5Ub',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 6, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 6, \"t\": \"X\"}, {\"source\": 9, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 9, \"t\": \"X\"}, {\"source\": 1, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}, {\"source\": 2, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 2, \"t\": \"X\"}, {\"source\": 16, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 16, \"t\": \"X\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -5052,7 +5054,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-vh0XAu3c\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-cBXRSh0r\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -5416,7 +5418,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-vh0XAu3c',\n",
+       "showGraph('#graph-output-cBXRSh0r',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 5, \"t\": \"Z\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 7, \"t\": \"Z\"}, {\"source\": 9, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 11, \"t\": \"Z\"}, {\"source\": 11, \"target\": 9, \"t\": \"Z\"}, {\"source\": 11, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 11, \"t\": \"Z\"}, {\"source\": 11, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 11, \"t\": \"Z\"}, {\"source\": 8, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 8, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -5431,7 +5433,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-uXwzJznd\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-mvQy8duQ\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -5795,7 +5797,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-uXwzJznd',\n",
+       "showGraph('#graph-output-mvQy8duQ',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 4, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 8, \"t\": \"X\"}, {\"source\": 8, \"target\": 4, \"t\": \"X\"}, {\"source\": 8, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 8, \"t\": \"X\"}, {\"source\": 8, \"target\": 12, \"t\": \"X\"}, {\"source\": 10, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 14, \"t\": \"X\"}, {\"source\": 12, \"target\": 8, \"t\": \"X\"}, {\"source\": 12, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 12, \"t\": \"X\"}, {\"source\": 14, \"target\": 10, \"t\": \"X\"}, {\"source\": 14, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 17, \"t\": \"X\"}, {\"source\": 17, \"target\": 14, \"t\": \"X\"}, {\"source\": 9, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 9, \"t\": \"X\"}, {\"source\": 1, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -5810,7 +5812,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-JoTq8LDc\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-mbaAihTO\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -6174,7 +6176,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-JoTq8LDc',\n",
+       "showGraph('#graph-output-mbaAihTO',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 17, \"target\": 14, \"t\": \"Z\"}, {\"source\": 14, \"target\": 17, \"t\": \"Z\"}, {\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 3, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 5, \"t\": \"Z\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 7, \"t\": \"Z\"}, {\"source\": 4, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 4, \"t\": \"Z\"}, {\"source\": 10, \"target\": 14, \"t\": \"Z\"}, {\"source\": 14, \"target\": 10, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -6205,7 +6207,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-XIlR8bQB\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-av1x89gh\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -6569,7 +6571,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-XIlR8bQB',\n",
+       "showGraph('#graph-output-av1x89gh',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -6597,7 +6599,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-iq8ttVJr\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-qF2SMlK3\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -6961,7 +6963,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-iq8ttVJr',\n",
+       "showGraph('#graph-output-qF2SMlK3',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 15, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 15, \"t\": \"X\"}, {\"source\": 4, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 4, \"t\": \"X\"}, {\"source\": 6, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 6, \"t\": \"X\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -6976,7 +6978,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-ZpUsYBvU\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-eQ4YQufA\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -7340,7 +7342,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-ZpUsYBvU',\n",
+       "showGraph('#graph-output-eQ4YQufA',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 3, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 5, \"t\": \"Z\"}, {\"source\": 13, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 13, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -7355,7 +7357,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-FSrpxlJ4\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-TtbVE3mZ\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -7719,7 +7721,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-FSrpxlJ4',\n",
+       "showGraph('#graph-output-TtbVE3mZ',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 16, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 16, \"t\": \"X\"}, {\"source\": 1, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}, {\"source\": 6, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 6, \"t\": \"X\"}, {\"source\": 2, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 2, \"t\": \"X\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -7734,7 +7736,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-WixF159B\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-1H9Ry1BI\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -8098,7 +8100,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-WixF159B',\n",
+       "showGraph('#graph-output-1H9Ry1BI',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 9, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 9, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 5, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -8113,7 +8115,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-s1GYJkN1\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-97M7ktIP\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -8477,7 +8479,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-s1GYJkN1',\n",
+       "showGraph('#graph-output-97M7ktIP',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 10, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 17, \"t\": \"X\"}, {\"source\": 17, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 10, \"t\": \"X\"}, {\"source\": 4, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 4, \"t\": \"X\"}, {\"source\": 1, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -8492,7 +8494,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-yHiT7TTV\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-i2b1J2Yn\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -8856,7 +8858,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-yHiT7TTV',\n",
+       "showGraph('#graph-output-i2b1J2Yn',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 3, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 5, \"t\": \"Z\"}, {\"source\": 17, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 17, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -8896,7 +8898,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-rRU4kSxP\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-rMRH1mtT\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -9260,7 +9262,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-rRU4kSxP',\n",
+       "showGraph('#graph-output-rMRH1mtT',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"1\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 200, 150, 50,\n",
        "  10.0, false, false, '');\n",
        "</script>"
@@ -9275,7 +9277,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-pV34CjZI\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-YNR311rS\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -9639,7 +9641,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-pV34CjZI',\n",
+       "showGraph('#graph-output-YNR311rS',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"2\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 200, 150, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -9678,7 +9680,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-e2MtVwrt\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-ghxn1f0V\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -10042,7 +10044,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-e2MtVwrt',\n",
+       "showGraph('#graph-output-ghxn1f0V',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"2\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 0, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 0, \"t\": \"Z\"}]}'), 200, 150, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -10057,7 +10059,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-VPC6ZGC4\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-ZYb6bGlU\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -10421,7 +10423,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-VPC6ZGC4',\n",
+       "showGraph('#graph-output-ZYb6bGlU',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"2\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 2, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 2, \"t\": \"Z\"}]}'), 200, 150, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -10461,7 +10463,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-7ovT72xb\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-6Tp04b0r\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -10825,7 +10827,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-7ovT72xb',\n",
+       "showGraph('#graph-output-6Tp04b0r',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 300, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 450, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 450, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 500, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 500, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 550, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 550, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 550, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"6\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"14\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 600, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -10868,7 +10870,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-wdSGXzh7\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-Y8s8k6ID\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -11232,7 +11234,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-wdSGXzh7',\n",
+       "showGraph('#graph-output-Y8s8k6ID',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 300, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 450, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 450, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 500, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 500, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 550, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 550, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 550, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"6\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": 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{\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 3, \"t\": \"X\"}, {\"source\": 5, \"target\": 8, \"t\": \"X\"}, {\"source\": 6, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 6, \"t\": \"X\"}, {\"source\": 20, \"target\": 18, \"t\": \"X\"}, {\"source\": 18, \"target\": 20, \"t\": \"X\"}, {\"source\": 8, \"target\": 5, \"t\": \"X\"}, {\"source\": 8, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 8, \"t\": \"X\"}, {\"source\": 8, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 8, \"t\": \"X\"}, {\"source\": 11, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 11, \"t\": \"Z\"}, {\"source\": 11, \"target\": 17, \"t\": \"X\"}, {\"source\": 15, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 15, \"t\": \"X\"}, {\"source\": 17, \"target\": 11, \"t\": \"X\"}, {\"source\": 17, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 17, \"t\": \"X\"}, {\"source\": 17, \"target\": 18, \"t\": \"X\"}, {\"source\": 18, \"target\": 17, \"t\": \"X\"}, {\"source\": 4, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 4, \"t\": \"Z\"}, {\"source\": 12, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 12, \"t\": \"Z\"}]}'), 600, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -11247,7 +11249,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-0XMIg22K\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-2sp4yjTz\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -11611,7 +11613,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-0XMIg22K',\n",
+       "showGraph('#graph-output-2sp4yjTz',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, 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{\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 2, \"target\": 6, \"t\": \"Y\"}, {\"source\": 6, \"target\": 2, \"t\": \"Y\"}, {\"source\": 4, \"target\": 0, \"t\": \"Y\"}, {\"source\": 0, \"target\": 4, \"t\": \"Y\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 10, \"t\": \"Y\"}, {\"source\": 6, \"target\": 7, \"t\": \"Y\"}, {\"source\": 7, \"target\": 6, \"t\": \"Y\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 9, \"t\": \"Y\"}, {\"source\": 9, \"target\": 7, \"t\": \"Y\"}, {\"source\": 9, \"target\": 10, \"t\": \"Y\"}, {\"source\": 10, \"target\": 9, \"t\": \"Y\"}, {\"source\": 9, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 9, \"t\": \"Z\"}, {\"source\": 10, \"target\": 4, \"t\": \"Y\"}, {\"source\": 10, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 10, \"t\": \"Z\"}, {\"source\": 18, \"target\": 17, \"t\": \"Z\"}, {\"source\": 17, \"target\": 18, \"t\": \"Z\"}, {\"source\": 18, \"target\": 19, \"t\": \"Z\"}, {\"source\": 19, \"target\": 18, \"t\": \"Z\"}, {\"source\": 18, \"target\": 20, \"t\": \"Z\"}, {\"source\": 20, \"target\": 18, \"t\": \"Z\"}, {\"source\": 3, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 3, \"t\": \"Z\"}, {\"source\": 5, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 5, \"t\": \"Z\"}, {\"source\": 11, \"target\": 17, \"t\": \"Z\"}, {\"source\": 17, \"target\": 11, \"t\": \"Z\"}, {\"source\": 13, \"target\": 19, \"t\": \"Z\"}, {\"source\": 19, \"target\": 13, \"t\": \"Z\"}]}'), 600, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -11626,7 +11628,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-8GRqzBVV\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-q6wIXFG9\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -11990,7 +11992,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-8GRqzBVV',\n",
+       "showGraph('#graph-output-q6wIXFG9',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, 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{\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 3, \"t\": \"X\"}, {\"source\": 5, \"target\": 8, \"t\": \"X\"}, {\"source\": 6, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 6, \"t\": \"X\"}, {\"source\": 21, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 21, \"t\": \"X\"}, {\"source\": 8, \"target\": 5, \"t\": \"X\"}, {\"source\": 8, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 8, \"t\": \"X\"}, {\"source\": 8, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 8, \"t\": \"X\"}, {\"source\": 15, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 15, \"t\": \"X\"}, {\"source\": 12, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 12, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}]}'), 600, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -12005,7 +12007,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-Zmro6vx5\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-8bf2Cg5s\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -12369,7 +12371,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-Zmro6vx5',\n",
+       "showGraph('#graph-output-8bf2Cg5s',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, 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false, \"vdata\": []}, {\"name\": \"19\", \"x\": 500, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 550, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 550, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 550, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"6\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": 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{\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 1, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}, {\"source\": 2, \"target\": 6, \"t\": \"Y\"}, {\"source\": 6, \"target\": 2, \"t\": \"Y\"}, {\"source\": 4, \"target\": 0, \"t\": \"Y\"}, {\"source\": 0, \"target\": 4, \"t\": \"Y\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 10, \"t\": \"Y\"}, {\"source\": 6, \"target\": 7, \"t\": \"Y\"}, {\"source\": 7, \"target\": 6, \"t\": \"Y\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 9, \"t\": \"Y\"}, {\"source\": 9, \"target\": 7, \"t\": \"Y\"}, {\"source\": 9, \"target\": 10, \"t\": \"Y\"}, {\"source\": 10, \"target\": 9, \"t\": \"Y\"}, {\"source\": 9, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 9, \"t\": \"Z\"}, {\"source\": 10, \"target\": 4, \"t\": \"Y\"}, {\"source\": 10, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 10, \"t\": \"Z\"}, {\"source\": 12, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 14, \"t\": \"Z\"}, {\"source\": 14, \"target\": 12, \"t\": \"X\"}, {\"source\": 14, \"target\": 15, \"t\": \"X\"}, {\"source\": 15, \"target\": 14, \"t\": \"Z\"}, {\"source\": 16, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 17, \"t\": \"Z\"}, {\"source\": 17, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 21, \"t\": \"Z\"}, {\"source\": 21, \"target\": 16, \"t\": \"Z\"}, {\"source\": 11, \"target\": 17, \"t\": \"Z\"}, {\"source\": 17, \"target\": 11, \"t\": \"Z\"}]}'), 600, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -12384,7 +12386,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-rO89VWio\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-J6advSRG\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -12748,7 +12750,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-rO89VWio',\n",
+       "showGraph('#graph-output-J6advSRG',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, 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{\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 6, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 6, \"t\": \"X\"}, {\"source\": 11, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 11, \"t\": \"Z\"}, {\"source\": 11, \"target\": 17, \"t\": \"X\"}, {\"source\": 13, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 19, \"t\": \"X\"}, {\"source\": 15, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 15, \"t\": \"X\"}, {\"source\": 17, \"target\": 11, \"t\": \"X\"}, {\"source\": 17, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 17, \"t\": \"X\"}, {\"source\": 17, \"target\": 18, \"t\": \"X\"}, {\"source\": 18, \"target\": 17, \"t\": \"X\"}, {\"source\": 19, \"target\": 13, \"t\": \"X\"}, {\"source\": 19, \"target\": 18, \"t\": \"X\"}, {\"source\": 18, \"target\": 19, \"t\": \"X\"}, {\"source\": 19, \"target\": 22, \"t\": \"X\"}, {\"source\": 22, \"target\": 19, \"t\": \"X\"}, {\"source\": 4, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 4, \"t\": \"Z\"}, {\"source\": 7, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 7, \"t\": \"X\"}, {\"source\": 12, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 12, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}]}'), 600, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -12763,7 +12765,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-C9FwOF4x\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-4YftlOii\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -13127,7 +13129,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-C9FwOF4x',\n",
+       "showGraph('#graph-output-4YftlOii',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 300, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 450, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 450, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 500, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 500, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 550, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 550, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 550, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"6\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": 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{\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 1, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}, {\"source\": 2, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 2, \"t\": \"Z\"}, {\"source\": 7, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 9, \"t\": \"Z\"}, {\"source\": 22, \"target\": 19, \"t\": \"Z\"}, {\"source\": 19, \"target\": 22, \"t\": \"Z\"}, {\"source\": 9, \"target\": 7, \"t\": \"Z\"}, {\"source\": 9, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 9, \"t\": \"Z\"}, {\"source\": 3, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 3, \"t\": \"Z\"}, {\"source\": 4, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 4, \"t\": \"Z\"}, {\"source\": 5, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 5, \"t\": \"Z\"}, {\"source\": 13, \"target\": 19, \"t\": \"Z\"}, {\"source\": 19, \"target\": 13, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}]}'), 600, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -13159,7 +13161,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-4C5WB6o0\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-dqWByjyL\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -13523,7 +13525,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-4C5WB6o0',\n",
+       "showGraph('#graph-output-dqWByjyL',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"1\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 200, 150, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -13538,7 +13540,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-iQRVBhCa\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-6QDEMZk5\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -13902,7 +13904,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-iQRVBhCa',\n",
+       "showGraph('#graph-output-6QDEMZk5',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"1\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 1, \"target\": 0, \"t\": \"Y\"}, {\"source\": 0, \"target\": 1, \"t\": \"Y\"}, {\"source\": 1, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 1, \"t\": \"X\"}]}'), 200, 150, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -13936,7 +13938,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-rKQJVTe9\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-wMvHqNBe\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -14300,7 +14302,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-rKQJVTe9',\n",
+       "showGraph('#graph-output-wMvHqNBe',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 28.571428571428573, \"y\": 57.142857142857146, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 28.571428571428573, \"y\": 85.71428571428572, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 28.571428571428573, \"y\": 114.28571428571429, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 28.571428571428573, \"y\": 142.85714285714286, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 57.142857142857146, \"y\": 114.28571428571429, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 85.71428571428572, \"y\": 114.28571428571429, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 85.71428571428572, \"y\": 85.71428571428572, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", 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{\"name\": \"22\", \"x\": 400.0, \"y\": 142.85714285714286, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"23\", \"x\": 400.0, \"y\": 57.142857142857146, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"24\", \"x\": 428.5714285714286, \"y\": 57.142857142857146, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"25\", \"x\": 457.14285714285717, \"y\": 85.71428571428572, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"26\", \"x\": 485.7142857142857, \"y\": 85.71428571428572, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"27\", \"x\": 485.7142857142857, \"y\": 114.28571428571429, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"28\", \"x\": 514.2857142857143, \"y\": 57.142857142857146, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"29\", \"x\": 514.2857142857143, \"y\": 114.28571428571429, \"t\": 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\"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"17\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"23\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"27\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"20\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"21\", \"target\": \"25\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"22\", \"target\": \"23\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"22\", \"target\": \"38\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"23\", \"target\": \"24\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"24\", \"target\": \"28\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"25\", \"target\": \"26\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"26\", \"target\": \"27\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"26\", \"target\": \"32\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"29\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"28\", \"target\": \"29\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"28\", \"target\": \"31\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"29\", \"target\": \"30\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"30\", \"target\": \"33\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"31\", \"target\": \"32\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"31\", \"target\": \"36\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"32\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"33\", \"target\": \"42\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"34\", \"target\": \"35\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"35\", \"target\": \"37\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"36\", \"target\": \"37\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"36\", \"target\": \"39\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"37\", \"target\": \"41\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"38\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"39\", \"target\": \"40\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 800.0, 171.42857142857144, 28.571428571428573,\n",
        "  5.714285714285715, false, true, '');\n",
        "</script>"
@@ -14315,7 +14317,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-EF8HU0dN\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-2Bd9OUhL\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -14679,7 +14681,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-EF8HU0dN',\n",
+       "showGraph('#graph-output-2Bd9OUhL',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 28.571428571428573, \"y\": 57.142857142857146, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 28.571428571428573, \"y\": 85.71428571428572, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 28.571428571428573, \"y\": 114.28571428571429, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 28.571428571428573, \"y\": 142.85714285714286, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 57.142857142857146, \"y\": 114.28571428571429, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 85.71428571428572, \"y\": 114.28571428571429, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 85.71428571428572, \"y\": 85.71428571428572, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 114.28571428571429, \"y\": 142.85714285714286, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 114.28571428571429, \"y\": 114.28571428571429, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 142.85714285714286, \"y\": 85.71428571428572, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 171.42857142857144, \"y\": 114.28571428571429, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 200.0, \"y\": 57.142857142857146, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 200.0, \"y\": 142.85714285714286, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 228.57142857142858, \"y\": 142.85714285714286, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 228.57142857142858, \"y\": 114.28571428571429, \"t\": 1, \"phase\": 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{\"name\": \"22\", \"x\": 400.0, \"y\": 142.85714285714286, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"23\", \"x\": 400.0, \"y\": 57.142857142857146, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"24\", \"x\": 428.5714285714286, \"y\": 57.142857142857146, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"25\", \"x\": 457.14285714285717, \"y\": 85.71428571428572, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"26\", \"x\": 485.7142857142857, \"y\": 85.71428571428572, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"27\", \"x\": 485.7142857142857, \"y\": 114.28571428571429, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"28\", \"x\": 514.2857142857143, \"y\": 57.142857142857146, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"29\", \"x\": 514.2857142857143, \"y\": 114.28571428571429, \"t\": 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        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
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        "\n",
-       "showGraph('#graph-output-Wis5b5Uc',\n",
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-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-J2YFModX\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-CDhkMZ48\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -15457,7 +15459,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-J2YFModX',\n",
+       "showGraph('#graph-output-CDhkMZ48',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 50, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"3\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 150, \"y\": 250, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"27\", \"x\": 200, \"y\": 200, \"t\": 1, \"phase\": \"3\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"34\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"38\", \"x\": 200, \"y\": 250, \"t\": 1, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"40\", \"x\": 250, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"41\", \"x\": 250, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"42\", \"x\": 250, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"43\", \"x\": 250, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"44\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"13\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"27\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"13\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"44\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"38\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"42\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"34\", \"target\": \"41\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"38\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"40\", \"target\": \"44\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 300, 300, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -15472,7 +15474,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-y9TasGED\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-9ZNNTFiU\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -15836,7 +15838,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-y9TasGED',\n",
+       "showGraph('#graph-output-9ZNNTFiU',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 50, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"3\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 150, \"y\": 250, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"27\", \"x\": 200, \"y\": 200, \"t\": 1, \"phase\": \"3\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"34\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"38\", \"x\": 200, \"y\": 250, \"t\": 1, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"40\", \"x\": 250, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"41\", \"x\": 250, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"42\", \"x\": 250, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"43\", \"x\": 250, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"44\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"13\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"27\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"13\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"44\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"38\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"42\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"34\", \"target\": \"41\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"38\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"40\", \"target\": \"44\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 4, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 34, \"t\": \"X\"}, {\"source\": 34, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 4, \"t\": \"Z\"}, {\"source\": 0, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 0, \"t\": \"Z\"}, {\"source\": 41, \"target\": 34, \"t\": \"Z\"}, {\"source\": 34, \"target\": 41, \"t\": \"Z\"}]}'), 300, 300, 50,\n",
        "  10.0, false, false, '');\n",
        "</script>"
@@ -15851,7 +15853,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-uiNKVneX\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-46G485iI\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -16215,7 +16217,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-uiNKVneX',\n",
+       "showGraph('#graph-output-46G485iI',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 50, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"3\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 150, \"y\": 250, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"27\", \"x\": 200, \"y\": 200, \"t\": 1, \"phase\": \"3\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"34\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"38\", \"x\": 200, \"y\": 250, \"t\": 1, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"40\", \"x\": 250, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"41\", \"x\": 250, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"42\", \"x\": 250, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"43\", \"x\": 250, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"44\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"13\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"27\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"13\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"44\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"38\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"42\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"34\", \"target\": \"41\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"38\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"40\", \"target\": \"44\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 34, \"target\": 41, \"t\": \"X\"}, {\"source\": 41, \"target\": 34, \"t\": \"X\"}, {\"source\": 34, \"target\": 5, \"t\": \"Y\"}, {\"source\": 5, \"target\": 34, \"t\": \"Y\"}, {\"source\": 34, \"target\": 27, \"t\": \"X\"}, {\"source\": 27, \"target\": 34, \"t\": \"Z\"}, {\"source\": 34, \"target\": 4, \"t\": \"Y\"}, {\"source\": 4, \"target\": 34, \"t\": \"Y\"}, {\"source\": 4, \"target\": 2, \"t\": \"Y\"}, {\"source\": 2, \"target\": 4, \"t\": \"Y\"}, {\"source\": 4, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 4, \"t\": \"Z\"}, {\"source\": 5, \"target\": 1, \"t\": \"X\"}, {\"source\": 1, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 27, \"t\": \"X\"}, {\"source\": 27, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 5, \"t\": \"Z\"}]}'), 300, 300, 50,\n",
        "  10.0, false, false, '');\n",
        "</script>"
@@ -16252,18 +16254,22 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Clifford+T Examples"
+    "# Errors and corrections\n",
+    "\n",
+    "Before moving on to Clifford+T examples, we'll show how to introduce errors into ZX-diagrams, correct them, and prove correctness.\n",
+    "\n",
+    "We can use the `PauliWeb` class to represent arbitrary Pauli errors on half edges. Note this will _not_ usually be a valid, closed Pauli web, but we can still use the `PauliWeb` class to hold this data."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 16,
+   "execution_count": 26,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-ZcsXLAFD\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-xT7LRFtp\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -16627,9 +16633,9 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-ZcsXLAFD',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 26.666666666666668, \"y\": 53.333333333333336, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 26.666666666666668, \"y\": 80.0, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 26.666666666666668, \"y\": 106.66666666666667, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 53.333333333333336, \"y\": 80.0, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 53.333333333333336, \"y\": 53.333333333333336, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 80.0, \"y\": 53.333333333333336, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 80.0, \"y\": 80.0, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 106.66666666666667, \"y\": 53.333333333333336, \"t\": 2, \"phase\": 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{\"name\": \"62\", \"x\": 773.3333333333334, \"y\": 53.333333333333336, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"63\", \"x\": 773.3333333333334, \"y\": 80.0, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"64\", \"x\": 773.3333333333334, \"y\": 106.66666666666667, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, 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-       "  5.333333333333334, false, false, '');\n",
+       "showGraph('#graph-output-xT7LRFtp',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 50, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 250, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 300, \"y\": 200, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 350, \"y\": 250, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 400, \"y\": 250, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 400, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 450, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 450, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 500, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 550, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 550, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 600, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 600, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 600, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"23\", \"x\": 600, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"23\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"17\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 650, 300, 50,\n",
+       "  10.0, false, true, '');\n",
        "</script>"
       ],
       "text/plain": [
@@ -16638,25 +16644,11 @@
      },
      "metadata": {},
      "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "# Generate a random CNOT, H, T circuit\n",
-    "random.seed(1330)\n",
-    "c = zx.generate.CNOT_HAD_PHASE_circuit(qubits=3, depth=40)\n",
-    "# for g in c.gates: print(g)\n",
-    "zx.draw(c)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 17,
-   "metadata": {},
-   "outputs": [
+    },
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-R9wzYyYv\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-fEOM57LR\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -17020,9 +17012,9 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-R9wzYyYv',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 200, \"y\": 100, \"t\": 2, \"phase\": \"7\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"43\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"54\", \"x\": 250, \"y\": 150, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"58\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"62\", \"x\": 300, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"63\", \"x\": 300, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"64\", \"x\": 300, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"54\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"54\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"43\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"62\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"63\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"58\", \"target\": \"64\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 350, 250, 50,\n",
-       "  10.0, false, true, '');\n",
+       "showGraph('#graph-output-fEOM57LR',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 50, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 250, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 300, \"y\": 200, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 350, \"y\": 250, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 400, \"y\": 250, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 400, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 450, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 450, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 500, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 550, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 550, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 600, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 600, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 600, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"23\", \"x\": 600, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"23\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"17\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 11, \"target\": 0, \"t\": \"Z\"}, {\"source\": 7, \"target\": 12, \"t\": \"Z\"}, {\"source\": 8, \"target\": 10, \"t\": \"Z\"}, {\"source\": 14, \"target\": 10, \"t\": \"Z\"}, {\"source\": 12, \"target\": 11, \"t\": \"X\"}, {\"source\": 14, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 14, \"t\": \"Y\"}, {\"source\": 15, \"target\": 17, \"t\": \"Z\"}, {\"source\": 18, \"target\": 22, \"t\": \"X\"}]}'), 650, 300, 50,\n",
+       "  10.0, false, false, '');\n",
        "</script>"
       ],
       "text/plain": [
@@ -17034,81 +17026,88 @@
     }
    ],
    "source": [
-    "# Convert to a ZX diagram and call the full_reduce procedure on it (PyZX's main ZX diagram optimisation pass)\n",
-    "g = c.to_graph()\n",
-    "zx.full_reduce(g)\n",
-    "zx.to_rg(g)\n",
-    "\n",
-    "# Normalise compacts the circuit visually and ensures every input/output is connected to a Z spider\n",
-    "g.normalize()\n",
-    "\n",
-    "# Compute the time-ordering on nodes (which is only important for the non-Clifford nodes) and compute the Pauli\n",
-    "# webs for every node.\n",
-    "order, zwebs, xwebs = compute_pauli_webs(g)\n",
+    "random.seed(1337)\n",
+    "g = zx.generate.CNOT_HAD_PHASE_circuit(qubits=4, depth=10, clifford=True).to_graph(compress_rows=False)\n",
+    "zx.draw(g, labels=True)\n",
     "\n",
-    "# Draw the simplified ZX diagram. Note blue edges correspond to edges with Hadamard gates\n",
-    "zx.draw(g, labels=True)"
+    "# introduce random Pauli errors on edges and draw them\n",
+    "errors = PauliWeb(g)\n",
+    "for e in g.edges():\n",
+    "    s,t = g.edge_st(e)\n",
+    "    if random.random() > 0.75:\n",
+    "        errors.add_half_edge((s,t), random.choice(['X','Y','Z']))\n",
+    "    if random.random() > 0.75:\n",
+    "        errors.add_half_edge((t,s), random.choice(['X','Y','Z']))\n",
+    "zx.draw(g, pauli_web=errors)"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 18,
+   "execution_count": 32,
    "metadata": {},
    "outputs": [
     {
-     "data": {
-      "text/plain": [
-       "{(3, 1): 'Y',\n",
-       " (1, 3): 'Y',\n",
-       " (3, 43): 'Z',\n",
-       " (43, 3): 'Z',\n",
-       " (3, 5): 'Y',\n",
-       " (5, 3): 'Y',\n",
-       " (3, 58): 'Z',\n",
-       " (58, 3): 'X',\n",
-       " (3, 54): 'Z',\n",
-       " (54, 3): 'Z',\n",
-       " (3, 14): 'Y',\n",
-       " (14, 3): 'Y',\n",
-       " (5, 16): 'Y',\n",
-       " (16, 5): 'Y',\n",
-       " (5, 2): 'Y',\n",
-       " (2, 5): 'Y',\n",
-       " (5, 43): 'X',\n",
-       " (43, 5): 'Z',\n",
-       " (5, 58): 'X',\n",
-       " (58, 5): 'X',\n",
-       " (5, 54): 'X',\n",
-       " (54, 5): 'Z',\n",
-       " (14, 58): 'X',\n",
-       " (58, 14): 'X',\n",
-       " (14, 16): 'Y',\n",
-       " (16, 14): 'Y',\n",
-       " (16, 43): 'Z',\n",
-       " (43, 16): 'Z',\n",
-       " (16, 58): 'Z',\n",
-       " (58, 16): 'X'}"
-      ]
-     },
-     "execution_count": 18,
-     "metadata": {},
-     "output_type": "execute_result"
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Z-web 20: commutes\n",
+      "X-web 20: anti-commutes, corrected\n",
+      "Z-web 21: commutes\n",
+      "X-web 21: commutes\n",
+      "Z-web 22: anti-commutes, corrected\n",
+      "X-web 22: commutes\n",
+      "Z-web 23: commutes\n",
+      "X-web 23: commutes\n"
+     ]
     }
    ],
    "source": [
-    "pw = zwebs[43]\n",
-    "pw.half_edges()"
+    "# To find the appropriate corrections, we should introduce Pauli X and Z gates at the outputs\n",
+    "# to make all 8 output webs commute with `errors`.\n",
+    "\n",
+    "order, zwebs, xwebs = compute_pauli_webs(g)\n",
+    "output_es = [(o, next(iter(g.neighbors(o)))) for o in g.outputs()]\n",
+    "\n",
+    "\n",
+    "errors1 = errors.copy()\n",
+    "\n",
+    "# for each of the outputs, if the errors anti-commute with the associated Z-web, introduce\n",
+    "# an X-error at that output to make it commute, and do similar for X-webs.\n",
+    "for o,n in output_es:\n",
+    "    print(f'Z-web {o}: ', end='')\n",
+    "    if zwebs[o].commutes_with(errors):\n",
+    "        print('commutes')\n",
+    "    else:\n",
+    "        print('anti-commutes', end='')\n",
+    "        errors1.add_half_edge((o,n), 'X')\n",
+    "        if zwebs[o].commutes_with(errors1):\n",
+    "            print(', corrected')\n",
+    "    print(f'X-web {o}: ', end='')\n",
+    "    if xwebs[o].commutes_with(errors):\n",
+    "        print('commutes')\n",
+    "    else:\n",
+    "        print('anti-commutes', end='')\n",
+    "        errors1.add_half_edge((o,n), 'Z')\n",
+    "        if xwebs[o].commutes_with(errors1):\n",
+    "            print(', corrected')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Clifford+T Examples"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 19,
+   "execution_count": 18,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-rsq3Ldo3\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-ruRdvBwG\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -17472,9 +17471,9 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-rsq3Ldo3',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 200, \"y\": 100, \"t\": 2, \"phase\": \"7\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"43\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"54\", \"x\": 250, \"y\": 150, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"58\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"62\", \"x\": 300, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"63\", \"x\": 300, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"64\", \"x\": 300, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"54\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"54\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"43\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"62\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"63\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"58\", \"target\": \"64\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Y\"}, {\"source\": 1, \"target\": 3, \"t\": \"Y\"}, {\"source\": 3, \"target\": 43, \"t\": \"Z\"}, {\"source\": 43, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 5, \"t\": \"Y\"}, {\"source\": 5, \"target\": 3, \"t\": \"Y\"}, {\"source\": 3, \"target\": 58, \"t\": \"Z\"}, {\"source\": 58, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 54, \"t\": \"Z\"}, {\"source\": 54, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 14, \"t\": \"Y\"}, {\"source\": 14, \"target\": 3, \"t\": \"Y\"}, {\"source\": 5, \"target\": 16, \"t\": \"Y\"}, {\"source\": 16, \"target\": 5, \"t\": \"Y\"}, {\"source\": 5, \"target\": 2, \"t\": \"Y\"}, {\"source\": 2, \"target\": 5, \"t\": \"Y\"}, {\"source\": 5, \"target\": 43, \"t\": \"X\"}, {\"source\": 43, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 58, \"t\": \"X\"}, {\"source\": 58, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 54, \"t\": \"X\"}, {\"source\": 54, \"target\": 5, \"t\": \"Z\"}, {\"source\": 14, \"target\": 58, \"t\": \"X\"}, {\"source\": 58, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 16, \"t\": \"Y\"}, {\"source\": 16, \"target\": 14, \"t\": \"Y\"}, {\"source\": 16, \"target\": 43, \"t\": \"Z\"}, {\"source\": 43, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 58, \"t\": \"Z\"}, {\"source\": 58, \"target\": 16, \"t\": \"X\"}]}'), 350, 250, 50,\n",
-       "  10.0, false, true, '');\n",
+       "showGraph('#graph-output-ruRdvBwG',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 26.666666666666668, \"y\": 53.333333333333336, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 26.666666666666668, \"y\": 80.0, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 26.666666666666668, \"y\": 106.66666666666667, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 53.333333333333336, \"y\": 80.0, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 53.333333333333336, \"y\": 53.333333333333336, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 80.0, \"y\": 53.333333333333336, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 80.0, \"y\": 80.0, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 106.66666666666667, \"y\": 53.333333333333336, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 106.66666666666667, \"y\": 106.66666666666667, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 133.33333333333334, \"y\": 106.66666666666667, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 160.0, \"y\": 80.0, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 160.0, \"y\": 106.66666666666667, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 186.66666666666669, \"y\": 106.66666666666667, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 186.66666666666669, \"y\": 53.333333333333336, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 186.66666666666669, \"y\": 80.0, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 213.33333333333334, \"y\": 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{\"name\": \"62\", \"x\": 773.3333333333334, \"y\": 53.333333333333336, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"63\", \"x\": 773.3333333333334, \"y\": 80.0, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"64\", \"x\": 773.3333333333334, \"y\": 106.66666666666667, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"9\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"21\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"24\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"20\", \"target\": \"23\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"21\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"22\", \"target\": \"25\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"23\", \"target\": \"30\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"24\", \"target\": \"26\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"25\", \"target\": \"26\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"25\", \"target\": \"27\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"26\", \"target\": \"28\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"28\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"31\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"28\", \"target\": \"29\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"29\", \"target\": \"30\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"29\", \"target\": \"34\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"30\", \"target\": \"33\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"31\", \"target\": \"32\", 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1}, {\"source\": \"40\", \"target\": \"41\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"41\", \"target\": \"42\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"41\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"42\", \"target\": \"47\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"44\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"44\", \"target\": \"45\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"45\", \"target\": \"46\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"46\", \"target\": \"47\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"46\", \"target\": \"49\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"47\", \"target\": \"48\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"48\", \"target\": \"49\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"48\", \"target\": \"50\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"49\", \"target\": \"53\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"50\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"51\", \"target\": \"52\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"52\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"53\", \"target\": \"56\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"55\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"55\", \"target\": \"57\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"56\", \"target\": \"57\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"56\", \"target\": \"60\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"57\", \"target\": \"59\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"58\", \"target\": \"59\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"58\", \"target\": \"64\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"59\", \"target\": \"61\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"60\", \"target\": \"61\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"60\", \"target\": \"62\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"61\", \"target\": \"63\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 800.0, 133.33333333333334, 26.666666666666668,\n",
+       "  5.333333333333334, false, false, '');\n",
        "</script>"
       ],
       "text/plain": [
@@ -17486,29 +17485,22 @@
     }
    ],
    "source": [
-    "# Once the Pauli webs have been computed, a specific web can be highlighted by `zx.draw` by passing it in as\n",
-    "# an optional argument. Note that webs change color when they cross Hadamard edges.\n",
-    "zx.draw(g, labels=True, pauli_web=zwebs[43])"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "We now show how this works in some simpler cases. The first is a single T gate.\n",
-    "\n",
-    "The T gate becomes a single, 1-legged phase gadget, connected to the input. This can be implemented by Z-merging a T magic state, then doing either an X or a Y measurement, depending on the parity of the Pauli web."
+    "# Generate a random CNOT, H, T circuit\n",
+    "random.seed(1330)\n",
+    "c = zx.generate.CNOT_HAD_PHASE_circuit(qubits=3, depth=40)\n",
+    "# for g in c.gates: print(g)\n",
+    "zx.draw(c)"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 20,
+   "execution_count": 19,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-HDIp0Foz\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-Tw6m2GYo\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -17872,9 +17864,9 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-HDIp0Foz',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"1\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 200, 150, 50,\n",
-       "  10.0, false, false, '');\n",
+       "showGraph('#graph-output-Tw6m2GYo',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 200, \"y\": 100, \"t\": 2, \"phase\": \"7\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"43\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"54\", \"x\": 250, \"y\": 150, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"58\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"62\", \"x\": 300, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"63\", \"x\": 300, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"64\", \"x\": 300, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"54\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"54\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"43\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"62\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"63\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"58\", \"target\": \"64\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 350, 250, 50,\n",
+       "  10.0, false, true, '');\n",
        "</script>"
       ],
       "text/plain": [
@@ -17883,11 +17875,84 @@
      },
      "metadata": {},
      "output_type": "display_data"
-    },
+    }
+   ],
+   "source": [
+    "# Convert to a ZX diagram and call the full_reduce procedure on it (PyZX's main ZX diagram optimisation pass)\n",
+    "g = c.to_graph()\n",
+    "zx.full_reduce(g)\n",
+    "zx.to_rg(g)\n",
+    "\n",
+    "# Normalise compacts the circuit visually and ensures every input/output is connected to a Z spider\n",
+    "g.normalize()\n",
+    "\n",
+    "# Compute the time-ordering on nodes (which is only important for the non-Clifford nodes) and compute the Pauli\n",
+    "# webs for every node.\n",
+    "order, zwebs, xwebs = compute_pauli_webs(g)\n",
+    "\n",
+    "# Draw the simplified ZX diagram. Note blue edges correspond to edges with Hadamard gates\n",
+    "zx.draw(g, labels=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{(3, 1): 'Y',\n",
+       " (1, 3): 'Y',\n",
+       " (3, 43): 'Z',\n",
+       " (43, 3): 'Z',\n",
+       " (3, 5): 'Y',\n",
+       " (5, 3): 'Y',\n",
+       " (3, 58): 'Z',\n",
+       " (58, 3): 'X',\n",
+       " (3, 54): 'Z',\n",
+       " (54, 3): 'Z',\n",
+       " (3, 14): 'Y',\n",
+       " (14, 3): 'Y',\n",
+       " (5, 16): 'Y',\n",
+       " (16, 5): 'Y',\n",
+       " (5, 2): 'Y',\n",
+       " (2, 5): 'Y',\n",
+       " (5, 43): 'X',\n",
+       " (43, 5): 'Z',\n",
+       " (5, 58): 'X',\n",
+       " (58, 5): 'X',\n",
+       " (5, 54): 'X',\n",
+       " (54, 5): 'Z',\n",
+       " (14, 58): 'X',\n",
+       " (58, 14): 'X',\n",
+       " (14, 16): 'Y',\n",
+       " (16, 14): 'Y',\n",
+       " (16, 43): 'Z',\n",
+       " (43, 16): 'Z',\n",
+       " (16, 58): 'Z',\n",
+       " (58, 16): 'X'}"
+      ]
+     },
+     "execution_count": 20,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "pw = zwebs[43]\n",
+    "pw.half_edges()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "metadata": {},
+   "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-FPXSt9tt\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-DqXCzq87\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -18251,8 +18316,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-FPXSt9tt',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"1\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 0, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 0, \"t\": \"Z\"}]}'), 200, 150, 50,\n",
+       "showGraph('#graph-output-DqXCzq87',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 200, \"y\": 100, \"t\": 2, \"phase\": \"7\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"43\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"54\", \"x\": 250, \"y\": 150, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"58\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"62\", \"x\": 300, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"63\", \"x\": 300, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"64\", \"x\": 300, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"54\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"54\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"43\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"62\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"63\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"58\", \"target\": \"64\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Y\"}, {\"source\": 1, \"target\": 3, \"t\": \"Y\"}, {\"source\": 3, \"target\": 43, \"t\": \"Z\"}, {\"source\": 43, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 5, \"t\": \"Y\"}, {\"source\": 5, \"target\": 3, \"t\": \"Y\"}, {\"source\": 3, \"target\": 58, \"t\": \"Z\"}, {\"source\": 58, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 54, \"t\": \"Z\"}, {\"source\": 54, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 14, \"t\": \"Y\"}, {\"source\": 14, \"target\": 3, \"t\": \"Y\"}, {\"source\": 5, \"target\": 16, \"t\": \"Y\"}, {\"source\": 16, \"target\": 5, \"t\": \"Y\"}, {\"source\": 5, \"target\": 2, \"t\": \"Y\"}, {\"source\": 2, \"target\": 5, \"t\": \"Y\"}, {\"source\": 5, \"target\": 43, \"t\": \"X\"}, {\"source\": 43, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 58, \"t\": \"X\"}, {\"source\": 58, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 54, \"t\": \"X\"}, {\"source\": 54, \"target\": 5, \"t\": \"Z\"}, {\"source\": 14, \"target\": 58, \"t\": \"X\"}, {\"source\": 58, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 16, \"t\": \"Y\"}, {\"source\": 16, \"target\": 14, \"t\": \"Y\"}, {\"source\": 16, \"target\": 43, \"t\": \"Z\"}, {\"source\": 43, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 58, \"t\": \"Z\"}, {\"source\": 58, \"target\": 16, \"t\": \"X\"}]}'), 350, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -18265,36 +18330,815 @@
     }
    ],
    "source": [
-    "c = zx.qasm(\"\"\"\n",
-    "qreg q[1];\n",
-    "t q[0];\n",
-    "\"\"\")\n",
-    "zx.draw(c)\n",
-    "\n",
-    "g = c.to_graph()\n",
-    "zx.full_reduce(g)\n",
-    "order, zwebs, xwebs = compute_pauli_webs(g)\n",
-    "\n",
-    "# highlight the web associated to the T spider\n",
-    "zx.draw(g, labels=True, pauli_web=zwebs[1])"
+    "# Once the Pauli webs have been computed, a specific web can be highlighted by `zx.draw` by passing it in as\n",
+    "# an optional argument. Note that webs change color when they cross Hadamard edges.\n",
+    "zx.draw(g, labels=True, pauli_web=zwebs[43])"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The next example is a circuit with some CNOT gates and a T gate, which can be simplified to a single phase gadget. I'm doing this manually here, since the automated simplifier comes up with a different answer (which is equivalent to this one, up to local Cliffords, but less clear what is going on)."
+    "We now show how this works in some simpler cases. The first is a single T gate.\n",
+    "\n",
+    "The T gate becomes a single, 1-legged phase gadget, connected to the input. This can be implemented by Z-merging a T magic state, then doing either an X or a Y measurement, depending on the parity of the Pauli web."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 21,
+   "execution_count": 22,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-kyA8pdMW\"></div>\n",
+       "<script type=\"module\">\n",
+       "var d3;\n",
+       "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
+       "var _settings_colors = JSON.parse('{\"edge\": \"#000000\", \"Hedge\": \"#0088ff\", \"Xedge\": \"#999999\", \"boundary\": \"#000000\", \"X\": \"#ff8888\", \"Y\": \"#aabbff\", \"Z\": \"#ccffcc\", \"H\": \"#ffff66\", \"W\": \"#000000\", \"Zalt\": \"#ccffcc\", \"Walt\": \"#000000\", \"Xweb\": \"#ff8888\", \"Yweb\": \"#aabbff\", \"Zweb\": \"#99dd99\"}');\n",
+       "// PyZX - Python library for quantum circuit rewriting \n",
+       "//        and optimisation using the ZX-calculus\n",
+       "// Copyright (C) 2018 - Aleks Kissinger and John van de Wetering\n",
+       "\n",
+       "// Licensed under the Apache License, Version 2.0 (the \"License\");\n",
+       "// you may not use this file except in compliance with the License.\n",
+       "// You may obtain a copy of the License at\n",
+       "\n",
+       "//    http://www.apache.org/licenses/LICENSE-2.0\n",
+       "\n",
+       "// Unless required by applicable law or agreed to in writing, software\n",
+       "// distributed under the License is distributed on an \"AS IS\" BASIS,\n",
+       "// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
+       "// See the License for the specific language governing permissions and\n",
+       "// limitations under the License.\n",
+       "\n",
+       "// styling functions\n",
+       "function nodeColor(t) {\n",
+       "    if (t == 0) return _settings_colors['boundary'];\n",
+       "    else if (t == 1) return _settings_colors['Z']; // \"#ccffcc\";\n",
+       "    else if (t == 2) return _settings_colors['X']; // \"#ff8888\";\n",
+       "    else if (t == 3) return _settings_colors['H']; // \"yellow\";\n",
+       "    else if (t == 4) return _settings_colors['W']; // \"black\";\n",
+       "    else if (t == 5) return _settings_colors['Walt']; // \"black\";\n",
+       "    else if (t == 6) return _settings_colors['Zalt']; // \"#ccffcc\";\n",
+       "}\n",
+       "\n",
+       "function edgeColor(t) {\n",
+       "    if (t == 1) return _settings_colors['edge']; //\"black\";\n",
+       "    else if (t == 2) return _settings_colors['Hedge']; // \"#08f\";\n",
+       "    else if (t == 3) return _settings_colors['Xedge']; // \"gray\";\n",
+       "}\n",
+       "\n",
+       "function webColor(t) {\n",
+       "    if (t == 'X') return _settings_colors['Xweb'];\n",
+       "    else if (t == 'Y') return _settings_colors['Yweb'];\n",
+       "    else if (t == 'Z') return _settings_colors['Zweb'];\n",
+       "    else if (t == 'I') return '#dddddd';\n",
+       "}\n",
+       "\n",
+       "function nodeStyle(selected) {\n",
+       "    return selected ? \"stroke-width: 2px; stroke: #00f\" : \"stroke-width: 1.5px\";\n",
+       "}\n",
+       "\n",
+       "var symbolGround = {\n",
+       "    draw: function(context, size){\n",
+       "        let s = size/2;\n",
+       "\n",
+       "        context.moveTo(0,-s);\n",
+       "        context.lineTo(0,0);\n",
+       "\n",
+       "        context.moveTo(-s,0);\n",
+       "        context.lineTo(s,0);\n",
+       "\n",
+       "        context.moveTo(-2*s/3,s/3);\n",
+       "        context.lineTo(2*s/3,s/3);\n",
+       "\n",
+       "        context.moveTo(-s/3,2*s/3);\n",
+       "        context.lineTo(s/3,2*s/3);\n",
+       "    }\n",
+       "}\n",
+       "\n",
+       "function showGraph(tag, graph, width, height, scale, node_size, auto_hbox, show_labels, scalar_str) {\n",
+       "    var ntab = {};\n",
+       "\n",
+       "    var groundOffset = 2.5 * node_size;\n",
+       "\n",
+       "    graph.nodes.forEach(function(d) {\n",
+       "        ntab[d.name] = d;\n",
+       "        d.selected = false;\n",
+       "        d.previouslySelected = false;\n",
+       "        d.nhd = [];\n",
+       "    });\n",
+       "\n",
+       "    var spiders_and_boundaries = graph.nodes.filter(function(d) {\n",
+       "        return d.t != 3;\n",
+       "    });\n",
+       "\n",
+       "    graph.links.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "        s.nhd.push(t);\n",
+       "        t.nhd.push(s);\n",
+       "    });\n",
+       "\n",
+       "    graph.pauli_web.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "    });\n",
+       "\n",
+       "    var shiftKey;\n",
+       "\n",
+       "    // SETUP SVG ITEMS\n",
+       "\n",
+       "    var svg = d3.select(tag)\n",
+       "    //.attr(\"tabindex\", 1)\n",
+       "        .on(\"keydown.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "        .on(\"keyup.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "    //.each(function() { this.focus(); })\n",
+       "        .append(\"svg\")\n",
+       "        .attr(\"style\", \"max-width: none; max-height: none\")\n",
+       "        .attr(\"width\", width)\n",
+       "        .attr(\"height\", height);\n",
+       "\n",
+       "    var web = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"web\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.pauli_web)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return webColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 7px\");\n",
+       "\n",
+       "    var link = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"link\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.links)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return edgeColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\");\n",
+       "\n",
+       "    var brush = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"brush\");\n",
+       "\n",
+       "    var node = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"node\")\n",
+       "        .selectAll(\"g\")\n",
+       "        .data(graph.nodes)\n",
+       "        .enter().append(\"g\")\n",
+       "        .attr(\"transform\", function(d) {\n",
+       "            return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "        });\n",
+       "\n",
+       "    // Draw a ground symbol connected to the node.\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", \"M 0 0 L 0 \"+(groundOffset))\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", d3.symbol().type(symbolGround).size(node_size*1.5))\n",
+       "        .attr(\"transform\", \"translate(0,\"+groundOffset+\")\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.t != 3 && d.t != 5 && d.t != 6; })\n",
+       "        .append(\"circle\")\n",
+       "        .attr(\"r\", function(d) {\n",
+       "            if (d.t == 0) return 0.5 * node_size;\n",
+       "            else if (d.t == 4) return 0.25 * node_size;\n",
+       "            else return node_size;\n",
+       "        })\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    var hbox = node.filter(function(d) { return d.t == 3; });\n",
+       "\n",
+       "    hbox.append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    // draw a triangle for d.t == 5\n",
+       "    node.filter(function(d) { return d.t == 5; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"d\", \"M 0 0 L \"+node_size+\" \"+node_size+\" L -\"+node_size+\" \"+node_size+\" Z\")\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\")\n",
+       "        .attr(\"transform\", \"translate(\" + (-node_size/2) + \", 0) rotate(-90)\");\n",
+       "\n",
+       "    // draw a square for Z box: d.t == 6\n",
+       "    node.filter(function(d) { return d.t == 6; })\n",
+       "        .append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.phase != ''; })\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", 0.7 * node_size + 14)\n",
+       "        .text(function (d) { return d.phase })\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"12px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#00d\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "\n",
+       "    if (show_labels) {\n",
+       "        node.append(\"text\")\n",
+       "            .attr(\"y\", -0.7 * node_size - 8)\n",
+       "            .text(function (d) { return d.name; })\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "            .attr(\"font-size\", \"10px\")\n",
+       "            .attr(\"font-family\", \"monospace\")\n",
+       "            .attr(\"fill\", \"#999\")\n",
+       "            .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "    }\n",
+       "\n",
+       "    // Display the chosen data fields over the node.\n",
+       "    node.filter(d => d.vdata.length > 0)\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", d => -0.7 * node_size - 14 - 10 * d.vdata.length)\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"8px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#c66\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;')\n",
+       "        .selectAll(\"tspan\")\n",
+       "        .data(d => d.vdata)\n",
+       "        .enter()\n",
+       "        .append(\"tspan\")\n",
+       "        .attr(\"x\", \"0\")\n",
+       "        .attr(\"dy\", \"1.2em\")\n",
+       "        .text(x => x.join(\": \"));\n",
+       "\n",
+       "    if (scalar_str != \"\") {\n",
+       "        svg.append(\"text\")\n",
+       "            .text(scalar_str)\n",
+       "            .attr(\"x\", 60).attr(\"y\", 40)\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "    }\n",
+       "\n",
+       "    function update_hboxes() {\n",
+       "        if (auto_hbox) {\n",
+       "            var pos = {};\n",
+       "            hbox.attr(\"transform\", function(d) {\n",
+       "                // calculate barycenter of non-hbox neighbours, then nudge a bit\n",
+       "                // to the NE.\n",
+       "                var x=0,y=0,sz=0;\n",
+       "                for (var i = 0; i < d.nhd.length; ++i) {\n",
+       "                    if (d.nhd[i].t != 3) {\n",
+       "                        sz++;\n",
+       "                        x += d.nhd[i].x;\n",
+       "                        y += d.nhd[i].y;\n",
+       "                    }\n",
+       "                }\n",
+       "\n",
+       "                offset = 0.25 * scale;\n",
+       "\n",
+       "                if (sz != 0) {\n",
+       "                    x = (x/sz) + offset;\n",
+       "                    y = (y/sz) - offset;\n",
+       "\n",
+       "                    while (pos[[x,y]]) {\n",
+       "                        x += offset;\n",
+       "                    }\n",
+       "                    d.x = x;\n",
+       "                    d.y = y;\n",
+       "                    pos[[x,y]] = true;\n",
+       "                }\n",
+       "\n",
+       "                return \"translate(\"+d.x+\",\"+d.y+\")\";\n",
+       "            });\n",
+       "        }\n",
+       "    }\n",
+       "\n",
+       "    update_hboxes();\n",
+       "\n",
+       "    var link_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = d.target.x, y1 = d.source.y, y2 = d.target.y;\n",
+       "        if (d.num_parallel == 1) {\n",
+       "            return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        } else {\n",
+       "            var dx = x2 - x1, dy = y2 - y1;\n",
+       "            var midx = 0.5 * (x1 + x2), midy = 0.5 * (y1 + y2);\n",
+       "            var pos = (d.index / (d.num_parallel-1)) - 0.5;\n",
+       "            var cx = midx - pos * dy;\n",
+       "            var cy = midy + pos * dx;\n",
+       "            return `M ${x1} ${y1} Q ${cx} ${cy}, ${x2} ${y2}`;\n",
+       "            // return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        }\n",
+       "    };\n",
+       "    link.attr(\"d\", link_curve);\n",
+       "\n",
+       "\n",
+       "    var web_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = (x1 + d.target.x)/2, y1 = d.source.y, y2 = (y1 + d.target.y)/2;\n",
+       "        return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "    }\n",
+       "    web.attr(\"d\", web_curve);\n",
+       "\n",
+       "    // EVENTS FOR DRAGGING AND SELECTION\n",
+       "\n",
+       "    node.on(\"mousedown\", function(d) {\n",
+       "        if (shiftKey) {\n",
+       "            d3.select(this).selectAll(\".selectable\").attr(\"style\", nodeStyle(d.selected = !d.selected));\n",
+       "            d3.event.stopImmediatePropagation();\n",
+       "        } else if (!d.selected) {\n",
+       "            node.selectAll(\".selectable\").attr(\"style\", function(p) { return nodeStyle(p.selected = d === p); });\n",
+       "        }\n",
+       "    })\n",
+       "        .call(d3.drag().on(\"drag\", function(d) {\n",
+       "            var dx = d3.event.dx;\n",
+       "            var dy = d3.event.dy;\n",
+       "            // node.filter(function(d) { return d.selected; })\n",
+       "            //     .attr(\"cx\", function(d) { return d.x += dx; })\n",
+       "            //     .attr(\"cy\", function(d) { return d.y += dy; });\n",
+       "            node.filter(function(d) { return d.selected; })\n",
+       "                .attr(\"transform\", function(d) {\n",
+       "                    d.x += dx;\n",
+       "                    d.y += dy;\n",
+       "                    return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "                });\n",
+       "\n",
+       "            update_hboxes();\n",
+       "\n",
+       "            link.filter(function(d) { return d.source.selected || d.target.selected ||\n",
+       "                    (auto_hbox && d.source.t == 3); })\n",
+       "                .attr(\"d\", link_curve);\n",
+       "            web.filter(function(d) { return d.source.selected || d.target.selected; })\n",
+       "                .attr(\"d\", web_curve);\n",
+       "        }));\n",
+       "\n",
+       "    brush.call(d3.brush().keyModifiers(false)\n",
+       "        .extent([[0, 0], [width, height]])\n",
+       "        .on(\"start\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(\n",
+       "                        d.selected = d.previouslySelected = shiftKey &&\n",
+       "                        d.selected);\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"brush\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                var selection = d3.event.selection;\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(d.selected = d.previouslySelected ^\n",
+       "                        (selection != null\n",
+       "                            && selection[0][0] <= d.x && d.x < selection[1][0]\n",
+       "                            && selection[0][1] <= d.y && d.y < selection[1][1]));\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"end\", function() {\n",
+       "            if (d3.event.selection != null) {\n",
+       "                d3.select(this).call(d3.event.target.move, null);\n",
+       "            }\n",
+       "        }));\n",
+       "}\n",
+       "\n",
+       "\n",
+       "showGraph('#graph-output-kyA8pdMW',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"1\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 200, 150, 50,\n",
+       "  10.0, false, false, '');\n",
+       "</script>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-4ZS1zzrq\"></div>\n",
+       "<script type=\"module\">\n",
+       "var d3;\n",
+       "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
+       "var _settings_colors = JSON.parse('{\"edge\": \"#000000\", \"Hedge\": \"#0088ff\", \"Xedge\": \"#999999\", \"boundary\": \"#000000\", \"X\": \"#ff8888\", \"Y\": \"#aabbff\", \"Z\": \"#ccffcc\", \"H\": \"#ffff66\", \"W\": \"#000000\", \"Zalt\": \"#ccffcc\", \"Walt\": \"#000000\", \"Xweb\": \"#ff8888\", \"Yweb\": \"#aabbff\", \"Zweb\": \"#99dd99\"}');\n",
+       "// PyZX - Python library for quantum circuit rewriting \n",
+       "//        and optimisation using the ZX-calculus\n",
+       "// Copyright (C) 2018 - Aleks Kissinger and John van de Wetering\n",
+       "\n",
+       "// Licensed under the Apache License, Version 2.0 (the \"License\");\n",
+       "// you may not use this file except in compliance with the License.\n",
+       "// You may obtain a copy of the License at\n",
+       "\n",
+       "//    http://www.apache.org/licenses/LICENSE-2.0\n",
+       "\n",
+       "// Unless required by applicable law or agreed to in writing, software\n",
+       "// distributed under the License is distributed on an \"AS IS\" BASIS,\n",
+       "// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
+       "// See the License for the specific language governing permissions and\n",
+       "// limitations under the License.\n",
+       "\n",
+       "// styling functions\n",
+       "function nodeColor(t) {\n",
+       "    if (t == 0) return _settings_colors['boundary'];\n",
+       "    else if (t == 1) return _settings_colors['Z']; // \"#ccffcc\";\n",
+       "    else if (t == 2) return _settings_colors['X']; // \"#ff8888\";\n",
+       "    else if (t == 3) return _settings_colors['H']; // \"yellow\";\n",
+       "    else if (t == 4) return _settings_colors['W']; // \"black\";\n",
+       "    else if (t == 5) return _settings_colors['Walt']; // \"black\";\n",
+       "    else if (t == 6) return _settings_colors['Zalt']; // \"#ccffcc\";\n",
+       "}\n",
+       "\n",
+       "function edgeColor(t) {\n",
+       "    if (t == 1) return _settings_colors['edge']; //\"black\";\n",
+       "    else if (t == 2) return _settings_colors['Hedge']; // \"#08f\";\n",
+       "    else if (t == 3) return _settings_colors['Xedge']; // \"gray\";\n",
+       "}\n",
+       "\n",
+       "function webColor(t) {\n",
+       "    if (t == 'X') return _settings_colors['Xweb'];\n",
+       "    else if (t == 'Y') return _settings_colors['Yweb'];\n",
+       "    else if (t == 'Z') return _settings_colors['Zweb'];\n",
+       "    else if (t == 'I') return '#dddddd';\n",
+       "}\n",
+       "\n",
+       "function nodeStyle(selected) {\n",
+       "    return selected ? \"stroke-width: 2px; stroke: #00f\" : \"stroke-width: 1.5px\";\n",
+       "}\n",
+       "\n",
+       "var symbolGround = {\n",
+       "    draw: function(context, size){\n",
+       "        let s = size/2;\n",
+       "\n",
+       "        context.moveTo(0,-s);\n",
+       "        context.lineTo(0,0);\n",
+       "\n",
+       "        context.moveTo(-s,0);\n",
+       "        context.lineTo(s,0);\n",
+       "\n",
+       "        context.moveTo(-2*s/3,s/3);\n",
+       "        context.lineTo(2*s/3,s/3);\n",
+       "\n",
+       "        context.moveTo(-s/3,2*s/3);\n",
+       "        context.lineTo(s/3,2*s/3);\n",
+       "    }\n",
+       "}\n",
+       "\n",
+       "function showGraph(tag, graph, width, height, scale, node_size, auto_hbox, show_labels, scalar_str) {\n",
+       "    var ntab = {};\n",
+       "\n",
+       "    var groundOffset = 2.5 * node_size;\n",
+       "\n",
+       "    graph.nodes.forEach(function(d) {\n",
+       "        ntab[d.name] = d;\n",
+       "        d.selected = false;\n",
+       "        d.previouslySelected = false;\n",
+       "        d.nhd = [];\n",
+       "    });\n",
+       "\n",
+       "    var spiders_and_boundaries = graph.nodes.filter(function(d) {\n",
+       "        return d.t != 3;\n",
+       "    });\n",
+       "\n",
+       "    graph.links.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "        s.nhd.push(t);\n",
+       "        t.nhd.push(s);\n",
+       "    });\n",
+       "\n",
+       "    graph.pauli_web.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "    });\n",
+       "\n",
+       "    var shiftKey;\n",
+       "\n",
+       "    // SETUP SVG ITEMS\n",
+       "\n",
+       "    var svg = d3.select(tag)\n",
+       "    //.attr(\"tabindex\", 1)\n",
+       "        .on(\"keydown.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "        .on(\"keyup.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "    //.each(function() { this.focus(); })\n",
+       "        .append(\"svg\")\n",
+       "        .attr(\"style\", \"max-width: none; max-height: none\")\n",
+       "        .attr(\"width\", width)\n",
+       "        .attr(\"height\", height);\n",
+       "\n",
+       "    var web = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"web\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.pauli_web)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return webColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 7px\");\n",
+       "\n",
+       "    var link = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"link\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.links)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return edgeColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\");\n",
+       "\n",
+       "    var brush = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"brush\");\n",
+       "\n",
+       "    var node = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"node\")\n",
+       "        .selectAll(\"g\")\n",
+       "        .data(graph.nodes)\n",
+       "        .enter().append(\"g\")\n",
+       "        .attr(\"transform\", function(d) {\n",
+       "            return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "        });\n",
+       "\n",
+       "    // Draw a ground symbol connected to the node.\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", \"M 0 0 L 0 \"+(groundOffset))\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", d3.symbol().type(symbolGround).size(node_size*1.5))\n",
+       "        .attr(\"transform\", \"translate(0,\"+groundOffset+\")\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.t != 3 && d.t != 5 && d.t != 6; })\n",
+       "        .append(\"circle\")\n",
+       "        .attr(\"r\", function(d) {\n",
+       "            if (d.t == 0) return 0.5 * node_size;\n",
+       "            else if (d.t == 4) return 0.25 * node_size;\n",
+       "            else return node_size;\n",
+       "        })\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    var hbox = node.filter(function(d) { return d.t == 3; });\n",
+       "\n",
+       "    hbox.append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    // draw a triangle for d.t == 5\n",
+       "    node.filter(function(d) { return d.t == 5; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"d\", \"M 0 0 L \"+node_size+\" \"+node_size+\" L -\"+node_size+\" \"+node_size+\" Z\")\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\")\n",
+       "        .attr(\"transform\", \"translate(\" + (-node_size/2) + \", 0) rotate(-90)\");\n",
+       "\n",
+       "    // draw a square for Z box: d.t == 6\n",
+       "    node.filter(function(d) { return d.t == 6; })\n",
+       "        .append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.phase != ''; })\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", 0.7 * node_size + 14)\n",
+       "        .text(function (d) { return d.phase })\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"12px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#00d\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "\n",
+       "    if (show_labels) {\n",
+       "        node.append(\"text\")\n",
+       "            .attr(\"y\", -0.7 * node_size - 8)\n",
+       "            .text(function (d) { return d.name; })\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "            .attr(\"font-size\", \"10px\")\n",
+       "            .attr(\"font-family\", \"monospace\")\n",
+       "            .attr(\"fill\", \"#999\")\n",
+       "            .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "    }\n",
+       "\n",
+       "    // Display the chosen data fields over the node.\n",
+       "    node.filter(d => d.vdata.length > 0)\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", d => -0.7 * node_size - 14 - 10 * d.vdata.length)\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"8px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#c66\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;')\n",
+       "        .selectAll(\"tspan\")\n",
+       "        .data(d => d.vdata)\n",
+       "        .enter()\n",
+       "        .append(\"tspan\")\n",
+       "        .attr(\"x\", \"0\")\n",
+       "        .attr(\"dy\", \"1.2em\")\n",
+       "        .text(x => x.join(\": \"));\n",
+       "\n",
+       "    if (scalar_str != \"\") {\n",
+       "        svg.append(\"text\")\n",
+       "            .text(scalar_str)\n",
+       "            .attr(\"x\", 60).attr(\"y\", 40)\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "    }\n",
+       "\n",
+       "    function update_hboxes() {\n",
+       "        if (auto_hbox) {\n",
+       "            var pos = {};\n",
+       "            hbox.attr(\"transform\", function(d) {\n",
+       "                // calculate barycenter of non-hbox neighbours, then nudge a bit\n",
+       "                // to the NE.\n",
+       "                var x=0,y=0,sz=0;\n",
+       "                for (var i = 0; i < d.nhd.length; ++i) {\n",
+       "                    if (d.nhd[i].t != 3) {\n",
+       "                        sz++;\n",
+       "                        x += d.nhd[i].x;\n",
+       "                        y += d.nhd[i].y;\n",
+       "                    }\n",
+       "                }\n",
+       "\n",
+       "                offset = 0.25 * scale;\n",
+       "\n",
+       "                if (sz != 0) {\n",
+       "                    x = (x/sz) + offset;\n",
+       "                    y = (y/sz) - offset;\n",
+       "\n",
+       "                    while (pos[[x,y]]) {\n",
+       "                        x += offset;\n",
+       "                    }\n",
+       "                    d.x = x;\n",
+       "                    d.y = y;\n",
+       "                    pos[[x,y]] = true;\n",
+       "                }\n",
+       "\n",
+       "                return \"translate(\"+d.x+\",\"+d.y+\")\";\n",
+       "            });\n",
+       "        }\n",
+       "    }\n",
+       "\n",
+       "    update_hboxes();\n",
+       "\n",
+       "    var link_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = d.target.x, y1 = d.source.y, y2 = d.target.y;\n",
+       "        if (d.num_parallel == 1) {\n",
+       "            return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        } else {\n",
+       "            var dx = x2 - x1, dy = y2 - y1;\n",
+       "            var midx = 0.5 * (x1 + x2), midy = 0.5 * (y1 + y2);\n",
+       "            var pos = (d.index / (d.num_parallel-1)) - 0.5;\n",
+       "            var cx = midx - pos * dy;\n",
+       "            var cy = midy + pos * dx;\n",
+       "            return `M ${x1} ${y1} Q ${cx} ${cy}, ${x2} ${y2}`;\n",
+       "            // return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        }\n",
+       "    };\n",
+       "    link.attr(\"d\", link_curve);\n",
+       "\n",
+       "\n",
+       "    var web_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = (x1 + d.target.x)/2, y1 = d.source.y, y2 = (y1 + d.target.y)/2;\n",
+       "        return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "    }\n",
+       "    web.attr(\"d\", web_curve);\n",
+       "\n",
+       "    // EVENTS FOR DRAGGING AND SELECTION\n",
+       "\n",
+       "    node.on(\"mousedown\", function(d) {\n",
+       "        if (shiftKey) {\n",
+       "            d3.select(this).selectAll(\".selectable\").attr(\"style\", nodeStyle(d.selected = !d.selected));\n",
+       "            d3.event.stopImmediatePropagation();\n",
+       "        } else if (!d.selected) {\n",
+       "            node.selectAll(\".selectable\").attr(\"style\", function(p) { return nodeStyle(p.selected = d === p); });\n",
+       "        }\n",
+       "    })\n",
+       "        .call(d3.drag().on(\"drag\", function(d) {\n",
+       "            var dx = d3.event.dx;\n",
+       "            var dy = d3.event.dy;\n",
+       "            // node.filter(function(d) { return d.selected; })\n",
+       "            //     .attr(\"cx\", function(d) { return d.x += dx; })\n",
+       "            //     .attr(\"cy\", function(d) { return d.y += dy; });\n",
+       "            node.filter(function(d) { return d.selected; })\n",
+       "                .attr(\"transform\", function(d) {\n",
+       "                    d.x += dx;\n",
+       "                    d.y += dy;\n",
+       "                    return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "                });\n",
+       "\n",
+       "            update_hboxes();\n",
+       "\n",
+       "            link.filter(function(d) { return d.source.selected || d.target.selected ||\n",
+       "                    (auto_hbox && d.source.t == 3); })\n",
+       "                .attr(\"d\", link_curve);\n",
+       "            web.filter(function(d) { return d.source.selected || d.target.selected; })\n",
+       "                .attr(\"d\", web_curve);\n",
+       "        }));\n",
+       "\n",
+       "    brush.call(d3.brush().keyModifiers(false)\n",
+       "        .extent([[0, 0], [width, height]])\n",
+       "        .on(\"start\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(\n",
+       "                        d.selected = d.previouslySelected = shiftKey &&\n",
+       "                        d.selected);\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"brush\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                var selection = d3.event.selection;\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(d.selected = d.previouslySelected ^\n",
+       "                        (selection != null\n",
+       "                            && selection[0][0] <= d.x && d.x < selection[1][0]\n",
+       "                            && selection[0][1] <= d.y && d.y < selection[1][1]));\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"end\", function() {\n",
+       "            if (d3.event.selection != null) {\n",
+       "                d3.select(this).call(d3.event.target.move, null);\n",
+       "            }\n",
+       "        }));\n",
+       "}\n",
+       "\n",
+       "\n",
+       "showGraph('#graph-output-4ZS1zzrq',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"1\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 0, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 0, \"t\": \"Z\"}]}'), 200, 150, 50,\n",
+       "  10.0, false, true, '');\n",
+       "</script>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "c = zx.qasm(\"\"\"\n",
+    "qreg q[1];\n",
+    "t q[0];\n",
+    "\"\"\")\n",
+    "zx.draw(c)\n",
+    "\n",
+    "g = c.to_graph()\n",
+    "zx.full_reduce(g)\n",
+    "order, zwebs, xwebs = compute_pauli_webs(g)\n",
+    "\n",
+    "# highlight the web associated to the T spider\n",
+    "zx.draw(g, labels=True, pauli_web=zwebs[1])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The next example is a circuit with some CNOT gates and a T gate, which can be simplified to a single phase gadget. I'm doing this manually here, since the automated simplifier comes up with a different answer (which is equivalent to this one, up to local Cliffords, but less clear what is going on)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-I2ncLdAj\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-QB0CIuve\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -18658,7 +19502,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-I2ncLdAj',\n",
+       "showGraph('#graph-output-QB0CIuve',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 350, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 400, 250, 50,\n",
        "  10.0, false, false, '');\n",
        "</script>"
@@ -18673,7 +19517,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-pSNSUuww\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-NExmaneQ\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -19037,7 +19881,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-pSNSUuww',\n",
+       "showGraph('#graph-output-NExmaneQ',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 300, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 350, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 300, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 200, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 110.00000000000001, \"y\": 300.0, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 80.0, \"y\": 250.0, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 400, 350, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -19052,7 +19896,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-1tVHF8yw\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-FRsGpofX\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -19416,7 +20260,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-1tVHF8yw',\n",
+       "showGraph('#graph-output-FRsGpofX',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 300, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 350, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 300, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 200, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 110.00000000000001, \"y\": 300.0, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 80.0, \"y\": 250.0, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 16, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 17, \"t\": \"Z\"}, {\"source\": 17, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 18, \"t\": \"Z\"}, {\"source\": 18, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 16, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}, {\"source\": 1, \"target\": 18, \"t\": \"Z\"}, {\"source\": 18, \"target\": 1, \"t\": \"Z\"}, {\"source\": 2, \"target\": 17, \"t\": \"Z\"}, {\"source\": 17, \"target\": 2, \"t\": \"Z\"}]}'), 400, 350, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -19470,7 +20314,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3 (ipykernel)",
+   "display_name": "Python 3",
    "language": "python",
    "name": "python3"
   },
@@ -19484,7 +20328,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.10.12"
+   "version": "3.12.0"
   }
  },
  "nbformat": 4,
diff --git a/pyzx/pauliweb.py b/pyzx/pauliweb.py
index abb5f687..ba865e27 100644
--- a/pyzx/pauliweb.py
+++ b/pyzx/pauliweb.py
@@ -14,6 +14,8 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 
+from __future__ import annotations
+
 from .gflow import gflow
 from .utils import EdgeType, VertexType, vertex_is_zx
 from .graph.base import BaseGraph, VT, ET
@@ -39,74 +41,64 @@ def h_pauli(p: str) -> str:
 class PauliWeb(Generic[VT, ET]):
     """A Pauli web
     
-    This class stores a Pauli web in "correction set" format. That is, the edges in the web are
-    all the edges incident to the vertices in `c`. To account for Hadamard edges, edge typing is
-    assigned to "half-edges".
+    This is a labelling of edges of a given graph by Paulis from the set {'X', 'Y', 'Z'}. In order
+    to deal properly with Hadamard edges, we actually label all "half-edges", where a half-edge named
+    (v, w) is the end of the edge closest to `v`, and (w, v) is the end closest to `w`.
+
+    A Pauli web is "closed" if
+    (i)  every spider has adjacent half-edges labelled by a stabiliser or anti-stabiliser of that spider
+    (ii) for every half-edge (v,w) labelled by a Pauli P, the other half is labelled by P (for a normal edge)
+         and labelled HPH (for a Hadamard edge).
+    
+    If these conditions are violated, the Pauli web is called "open", and a spider violating (i) or edge violating
+    (ii) is called a boundary of the web.
     """
     def __init__(self, g: BaseGraph[VT,ET]):
         self.g = g
         self.es: Dict[Tuple[VT,VT], str] = dict()
-        self.vs: Set[VT] = set()
+    
+    def copy(self) -> PauliWeb:
+        pw = PauliWeb(self.g)
+        pw.es = self.es.copy()
+        return pw
     
     def add_half_edge(self, v_pair: Tuple[VT, VT], pauli: str):
         s, t = v_pair
-        self.vs.add(s)
-        p = self.es.get((s,t), 'I')
-        self.es[(s,t)] = multiply_paulis(p, pauli)
+        p = multiply_paulis(self.es.get((s,t), 'I'), pauli)
+        if p == 'I':
+            self.es.pop((s,t),'')
+        else:
+            self.es[(s,t)] = p
 
     def add_edge(self, v_pair: Tuple[VT, VT], pauli: str):
         s, t = v_pair
         et = self.g.edge_type(self.g.edge(s, t))
         self.add_half_edge((s,t), pauli)
         self.add_half_edge((t,s), pauli if et == EdgeType.SIMPLE else h_pauli(pauli))
-
-        # if spread_to_input: 
-        #     inp = self.g.inputs()
-        #     if ('Z' if self.g.type(s) == VertexType.Z else 'X') == pauli:
-        #         for v2 in self.g.neighbors(s):
-        #             if v2 in inp:
-        #                 self.add_edge((s, v2), pauli, spread_to_input=False)
-        #                 break
-
-        #     if ('Z' if self.g.type(t) == VertexType.Z else 'X') == pauli:
-        #         for v2 in self.g.neighbors(t):
-        #             if v2 in inp:
-        #                 self.add_edge((t, v2), pauli, spread_to_input=False)
-        #                 break
-
-    # def spread_to_boundary(self, inputs=True, outputs=True):
-    #     bnd = []
-    #     if inputs: bnd += self.g.inputs()
-    #     if outputs: bnd += self.g.outputs()
-
-    #     for i in bnd:
-    #         v = next(iter(self.g.neighbors(i)))
-    #         vt = self.g.type(v)
-    #         if vt == VertexType.Z:
-    #             p = 'Z'
-    #         elif vt == VertexType.X:
-    #             p = 'X'
-    #         else:
-    #             continue
-    #         adj = sum(1 for v1 in self.g.neighbors(v)
-    #                   if (v,v1) in self.es and self.es[(v,v1)] in [p, 'Y'])
-    #         if adj % 2 == 1:
-    #             self.add_edge((v, i), p)
-
-
-    # def add_vertex(self, v: VT, spread_to_input: bool=False):
-    #     p = 'X' if self.g.type(v) == VertexType.Z else 'Z'
-    #     for v1 in self.g.neighbors(v):
-    #         self.add_edge((v, v1), p, spread_to_input=spread_to_input)
     
     def vertices(self):
-        return self.vs
+        return set(v for (v,_) in self.es)
 
     def half_edges(self) -> Dict[Tuple[VT,VT],str]:
         return self.es
     
     def __repr__(self):
-        return 'PauliWeb' + str(self.vs)
+        return 'PauliWeb' + str(self.vertices())
+    
+    def __mul__(self, other: PauliWeb):
+        pw = self.copy()
+        for e,p in other.es.items():
+            pw.add_half_edge(e, p)
+        return pw
+    
+    def commutes_with(self, other: PauliWeb):
+        comm = True
+        for e,p1 in self.es.items():
+            p2 = other.es.get(e, 'I')
+            if p1 != 'I' and p2 != 'I' and p1 != p2:
+                comm = not comm
+        return comm
+
 
 def transpose_corrections(c: Dict[VT, Set[VT]]) -> Dict[VT, Set[VT]]:
     ct: Dict[VT, Set[VT]] = dict()