From a2775a2fec311485ec482e2a9c48eeb7b0922c80 Mon Sep 17 00:00:00 2001
From: Aleks Kissinger <aleks0@gmail.com>
Date: Tue, 12 Nov 2024 12:43:58 +0000
Subject: [PATCH] ...

---
 scratchpads/ak.ipynb | 125 ++++++++++++++++++++++++++++---------------
 1 file changed, 82 insertions(+), 43 deletions(-)

diff --git a/scratchpads/ak.ipynb b/scratchpads/ak.ipynb
index 93293369..259ff3dd 100644
--- a/scratchpads/ak.ipynb
+++ b/scratchpads/ak.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 28,
+   "execution_count": 1,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -18,6 +18,8 @@
     "import random\n",
     "from pyzx.gflow import gflow\n",
     "from pyzx.utils import vertex_is_zx\n",
+    "from pyzx.graph.base import BaseGraph\n",
+    "from pyzx import VertexType, EdgeType\n",
     "# zx.settings.drawing_backend = 'matplotlib'\n",
     "\n",
     "Z = zx.VertexType.Z\n",
@@ -29,7 +31,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 2,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -40,12 +42,15 @@
     "            if v not in pt: pt[v] = set()\n",
     "            pt[v].add(k)\n",
     "    return pt\n",
-    "def extend(g, p):\n",
+    "def extend(g, s):\n",
+    "    s1 = s.copy()\n",
+    "    for v in s:\n",
+    "        s1 |= {v for v in g.neighbors(v) if vertex_is_zx(g.type(v))}\n",
+    "    return s1\n",
+    "def extend_dict(g, p):\n",
     "    pe = dict()\n",
     "    for k,s in p.items():\n",
-    "        pe[k] = s.copy()\n",
-    "        for v in s:\n",
-    "            pe[k] |= {v for v in g.neighbors(v) if vertex_is_zx(g.type(v))}\n",
+    "        pe[k] = extend(g, s)\n",
     "    return pe\n",
     "def is_dag(p):\n",
     "    pt = transpose(p)\n",
@@ -53,18 +58,32 @@
     "        for v in s:\n",
     "            if k != v and k in pt and v in pt[k]:\n",
     "                return False\n",
-    "    return True\n"
+    "    return True\n",
+    "\n",
+    "def is_clifford(p):\n",
+    "    return p in (0, 1, Fraction(1, 2), Fraction(3,2))\n",
+    "\n",
+    "def gadgetize(g: BaseGraph):\n",
+    "    for v in list(g.vertices()):\n",
+    "        p = g.phase(v)\n",
+    "        if not is_clifford(p) and g.vertex_degree(v) > 1:\n",
+    "            x = g.add_vertex(VertexType.Z, -1, g.row(v))\n",
+    "            y = g.add_vertex(VertexType.Z, -2, g.row(v))\n",
+    "            g.add_edge((x, y), EdgeType.HADAMARD)\n",
+    "            g.add_edge((v, x), EdgeType.HADAMARD)\n",
+    "            g.set_phase(y, p)\n",
+    "            g.set_phase(v, 0)\n"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 42,
+   "execution_count": 3,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-2BR7PWNG\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-gXQUVmtX\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -395,8 +414,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-2BR7PWNG',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 20, \"y\": 40, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 20, \"y\": 60, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 20, \"y\": 80, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 20, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 20, \"y\": 120, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 40, \"y\": 40, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 40, \"y\": 60, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 40, \"y\": 80, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 40, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 40, \"y\": 120, 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+       "showGraph('#graph-output-gXQUVmtX',\n",
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1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"28\", \"x\": 400, \"y\": 60, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"29\", \"x\": 420, \"y\": 40, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"30\", \"x\": 440, \"y\": 60, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"31\", \"x\": 460, \"y\": 40, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"32\", \"x\": 480, \"y\": 80, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"33\", \"x\": 480, \"y\": 40, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"34\", \"x\": 500, \"y\": 80, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"35\", \"x\": 520, \"y\": 40, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"36\", \"x\": 520, \"y\": 80, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"37\", \"x\": 540, \"y\": 40, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"38\", \"x\": 560, \"y\": 60, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"39\", \"x\": 560, \"y\": 80, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"40\", \"x\": 580, \"y\": 60, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"41\", \"x\": 580, \"y\": 40, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"42\", \"x\": 600, \"y\": 80, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"43\", \"x\": 620, \"y\": 40, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"44\", \"x\": 640, \"y\": 80, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"45\", \"x\": 640, \"y\": 40, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"46\", \"x\": 660, \"y\": 40, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"47\", \"x\": 660, \"y\": 80, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"48\", \"x\": 680, \"y\": 80, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"49\", \"x\": 700, \"y\": 80, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"50\", \"x\": 720, \"y\": 60, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"51\", \"x\": 720, \"y\": 80, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"52\", \"x\": 740, \"y\": 80, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"53\", \"x\": 740, \"y\": 60, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"54\", \"x\": 760, \"y\": 60, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"55\", \"x\": 780, \"y\": 60, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"56\", \"x\": 780, \"y\": 80, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"57\", \"x\": 800, \"y\": 40, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"58\", \"x\": 820, \"y\": 60, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"59\", \"x\": 820, \"y\": 80, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"60\", \"x\": 840, \"y\": 80, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"61\", \"x\": 840, \"y\": 60, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"62\", \"x\": 860, \"y\": 40, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"63\", \"x\": 880, \"y\": 40, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"64\", \"x\": 900, \"y\": 40, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"65\", \"x\": 920, \"y\": 80, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"66\", \"x\": 940, \"y\": 40, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"67\", \"x\": 960, \"y\": 40, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"68\", \"x\": 980, \"y\": 40, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"69\", \"x\": 1000, \"y\": 40, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"70\", \"x\": 1020, \"y\": 80, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"71\", \"x\": 1020, \"y\": 60, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"72\", \"x\": 1040, \"y\": 40, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"73\", \"x\": 1040, \"y\": 60, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"74\", \"x\": 1060, \"y\": 60, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"75\", \"x\": 1080, \"y\": 80, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"76\", \"x\": 1080, \"y\": 40, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"77\", \"x\": 1100, \"y\": 60, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"78\", \"x\": 1100, \"y\": 80, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"79\", \"x\": 1120, \"y\": 80, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"80\", \"x\": 1140, \"y\": 80, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"81\", \"x\": 1140, \"y\": 60, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"82\", \"x\": 1160, \"y\": 80, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"83\", \"x\": 1160, \"y\": 60, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"84\", \"x\": 1180, \"y\": 60, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"85\", \"x\": 1200, \"y\": 80, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"86\", \"x\": 1220, \"y\": 40, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"87\", \"x\": 1240, \"y\": 60, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"88\", \"x\": 1260, \"y\": 40, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"89\", \"x\": 1260, \"y\": 60, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"90\", \"x\": 1260, \"y\": 80, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"91\", \"x\": 1280, \"y\": 40, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"92\", \"x\": 1280, \"y\": 60, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"93\", \"x\": 1280, \"y\": 80, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"7\", \"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\": \"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\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"22\", \"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\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"23\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"20\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"21\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"21\", \"target\": \"32\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"22\", \"target\": \"25\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"23\", \"target\": \"24\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"24\", \"target\": \"26\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"25\", \"target\": \"28\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"26\", \"target\": \"27\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"28\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"29\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"28\", \"target\": \"30\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"29\", \"target\": \"31\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"30\", \"target\": \"38\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"31\", \"target\": \"33\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"32\", \"target\": \"33\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"32\", \"target\": \"34\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"33\", \"target\": \"35\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"34\", \"target\": \"36\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"35\", \"target\": \"36\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"35\", \"target\": \"37\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"36\", \"target\": \"39\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"37\", \"target\": \"41\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"38\", \"target\": \"39\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"38\", \"target\": \"40\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"39\", \"target\": \"42\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"40\", \"target\": \"41\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"40\", \"target\": \"50\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"41\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"42\", \"target\": \"44\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"45\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"44\", \"target\": \"45\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"44\", \"target\": \"47\", \"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\": \"57\", \"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\": \"49\", \"target\": \"51\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"50\", \"target\": \"51\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"50\", \"target\": \"53\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"51\", \"target\": \"52\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"52\", \"target\": \"53\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"52\", \"target\": \"56\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"53\", \"target\": \"54\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"55\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"55\", \"target\": \"56\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"55\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"56\", \"target\": \"59\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"57\", \"target\": \"62\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"58\", \"target\": \"59\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"58\", \"target\": \"61\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"59\", \"target\": \"60\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"60\", \"target\": \"61\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"60\", \"target\": \"65\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"61\", \"target\": \"71\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"62\", \"target\": \"63\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"63\", \"target\": \"64\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"64\", \"target\": \"66\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"65\", \"target\": \"70\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"66\", \"target\": \"67\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"67\", \"target\": \"68\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"68\", \"target\": \"69\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"69\", \"target\": \"72\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"70\", \"target\": \"71\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"70\", \"target\": \"75\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"71\", \"target\": \"73\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"72\", \"target\": \"73\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"72\", \"target\": \"76\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"73\", \"target\": \"74\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"74\", \"target\": \"77\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"75\", \"target\": \"76\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"75\", \"target\": \"78\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"76\", \"target\": \"86\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"77\", \"target\": \"78\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"77\", \"target\": \"81\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"78\", \"target\": \"79\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"79\", \"target\": \"80\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"80\", \"target\": \"81\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"80\", \"target\": \"82\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"81\", \"target\": \"83\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"82\", \"target\": \"83\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"82\", \"target\": \"85\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"83\", \"target\": \"84\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"84\", \"target\": \"87\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"85\", \"target\": \"90\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"86\", \"target\": \"88\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"87\", \"target\": \"89\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"88\", \"target\": \"91\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"89\", \"target\": \"92\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"90\", \"target\": \"93\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}]}'), 1300, 100, 20,\n",
        "  4.0, false, false, '');\n",
        "</script>"
       ],
@@ -410,30 +429,30 @@
     {
      "data": {
       "text/plain": [
-       "Graph(541 vertices, 657 edges)"
+       "Graph(94 vertices, 113 edges)"
       ]
      },
-     "execution_count": 42,
+     "execution_count": 3,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "random.seed(1337)\n",
-    "g = zx.generate.cliffordT(5, 400, p_t=0.2)\n",
+    "random.seed(1330)\n",
+    "g = zx.generate.cliffordT(3, 60, p_t=0.2)\n",
     "zx.draw(g)\n",
     "g"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 43,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-IBZEuamR\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-dkKujnpW\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -764,9 +783,9 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-IBZEuamR',\n",
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\"target\": \"46\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"42\", \"target\": \"44\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"44\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"61\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"44\", \"target\": \"50\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"44\", \"target\": \"56\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"44\", \"target\": \"46\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"44\", \"target\": \"47\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"44\", \"target\": \"51\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"45\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"45\", \"target\": \"46\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"45\", \"target\": \"49\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"45\", \"target\": \"51\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"46\", \"target\": \"49\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"46\", \"target\": \"48\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"47\", \"target\": \"48\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"47\", \"target\": \"60\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"47\", \"target\": \"51\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"48\", \"target\": \"53\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"49\", \"target\": \"55\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"50\", \"target\": \"51\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"50\", \"target\": \"63\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"51\", \"target\": \"52\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}]}'), 860, 180, 20,\n",
-       "  4.0, false, true, '');\n",
+       "showGraph('#graph-output-dkKujnpW',\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\": \"3\", \"x\": 100, \"y\": 250, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 300, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 250, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 250, \"y\": 250, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 400, \"y\": 300, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 400, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 400, \"y\": 250, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 450, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 450, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 450, \"y\": 300, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 150, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 150, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"23\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"3\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"24\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"25\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"7\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"26\", \"x\": 350, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"27\", \"x\": 350, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"4\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"10\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"8\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"16\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"18\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"10\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"22\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"24\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"12\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"26\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"13\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"20\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"22\", \"target\": \"23\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"24\", \"target\": \"25\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"26\", \"target\": \"27\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}]}'), 500, 350, 50,\n",
+       "  10.0, false, true, '');\n",
        "</script>"
       ],
       "text/plain": [
@@ -780,6 +799,7 @@
    "source": [
     "g1 = g.copy()\n",
     "zx.full_reduce(g1)\n",
+    "gadgetize(g1)\n",
     "g1.normalize()\n",
     "g1 = g1.copy()\n",
     "zx.to_rg(g1)\n",
@@ -790,57 +810,76 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 50,
+   "execution_count": 7,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "{5: 14, 6: 14, 7: 14, 8: 14, 13: 14, 9: 13, 10: 13, 11: 13, 58: 13, 12: 12, 15: 12, 17: 12, 18: 12, 57: 12, 59: 12, 14: 11, 16: 11, 19: 11, 20: 10, 21: 10, 22: 10, 23: 9, 24: 9, 25: 9, 27: 9, 26: 8, 28: 8, 29: 8, 30: 8, 31: 7, 40: 7, 62: 7, 32: 6, 33: 6, 34: 5, 35: 5, 36: 5, 37: 4, 38: 3, 39: 2, 41: 2, 42: 2, 44: 2, 60: 2, 61: 2, 43: 1, 45: 1, 46: 1, 47: 1, 48: 1, 51: 1, 63: 1, 49: 0, 50: 0}\n",
-      "{9: {8, 6}, 10: {8, 5, 6}, 11: {6}, 58: {8, 12, 5, 6}, 12: {58}, 15: {8, 10, 58}, 17: {5, 6, 8, 13, 58}, 18: {10, 58, 6, 7}, 57: {5, 6, 7, 10, 13, 14}, 59: {6, 8, 10, 16, 58}, 14: {57}, 16: {59}, 19: {10, 11, 58, 15}, 20: {19, 7}, 21: {6, 7, 9, 11, 13, 15, 17, 19, 58}, 22: {7, 9, 11, 17, 18, 19, 57, 59}, 23: {7, 11, 13, 17, 18, 20, 58, 59}, 24: {5, 15, 17, 19, 20, 21, 57, 59}, 25: {7, 8, 9, 15, 17, 18, 20, 22, 57, 59}, 27: {5, 9, 10, 17, 18, 19, 21, 22, 57, 58}, 26: {5, 6, 7, 13, 17, 18, 20, 21, 23, 58}, 28: {5, 9, 10, 11, 15, 19, 21, 22, 23, 25, 58, 57}, 29: {5, 6, 7, 8, 9, 18, 23}, 30: {6, 9, 10, 11, 13, 15, 18, 20, 21, 22, 24}, 31: {6, 7, 8, 9, 13, 17, 18, 21, 23, 24, 57, 58, 28, 29}, 40: {58, 6, 7, 8, 59, 11, 13, 21, 26, 27, 29}, 62: {6, 8, 10, 13, 15, 17, 18, 19, 23, 59, 28}, 32: {5, 7, 8, 10, 13, 17, 18, 20, 21, 22, 24, 25, 62}, 33: {58, 6, 7, 9, 10, 15, 17, 21, 22, 24, 26, 59, 28, 29, 62, 31}, 34: {32, 5, 8, 9, 17, 20, 21, 23, 24, 62, 26, 28, 29, 30, 31}, 35: {33, 5, 6, 13, 18, 19, 20, 23, 24, 25, 58, 57, 62, 31}, 36: {32, 7, 59, 10, 11, 13, 18, 20, 24, 57, 27, 62, 31}, 37: {34, 5, 6, 7, 11, 18, 19, 20, 62, 22, 23, 24, 57, 26, 59, 28, 30}, 38: {6, 7, 8, 9, 13, 20, 21, 22, 23, 24, 25, 28, 30, 34, 36, 37, 57, 58, 62}, 39: {5, 6, 7, 9, 10, 17, 18, 19, 24, 25, 26, 28, 32, 34, 35, 36, 38, 57, 58}, 41: {6, 7, 9, 11, 13, 15, 17, 19, 24, 26, 27, 28, 30, 31, 32, 33, 34, 38, 40, 58}, 42: {33, 36, 5, 6, 7, 38, 10, 13, 17, 18, 20, 24, 62, 27, 28, 29, 30, 31}, 44: {33, 35, 38, 40, 59, 10, 13, 15, 17, 19, 20, 57, 26, 27, 30}, 60: {9, 11, 13, 15, 19, 22, 23, 25, 26, 27, 32, 33, 36, 38, 40, 47, 57, 58, 62}, 61: {36, 5, 38, 11, 43, 17, 18, 19, 20, 25, 26, 59, 31}, 43: {61}, 45: {32, 33, 5, 6, 7, 40, 9, 41, 19, 20, 21, 22, 23, 57, 26, 27, 61, 31}, 46: {35, 8, 9, 10, 11, 42, 13, 15, 17, 22, 24, 27, 28, 62, 31}, 47: {60}, 48: {6, 7, 8, 17, 19, 22, 24, 27, 28, 29, 30, 33, 37, 38, 39, 40, 57, 61, 62}, 51: {6, 7, 8, 9, 10, 11, 19, 20, 23, 24, 25, 27, 28, 29, 32, 33, 35, 36, 37, 38, 39, 40, 57, 58, 59, 61}, 63: {6, 7, 8, 9, 11, 13, 15, 17, 23, 24, 25, 26, 28, 29, 30, 32, 36, 37, 39, 50, 58, 61}, 49: {6, 7, 8, 9, 15, 17, 18, 19, 20, 21, 23, 30, 32, 35, 36, 37, 38, 39, 40, 41, 46, 57, 59}, 50: {63}}\n",
-      "{9: {6, 8, 9, 11}, 10: {5, 6, 8, 9, 10, 11, 13}, 11: {11, 6}, 58: {5, 6, 8, 9, 10, 11, 12, 13, 58}, 15: {5, 8, 9, 10, 11, 12, 13, 15, 58}, 17: {5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 58}, 18: {5, 6, 7, 9, 10, 11, 12, 13, 15, 18, 58}, 57: {5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 57}, 59: {5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 58, 59}, 19: {5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 19, 21, 25, 26, 27, 29, 58}, 20: {7, 9, 13, 15, 18, 19, 20}, 21: {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 29, 58}, 22: {5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 29, 57, 59}, 23: {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 58, 59}, 24: {5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 57, 59}, 25: {5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 57, 59}, 27: {5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 57, 58}, 26: {5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 58}, 28: {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 57, 58}, 29: {5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29}, 30: {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30}, 31: {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 57, 58, 62}, 40: {5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 39, 40, 41, 42, 44, 58, 59, 62}, 62: {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 59, 62}, 32: {5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 62}, 33: {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 39, 41, 42, 44, 58, 59, 62}, 34: {5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 39, 40, 41, 42, 44, 62}, 35: {5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 39, 41, 42, 44, 57, 58, 62}, 36: {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 39, 40, 41, 42, 44, 57, 59, 62}, 37: {5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 44, 57, 59, 62}, 38: {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 48, 51, 57, 58, 62}, 39: {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 48, 51, 57, 58, 62}, 41: {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 48, 51, 58, 62}, 42: {5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 48, 51, 62}, 44: {5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 48, 51, 57, 59, 62}, 60: {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 47, 48, 51, 57, 58, 60, 62}, 61: {5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 48, 51, 59, 61}, 45: {5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 39, 40, 41, 42, 43, 44, 45, 48, 51, 57, 61, 62}, 46: {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 44, 46, 62}, 48: {6, 7, 8, 9, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 48, 51, 57, 61, 62}, 51: {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 48, 51, 57, 58, 59, 61, 62}, 63: {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 48, 50, 51, 58, 61, 62, 63}}\n"
+      "17 <- {16} + {17, 3}\n",
+      "19 <- {18} + {19, 4}\n",
+      "15 <- {3, 4, 6, 7, 17, 19} + {5, 8, 9, 10, 15, 16, 18}\n",
+      "21 <- {19, 4, 20, 6} + {3, 5, 9, 18, 21}\n",
+      "25 <- {24, 19, 4} + {9, 18, 25}\n",
+      "23 <- {4, 5, 19, 21, 22} + {3, 6, 8, 9, 18, 20, 23}\n",
+      "27 <- {3, 5, 6, 17, 21, 26} + {8, 9, 10, 16, 20, 27}\n"
      ]
-    },
-    {
-     "data": {
-      "text/plain": [
-       "False"
-      ]
-     },
-     "execution_count": 50,
-     "metadata": {},
-     "output_type": "execute_result"
     }
    ],
    "source": [
     "o, p = gflow(g1, focus=True, reverse=True, pauli=True)\n",
-    "print(o)\n",
-    "print(p)\n",
-    "ep = extend(g1, p)\n",
-    "print(ep)\n",
-    "is_dag(ep)"
+    "vs = [v for v in g1.vertices() if not is_clifford(g1.phase(v))]\n",
+    "list.sort(vs, key=lambda v: o[v], reverse=True)\n",
+    "for v in vs: print(f\"{v} <- {p[v]} + {extend(g1, p[v]).difference(p[v])}\")"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "{48: 14, 49: 14, 51: 14, 44: 14, 45: 14, 41: 13, 46: 13, 60: 13, 63: 13, 39: 12, 40: 12, 42: 12, 47: 12, 50: 12, 61: 12, 35: 11, 38: 11, 43: 11, 33: 10, 36: 10, 37: 10, 27: 9, 34: 9, 26: 8, 30: 8, 31: 8, 32: 8, 24: 7, 29: 7, 62: 7, 28: 6, 21: 5, 23: 5, 25: 5, 20: 4, 22: 4, 9: 3, 17: 3, 18: 3, 19: 3, 57: 3, 59: 3, 7: 2, 11: 2, 13: 2, 14: 2, 15: 2, 16: 2, 10: 1, 58: 1, 5: 0, 6: 0, 8: 0, 12: 0}\n",
-      "{49: {6, 7, 8, 9, 15, 17, 18, 19, 20, 21, 23, 30, 32, 35, 36, 37, 38, 39, 40, 41, 46, 57, 59}, 45: {32, 33, 5, 6, 7, 40, 41, 9, 19, 20, 21, 22, 23, 57, 26, 27, 61, 31}, 47: {60}, 50: {63}, 48: {6, 7, 8, 17, 19, 22, 24, 27, 28, 29, 30, 33, 37, 38, 39, 40, 57, 61, 62}, 51: {6, 7, 8, 9, 10, 11, 19, 20, 23, 24, 25, 27, 28, 29, 32, 33, 35, 36, 37, 38, 39, 40, 57, 58, 59, 61}, 63: {6, 7, 8, 9, 11, 13, 15, 17, 23, 24, 25, 26, 28, 29, 30, 32, 36, 37, 39, 50, 58, 61}, 41: {6, 7, 9, 11, 13, 15, 17, 19, 24, 26, 27, 28, 30, 31, 32, 33, 34, 38, 40, 58}, 44: {33, 35, 38, 40, 59, 10, 13, 15, 17, 19, 20, 57, 26, 27, 30}, 60: {9, 11, 13, 15, 19, 22, 23, 25, 26, 27, 32, 33, 36, 38, 40, 47, 57, 58, 62}, 46: {35, 8, 9, 42, 11, 10, 13, 15, 17, 22, 24, 27, 28, 62, 31}, 43: {61}, 39: {5, 6, 7, 9, 10, 17, 18, 19, 24, 25, 26, 28, 32, 34, 35, 36, 38, 57, 58}, 42: {33, 36, 5, 38, 7, 6, 10, 13, 17, 18, 20, 24, 62, 27, 28, 29, 30, 31}, 61: {36, 5, 38, 43, 11, 17, 18, 19, 20, 25, 26, 59, 31}, 35: {33, 5, 6, 13, 18, 19, 20, 23, 24, 25, 58, 57, 62, 31}, 38: {6, 7, 8, 9, 13, 20, 21, 22, 23, 24, 25, 28, 30, 34, 36, 37, 57, 58, 62}, 36: {32, 7, 59, 10, 11, 13, 18, 20, 24, 57, 27, 62, 31}, 40: {58, 6, 7, 27, 8, 11, 13, 21, 26, 59, 29}, 37: {34, 5, 6, 7, 11, 18, 19, 23, 20, 22, 30, 24, 57, 26, 59, 28, 62}, 33: {58, 6, 7, 9, 10, 15, 17, 21, 22, 24, 26, 59, 28, 29, 62, 31}, 34: {32, 5, 8, 9, 17, 20, 21, 23, 24, 62, 26, 28, 29, 30, 31}, 32: {5, 7, 8, 10, 13, 17, 18, 20, 21, 22, 24, 25, 62}, 30: {6, 9, 10, 11, 13, 15, 18, 20, 21, 22, 24}, 31: {6, 7, 8, 9, 13, 17, 18, 21, 23, 24, 57, 58, 28, 29}, 62: {6, 8, 10, 13, 15, 17, 18, 19, 23, 59, 28}, 24: {5, 15, 17, 19, 20, 21, 57, 59}, 26: {5, 6, 7, 13, 17, 18, 20, 21, 23, 58}, 27: {5, 9, 10, 17, 18, 19, 21, 22, 57, 58}, 28: {5, 9, 10, 11, 15, 19, 21, 22, 23, 25, 58, 57}, 29: {5, 6, 7, 8, 9, 18, 23}, 23: {7, 11, 13, 17, 18, 20, 58, 59}, 25: {7, 8, 9, 15, 17, 18, 20, 22, 57, 59}, 21: {6, 7, 9, 11, 13, 15, 17, 19, 58}, 22: {7, 9, 11, 17, 18, 19, 57, 59}, 20: {19, 7}, 14: {57}, 16: {59}, 18: {10, 58, 6, 7}, 57: {5, 6, 7, 10, 13, 14}, 19: {10, 11, 58, 15}, 17: {5, 6, 8, 13, 58}, 59: {6, 8, 10, 16, 58}, 15: {8, 10, 58}, 12: {58}, 58: {8, 12, 5, 6}, 10: {8, 5, 6}, 9: {8, 6}, 11: {6}}\n"
+      "{9: 4, 10: 4, 11: 4, 8: 3, 15: 3, 23: 3, 25: 3, 27: 3, 3: 2, 5: 2, 7: 2, 17: 2, 21: 2, 22: 2, 24: 2, 26: 2, 4: 1, 6: 1, 16: 1, 19: 1, 20: 1, 18: 0}\n"
      ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "{11: {3, 4, 5, 8, 15, 17, 19, 21, 23},\n",
+       " 7: {15},\n",
+       " 22: {23},\n",
+       " 24: {25},\n",
+       " 26: {27},\n",
+       " 27: {3, 5, 6, 17, 21, 26},\n",
+       " 10: {3, 5, 6, 17, 21},\n",
+       " 15: {3, 4, 6, 7, 17, 19},\n",
+       " 8: {4, 5, 19, 21},\n",
+       " 23: {4, 5, 19, 21, 22},\n",
+       " 16: {17},\n",
+       " 20: {21},\n",
+       " 25: {4, 19, 24},\n",
+       " 5: {4, 6, 19},\n",
+       " 9: {4, 19},\n",
+       " 21: {4, 6, 19, 20},\n",
+       " 17: {16},\n",
+       " 18: {19},\n",
+       " 19: {18}}"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
     }
    ],
    "source": [
     "o,p = gflow(g1, focus=True, reverse=False, pauli=True)\n",
     "pt = transpose(p)\n",
     "print(o)\n",
-    "print(pt)"
+    "pt"
    ]
   },
   {