Orbital rotations and quadratic Hamiltonians

-

In this page, we discuss orbital rotations and how they can be used to implement time evolution by a quadratic Hamiltonian.

+

This page discusses orbital rotations and how they can be used to implement time evolution by a quadratic Hamiltonian.

Orbital rotations

The orbital rotation is a fundamental operation in the simulation of a system of fermionic modes. An orbital rotation is described by an \(N \times N\) unitary matrix \(\mathbf{W}\) (here \(N\) is the number spatial orbitals), and we denote the corresponding operator as \(\mathcal{W}\). This operator has the following action on the fermionic creation operators \(\set{a^\dagger_{\sigma, i}}\):

diff --git a/explanations/orbital-rotation.ipynb b/explanations/orbital-rotation.ipynb index 8aad74446..aa96276e9 100644 --- a/explanations/orbital-rotation.ipynb +++ b/explanations/orbital-rotation.ipynb @@ -6,7 +6,7 @@ "source": [ "# Orbital rotations and quadratic Hamiltonians\n", "\n", - "In this page, we discuss orbital rotations and how they can be used to implement time evolution by a quadratic Hamiltonian.\n", + "This page discusses orbital rotations and how they can be used to implement time evolution by a quadratic Hamiltonian.\n", "\n", "## Orbital rotations\n", "\n", @@ -62,10 +62,10 @@ "execution_count": 1, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:14.729266Z", - "iopub.status.busy": "2024-04-24T06:38:14.729074Z", - "iopub.status.idle": "2024-04-24T06:38:15.361173Z", - "shell.execute_reply": "2024-04-24T06:38:15.360495Z" + "iopub.execute_input": "2024-04-24T07:11:38.021734Z", + "iopub.status.busy": "2024-04-24T07:11:38.021535Z", + "iopub.status.idle": "2024-04-24T07:11:38.638847Z", + "shell.execute_reply": "2024-04-24T07:11:38.638242Z" } }, "outputs": [], diff --git a/explanations/state-vectors-and-gates.ipynb b/explanations/state-vectors-and-gates.ipynb index b0353f367..525a3b2fd 100644 --- a/explanations/state-vectors-and-gates.ipynb +++ b/explanations/state-vectors-and-gates.ipynb @@ -26,10 +26,10 @@ "execution_count": 1, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:16.998500Z", - "iopub.status.busy": "2024-04-24T06:38:16.998307Z", - "iopub.status.idle": "2024-04-24T06:38:17.636829Z", - "shell.execute_reply": "2024-04-24T06:38:17.636143Z" + "iopub.execute_input": "2024-04-24T07:11:40.151148Z", + "iopub.status.busy": "2024-04-24T07:11:40.150954Z", + "iopub.status.idle": "2024-04-24T07:11:40.761322Z", + "shell.execute_reply": "2024-04-24T07:11:40.760688Z" } }, "outputs": [ @@ -74,10 +74,10 @@ "execution_count": 2, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:17.639496Z", - "iopub.status.busy": "2024-04-24T06:38:17.639038Z", - "iopub.status.idle": "2024-04-24T06:38:17.645831Z", - "shell.execute_reply": "2024-04-24T06:38:17.645267Z" + "iopub.execute_input": "2024-04-24T07:11:40.764095Z", + "iopub.status.busy": "2024-04-24T07:11:40.763511Z", + "iopub.status.idle": "2024-04-24T07:11:40.770429Z", + "shell.execute_reply": "2024-04-24T07:11:40.769866Z" } }, "outputs": [ @@ -118,10 +118,10 @@ "execution_count": 3, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:17.647970Z", - "iopub.status.busy": "2024-04-24T06:38:17.647796Z", - "iopub.status.idle": "2024-04-24T06:38:17.651871Z", - "shell.execute_reply": "2024-04-24T06:38:17.651355Z" + "iopub.execute_input": "2024-04-24T07:11:40.772636Z", + "iopub.status.busy": "2024-04-24T07:11:40.772317Z", + "iopub.status.idle": "2024-04-24T07:11:40.776546Z", + "shell.execute_reply": "2024-04-24T07:11:40.775984Z" } }, "outputs": [ @@ -155,10 +155,10 @@ "execution_count": 4, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:17.654175Z", - "iopub.status.busy": "2024-04-24T06:38:17.653825Z", - "iopub.status.idle": "2024-04-24T06:38:17.657942Z", - "shell.execute_reply": "2024-04-24T06:38:17.657284Z" + "iopub.execute_input": "2024-04-24T07:11:40.778903Z", + "iopub.status.busy": "2024-04-24T07:11:40.778521Z", + "iopub.status.idle": "2024-04-24T07:11:40.782526Z", + "shell.execute_reply": "2024-04-24T07:11:40.782049Z" } }, "outputs": [ @@ -197,10 +197,10 @@ "execution_count": 5, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:17.660390Z", - "iopub.status.busy": "2024-04-24T06:38:17.660007Z", - "iopub.status.idle": "2024-04-24T06:38:17.665743Z", - "shell.execute_reply": "2024-04-24T06:38:17.665256Z" + "iopub.execute_input": "2024-04-24T07:11:40.784698Z", + "iopub.status.busy": "2024-04-24T07:11:40.784373Z", + "iopub.status.idle": "2024-04-24T07:11:40.790752Z", + "shell.execute_reply": "2024-04-24T07:11:40.790182Z" } }, "outputs": [ @@ -243,10 +243,10 @@ "execution_count": 6, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:17.668114Z", - "iopub.status.busy": "2024-04-24T06:38:17.667755Z", - "iopub.status.idle": "2024-04-24T06:38:17.672691Z", - "shell.execute_reply": "2024-04-24T06:38:17.672122Z" + "iopub.execute_input": "2024-04-24T07:11:40.793072Z", + "iopub.status.busy": "2024-04-24T07:11:40.792741Z", + "iopub.status.idle": "2024-04-24T07:11:40.798031Z", + "shell.execute_reply": "2024-04-24T07:11:40.797467Z" } }, "outputs": [ diff --git a/how-to-guides/entanglement-forging.html b/how-to-guides/entanglement-forging.html index aaf0b2658..626636ddd 100644 --- a/how-to-guides/entanglement-forging.html +++ b/how-to-guides/entanglement-forging.html @@ -317,7 +317,7 @@

How to simulate entanglement forging 
-Energy at initialialization: -74.20656273321636
+Energy at initialialization: -74.20656273321637
@@ -359,10 +359,10 @@

How to simulate entanglement forging\n" diff --git a/how-to-guides/fermion-operator.html b/how-to-guides/fermion-operator.html index b45714470..8032550c1 100644 --- a/how-to-guides/fermion-operator.html +++ b/how-to-guides/fermion-operator.html @@ -306,17 +306,17 @@

How to use the FermionOperator class 
 FermionOperator({
-    (cre_b(2)): 0-0.25j,
-    (cre_a(3), des_a(0)): -0.5,
-    (cre_b(1), des_b(5), cre_a(4), des_a(3), des_b(3)): -0.25-0.25j,
-    (cre_a(0), des_a(3), cre_b(2)): 0+0.5j,
     (cre_a(0), des_a(3), des_a(3), des_b(3)): -0.125,
     (cre_b(1), des_b(5), cre_a(4)): 2+2j,
     (cre_b(1), des_b(5), cre_a(4), cre_b(2)): -1+1j,
+    (cre_a(3), des_a(0), des_a(3), des_b(3)): 0.0625,
+    (cre_b(1), des_b(5), cre_a(4), des_a(3), des_b(3)): -0.25-0.25j,
     (cre_a(3), des_a(0), cre_b(2)): 0-0.25j,
+    (cre_a(0), des_a(3), cre_b(2)): 0+0.5j,
     (cre_a(0), des_a(3)): 1,
-    (des_a(3), des_b(3)): 0.0625,
-    (cre_a(3), des_a(0), des_a(3), des_b(3)): 0.0625
+    (cre_a(3), des_a(0)): -0.5,
+    (cre_b(2)): 0-0.25j,
+    (des_a(3), des_b(3)): 0.0625
 })

@@ -345,17 +345,17 @@

How to use the FermionOperator class 
 FermionOperator({
-    (cre_b(2)): -5,
-    (cre_a(3), des_a(0)): 0+3j,
-    (cre_b(1), des_b(5), cre_a(4), des_a(3), des_b(3)): -1+1j,
-    (cre_a(0), des_a(3), cre_b(2)): 2,
     (cre_a(0), des_a(3), des_a(3), des_b(3)): 0+0.5j,
     (cre_b(1), des_b(5), cre_a(4)): 12-12j,
     (cre_b(1), des_b(5), cre_a(4), cre_b(2)): 4+4j,
+    (cre_a(3), des_a(0), des_a(3), des_b(3)): 0-0.25j,
+    (cre_b(1), des_b(5), cre_a(4), des_a(3), des_b(3)): -1+1j,
     (cre_a(3), des_a(0), cre_b(2)): -1,
+    (cre_a(0), des_a(3), cre_b(2)): 2,
     (cre_a(0), des_a(3)): 0-6j,
-    (des_a(3), des_b(3)): 0-1.25j,
-    (cre_a(3), des_a(0), des_a(3), des_b(3)): 0-0.25j
+    (cre_a(3), des_a(0)): 0+3j,
+    (cre_b(2)): -5,
+    (des_a(3), des_b(3)): 0-1.25j
 })
@@ -376,16 +376,16 @@

How to use the FermionOperator class 
 FermionOperator({
-    (cre_b(2), cre_a(0), des_a(3)): 2,
     (cre_a(3), des_a(0)): 0+3j,
-    (cre_b(1), cre_a(4), des_b(5), des_b(3), des_a(3)): -1+1j,
-    (cre_a(3), des_b(3), des_a(3), des_a(0)): 0+0.25j,
-    (cre_b(2), cre_a(3), des_a(0)): -1,
-    (cre_a(0), des_a(3)): 0-6j,
-    (cre_b(1), cre_a(4), des_b(5)): -12+12j,
     (cre_b(2), cre_b(1), cre_a(4), des_b(5)): 4+4j,
     (des_b(3), des_a(3)): 0+1.25j,
-    (cre_b(2)): -5
+    (cre_a(3), des_b(3), des_a(3), des_a(0)): 0+0.25j,
+    (cre_b(1), cre_a(4), des_b(5), des_b(3), des_a(3)): -1+1j,
+    (cre_b(2), cre_a(0), des_a(3)): 2,
+    (cre_a(0), des_a(3)): 0-6j,
+    (cre_b(2), cre_a(3), des_a(0)): -1,
+    (cre_b(2)): -5,
+    (cre_b(1), cre_a(4), des_b(5)): -12+12j
 })
@@ -454,11 +454,11 @@

How to use the FermionOperator class 
-array([0.        +0.j        , 0.        +0.j        ,
-       0.        +0.j        , 0.        +0.j        ,
-       0.24268319+0.17800037j, 0.        +0.j        ,
-       0.        +0.j        , 0.        +0.j        ,
-       0.        +0.j        ])
+array([ 0.        +0.j        ,  0.        +0.j        ,
+        0.        +0.j        ,  0.        +0.j        ,
+       -0.13117779+0.11588432j,  0.        +0.j        ,
+        0.        +0.j        ,  0.        +0.j        ,
+        0.        +0.j        ])

It can also be passed into most linear algebra routines in scipy.sparse.linalg.

diff --git a/how-to-guides/fermion-operator.ipynb b/how-to-guides/fermion-operator.ipynb index efc2b7d9e..046b3e7c4 100644 --- a/how-to-guides/fermion-operator.ipynb +++ b/how-to-guides/fermion-operator.ipynb @@ -29,10 +29,10 @@ "execution_count": 1, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:29.815948Z", - "iopub.status.busy": "2024-04-24T06:38:29.815750Z", - "iopub.status.idle": "2024-04-24T06:38:30.432103Z", - "shell.execute_reply": "2024-04-24T06:38:30.431410Z" + "iopub.execute_input": "2024-04-24T07:11:52.462303Z", + "iopub.status.busy": "2024-04-24T07:11:52.462109Z", + "iopub.status.idle": "2024-04-24T07:11:53.107495Z", + "shell.execute_reply": "2024-04-24T07:11:53.106886Z" } }, "outputs": [ @@ -76,10 +76,10 @@ "execution_count": 2, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:30.435159Z", - "iopub.status.busy": "2024-04-24T06:38:30.434642Z", - "iopub.status.idle": "2024-04-24T06:38:30.439179Z", - "shell.execute_reply": "2024-04-24T06:38:30.438687Z" + "iopub.execute_input": "2024-04-24T07:11:53.110465Z", + "iopub.status.busy": "2024-04-24T07:11:53.109821Z", + "iopub.status.idle": "2024-04-24T07:11:53.114213Z", + "shell.execute_reply": "2024-04-24T07:11:53.113652Z" } }, "outputs": [ @@ -110,10 +110,10 @@ "execution_count": 3, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:30.441845Z", - "iopub.status.busy": "2024-04-24T06:38:30.441260Z", - "iopub.status.idle": "2024-04-24T06:38:30.446114Z", - "shell.execute_reply": "2024-04-24T06:38:30.445507Z" + "iopub.execute_input": "2024-04-24T07:11:53.116351Z", + "iopub.status.busy": "2024-04-24T07:11:53.116141Z", + "iopub.status.idle": "2024-04-24T07:11:53.120393Z", + "shell.execute_reply": "2024-04-24T07:11:53.119833Z" } }, "outputs": [ @@ -121,17 +121,17 @@ "data": { "text/plain": [ "FermionOperator({\n", - " (cre_b(2)): 0-0.25j,\n", - " (cre_a(3), des_a(0)): -0.5,\n", - " (cre_b(1), des_b(5), cre_a(4), des_a(3), des_b(3)): -0.25-0.25j,\n", - " (cre_a(0), des_a(3), cre_b(2)): 0+0.5j,\n", " (cre_a(0), des_a(3), des_a(3), des_b(3)): -0.125,\n", " (cre_b(1), des_b(5), cre_a(4)): 2+2j,\n", " (cre_b(1), des_b(5), cre_a(4), cre_b(2)): -1+1j,\n", + " (cre_a(3), des_a(0), des_a(3), des_b(3)): 0.0625,\n", + " (cre_b(1), des_b(5), cre_a(4), des_a(3), des_b(3)): -0.25-0.25j,\n", " (cre_a(3), des_a(0), cre_b(2)): 0-0.25j,\n", + " (cre_a(0), des_a(3), cre_b(2)): 0+0.5j,\n", " (cre_a(0), des_a(3)): 1,\n", - " (des_a(3), des_b(3)): 0.0625,\n", - " (cre_a(3), des_a(0), des_a(3), des_b(3)): 0.0625\n", + " (cre_a(3), des_a(0)): -0.5,\n", + " (cre_b(2)): 0-0.25j,\n", + " (des_a(3), des_b(3)): 0.0625\n", "})" ] }, @@ -170,10 +170,10 @@ "execution_count": 4, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:30.448493Z", - "iopub.status.busy": "2024-04-24T06:38:30.448279Z", - "iopub.status.idle": "2024-04-24T06:38:30.452264Z", - "shell.execute_reply": "2024-04-24T06:38:30.451710Z" + "iopub.execute_input": "2024-04-24T07:11:53.122675Z", + "iopub.status.busy": "2024-04-24T07:11:53.122330Z", + "iopub.status.idle": "2024-04-24T07:11:53.126381Z", + "shell.execute_reply": "2024-04-24T07:11:53.125823Z" } }, "outputs": [ @@ -181,17 +181,17 @@ "data": { "text/plain": [ "FermionOperator({\n", - " (cre_b(2)): -5,\n", - " (cre_a(3), des_a(0)): 0+3j,\n", - " (cre_b(1), des_b(5), cre_a(4), des_a(3), des_b(3)): -1+1j,\n", - " (cre_a(0), des_a(3), cre_b(2)): 2,\n", " (cre_a(0), des_a(3), des_a(3), des_b(3)): 0+0.5j,\n", " (cre_b(1), des_b(5), cre_a(4)): 12-12j,\n", " (cre_b(1), des_b(5), cre_a(4), cre_b(2)): 4+4j,\n", + " (cre_a(3), des_a(0), des_a(3), des_b(3)): 0-0.25j,\n", + " (cre_b(1), des_b(5), cre_a(4), des_a(3), des_b(3)): -1+1j,\n", " (cre_a(3), des_a(0), cre_b(2)): -1,\n", + " (cre_a(0), des_a(3), cre_b(2)): 2,\n", " (cre_a(0), des_a(3)): 0-6j,\n", - " (des_a(3), des_b(3)): 0-1.25j,\n", - " (cre_a(3), des_a(0), des_a(3), des_b(3)): 0-0.25j\n", + " (cre_a(3), des_a(0)): 0+3j,\n", + " (cre_b(2)): -5,\n", + " (des_a(3), des_b(3)): 0-1.25j\n", "})" ] }, @@ -220,10 +220,10 @@ "execution_count": 5, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:30.454788Z", - "iopub.status.busy": "2024-04-24T06:38:30.454337Z", - "iopub.status.idle": "2024-04-24T06:38:30.458486Z", - "shell.execute_reply": "2024-04-24T06:38:30.457887Z" + "iopub.execute_input": "2024-04-24T07:11:53.128817Z", + "iopub.status.busy": "2024-04-24T07:11:53.128324Z", + "iopub.status.idle": "2024-04-24T07:11:53.132378Z", + "shell.execute_reply": "2024-04-24T07:11:53.131722Z" } }, "outputs": [ @@ -231,16 +231,16 @@ "data": { "text/plain": [ "FermionOperator({\n", - " (cre_b(2), cre_a(0), des_a(3)): 2,\n", " (cre_a(3), des_a(0)): 0+3j,\n", - " (cre_b(1), cre_a(4), des_b(5), des_b(3), des_a(3)): -1+1j,\n", - " (cre_a(3), des_b(3), des_a(3), des_a(0)): 0+0.25j,\n", - " (cre_b(2), cre_a(3), des_a(0)): -1,\n", - " (cre_a(0), des_a(3)): 0-6j,\n", - " (cre_b(1), cre_a(4), des_b(5)): -12+12j,\n", " (cre_b(2), cre_b(1), cre_a(4), des_b(5)): 4+4j,\n", " (des_b(3), des_a(3)): 0+1.25j,\n", - " (cre_b(2)): -5\n", + " (cre_a(3), des_b(3), des_a(3), des_a(0)): 0+0.25j,\n", + " (cre_b(1), cre_a(4), des_b(5), des_b(3), des_a(3)): -1+1j,\n", + " (cre_b(2), cre_a(0), des_a(3)): 2,\n", + " (cre_a(0), des_a(3)): 0-6j,\n", + " (cre_b(2), cre_a(3), des_a(0)): -1,\n", + " (cre_b(2)): -5,\n", + " (cre_b(1), cre_a(4), des_b(5)): -12+12j\n", "})" ] }, @@ -265,10 +265,10 @@ "execution_count": 6, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:30.461496Z", - "iopub.status.busy": "2024-04-24T06:38:30.461031Z", - "iopub.status.idle": "2024-04-24T06:38:30.464281Z", - "shell.execute_reply": "2024-04-24T06:38:30.463784Z" + "iopub.execute_input": "2024-04-24T07:11:53.134591Z", + "iopub.status.busy": "2024-04-24T07:11:53.134272Z", + "iopub.status.idle": "2024-04-24T07:11:53.137692Z", + "shell.execute_reply": "2024-04-24T07:11:53.137091Z" } }, "outputs": [ @@ -298,10 +298,10 @@ "execution_count": 7, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:30.466929Z", - "iopub.status.busy": "2024-04-24T06:38:30.466460Z", - "iopub.status.idle": "2024-04-24T06:38:30.470827Z", - "shell.execute_reply": "2024-04-24T06:38:30.470244Z" + "iopub.execute_input": "2024-04-24T07:11:53.139879Z", + "iopub.status.busy": "2024-04-24T07:11:53.139547Z", + "iopub.status.idle": "2024-04-24T07:11:53.143896Z", + "shell.execute_reply": "2024-04-24T07:11:53.143313Z" } }, "outputs": [ @@ -341,21 +341,21 @@ "execution_count": 8, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:30.473350Z", - "iopub.status.busy": "2024-04-24T06:38:30.472957Z", - "iopub.status.idle": "2024-04-24T06:38:30.479359Z", - "shell.execute_reply": "2024-04-24T06:38:30.478901Z" + "iopub.execute_input": "2024-04-24T07:11:53.146242Z", + "iopub.status.busy": "2024-04-24T07:11:53.145858Z", + "iopub.status.idle": "2024-04-24T07:11:53.151856Z", + "shell.execute_reply": "2024-04-24T07:11:53.151265Z" } }, "outputs": [ { "data": { "text/plain": [ - "array([0. +0.j , 0. +0.j ,\n", - " 0. +0.j , 0. +0.j ,\n", - " 0.24268319+0.17800037j, 0. +0.j ,\n", - " 0. +0.j , 0. +0.j ,\n", - " 0. +0.j ])" + "array([ 0. +0.j , 0. +0.j ,\n", + " 0. +0.j , 0. +0.j ,\n", + " -0.13117779+0.11588432j, 0. +0.j ,\n", + " 0. +0.j , 0. +0.j ,\n", + " 0. +0.j ])" ] }, "execution_count": 8, @@ -380,10 +380,10 @@ "execution_count": 9, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:38:30.481999Z", - "iopub.status.busy": "2024-04-24T06:38:30.481534Z", - "iopub.status.idle": "2024-04-24T06:38:30.493345Z", - "shell.execute_reply": "2024-04-24T06:38:30.492866Z" + "iopub.execute_input": "2024-04-24T07:11:53.154037Z", + "iopub.status.busy": "2024-04-24T07:11:53.153715Z", + "iopub.status.idle": "2024-04-24T07:11:53.165587Z", + "shell.execute_reply": "2024-04-24T07:11:53.165124Z" } }, "outputs": [ diff --git a/how-to-guides/lucj.html b/how-to-guides/lucj.html index f1adbd6cd..c38a1be4b 100644 --- a/how-to-guides/lucj.html +++ b/how-to-guides/lucj.html @@ -269,7 +269,7 @@

How to simulate the local unitary cluster Jastrow (LUCJ) ansatz 
-converged SCF energy = -77.4456267643961
+converged SCF energy = -77.4456267643962
@@ -277,7 +277,7 @@

How to simulate the local unitary cluster Jastrow (LUCJ) ansatz 
-CASCI E = -77.6290254326717  E(CI) = -3.57322412553862  S^2 = 0.0000000
+CASCI E = -77.6290254326717  E(CI) = -3.57322412553863  S^2 = 0.0000000

@@ -331,7 +331,7 @@

The unitary cluster Jastrow (UCJ) ansatz 
-E(CCSD) = -77.49387212754465  E_corr = -0.04824536314851344
+E(CCSD) = -77.49387212754468  E_corr = -0.04824536314851544
@@ -339,7 +339,7 @@

The unitary cluster Jastrow (UCJ) ansatz 
-Energy at initialization: -77.46975600021696
+Energy at initialization: -77.46975600021707

To facilitate variational optimization of the ansatz, UCJOperator implements methods for conversion to and from a vector of real-valued parameters. The precise relation between a parameter vector and the matrices of the UCJ operator is somewhat complicated. In short, the parameter vector stores the entries of the UCJ matrices in a non-redundant way (for the orbital rotations, the parameter vector actually stores the entries of their generators.)

@@ -378,10 +378,10 @@

The unitary cluster Jastrow (UCJ) ansatz\n" @@ -243,10 +243,10 @@ "execution_count": 4, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:39:20.370260Z", - "iopub.status.busy": "2024-04-24T06:39:20.369082Z", - "iopub.status.idle": "2024-04-24T06:39:53.494495Z", - "shell.execute_reply": "2024-04-24T06:39:53.493852Z" + "iopub.execute_input": "2024-04-24T07:12:43.757348Z", + "iopub.status.busy": "2024-04-24T07:12:43.757076Z", + "iopub.status.idle": "2024-04-24T07:13:16.006596Z", + "shell.execute_reply": "2024-04-24T07:13:16.005572Z" } }, "outputs": [ @@ -258,10 +258,10 @@ " message: STOP: TOTAL NO. of ITERATIONS REACHED LIMIT\n", " success: False\n", " status: 1\n", - " fun: -77.45743007109475\n", - " x: [ 1.253e+00 -6.338e-01 ... 4.416e-04 1.435e-02]\n", + " fun: -77.45741509166481\n", + " x: [ 1.261e+00 4.805e-01 ... 8.055e-03 8.157e-03]\n", " nit: 5\n", - " jac: [ 7.390e-05 -2.274e-04 ... -1.870e-03 -2.905e-03]\n", + " jac: [ 1.876e-04 6.963e-05 ... -1.320e-03 -3.342e-03]\n", " nfev: 423\n", " njev: 9\n", " hess_inv: <46x46 LbfgsInvHessProduct with dtype=float64>\n" @@ -311,10 +311,10 @@ "execution_count": 5, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:39:53.497777Z", - "iopub.status.busy": "2024-04-24T06:39:53.497481Z", - "iopub.status.idle": "2024-04-24T06:40:12.988724Z", - "shell.execute_reply": "2024-04-24T06:40:12.988113Z" + "iopub.execute_input": "2024-04-24T07:13:16.009766Z", + "iopub.status.busy": "2024-04-24T07:13:16.009475Z", + "iopub.status.idle": "2024-04-24T07:13:31.585400Z", + "shell.execute_reply": "2024-04-24T07:13:31.584778Z" } }, "outputs": [ @@ -325,34 +325,34 @@ "Number of parameters: 46\n", " message: Stop: Total number of iterations reached limit.\n", " success: False\n", - " fun: -77.46201475466209\n", - " x: [ 1.273e+00 -5.672e-01 ... 8.623e-02 -2.793e-01]\n", + " fun: -77.45818447456564\n", + " x: [ 1.287e+00 4.532e-01 ... 3.552e-02 -8.566e-03]\n", " nit: 5\n", - " jac: [-4.342e-03 -3.215e-04 ... 3.007e-03 6.633e-03]\n", - " nfev: 691\n", + " jac: [-5.629e-04 3.020e-04 ... 7.026e-04 -1.219e-03]\n", + " nfev: 634\n", " njev: 5\n", - " nlinop: 461\n", + " nlinop: 404\n", "\n", "Iteration 1\n", - " Energy: -77.45683742243246\n", - " Norm of gradient: 0.015397154171982655\n", - " Regularization hyperparameter: 0.024452735271862664\n", - " Variation hyperparameter: 0.47040722989098466\n", + " Energy: -77.45029302626645\n", + " Norm of gradient: 0.02020004370995382\n", + " Regularization hyperparameter: 0.8826379901342593\n", + " Variation hyperparameter: 0.6648183897123532\n", "Iteration 2\n", - " Energy: -77.45777592466433\n", - " Norm of gradient: 0.01079051324391571\n", - " Regularization hyperparameter: 0.00041680639591294135\n", - " Variation hyperparameter: 0.47482696753224485\n", + " Energy: -77.45045534010309\n", + " Norm of gradient: 0.018881646064850023\n", + " Regularization hyperparameter: 0.8823610881773326\n", + " Variation hyperparameter: 0.6648238632610717\n", "Iteration 3\n", - " Energy: -77.45781000647975\n", - " Norm of gradient: 0.010238695149053208\n", - " Regularization hyperparameter: 1.0327012686007995\n", - " Variation hyperparameter: 0.42969865263145535\n", + " Energy: -77.45711181898201\n", + " Norm of gradient: 0.01200729687112761\n", + " Regularization hyperparameter: 0.025598521789727345\n", + " Variation hyperparameter: 0.6940657521578105\n", "Iteration 4\n", - " Energy: -77.46177620525042\n", - " Norm of gradient: 0.024720368126669716\n", - " Regularization hyperparameter: 0.003152814192004464\n", - " Variation hyperparameter: 0.8946537857915279\n" + " Energy: -77.45816307257847\n", + " Norm of gradient: 0.007591266502855039\n", + " Regularization hyperparameter: 0.00041396927536031585\n", + " Variation hyperparameter: 0.6982897824464199\n" ] } ], diff --git a/objects.inv b/objects.inv index 424e197c13264baf3a4ea7ff524033ba05a111de..f421305ce04b79261fafdc3d219c7158cca34294 100644 GIT binary patch delta 5384 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zGSZh${<-YS8^h8s<#3sJjq3`j>wm^HD0_bdK%_ho@`?N?Pvk9WAiaO&X*?23|M706 zEwTe5V}HNO2eQjy)4%Wr2=$hpIIeZ%qgEB zmZ|qj{ynuc3ZkEr7K2-CO0g5FsvHXvim9gbtMZ2eRI{%`$_v_zPAf2q%75?tXEOd2 z5F?=DAx!DxV=%o+6Ui+Vf&8>@Ze77(7^n6Oortt#N^^N|7Lg6@E~EvlE!%Yd(}SgI zy)RoS)R08KDw>zFPLECT$vk08!?Sh&>Ziu3tdmz`%CqOgO>8&GtN!`4%P?Ix$GCbf zpZfU7#7%(XbuaUEfQcL)On<{o%PM)CHaW^xe=NEnsokL~*(p8b#iw)E-1zvG4@Bne zl4*bYcKcL{;hI3~X&Ko4Hu84ni=*_S#!w|}e?<4DxzG>#4y zYkkTdE{gKnPO}Zs-Pm3-BKPr%VX7yS|%PT5V<5et*|Sa<)>HRHqAz4m35oHb+MlX`fOlm~4I3Y@j_*dp4Mpx+e@9b~?54}0CaEQh` AImplementing Trotter simulation of the double-factorized Hamiltonian 
-Fidelity of evolved state w.r.t. initial state: 0.9315062301396794
+Fidelity of evolved state w.r.t. initial state: 0.931506230139706

Now, let’s test our implementation.

@@ -531,7 +531,7 @@

Implementing Trotter simulation of the double-factorized Hamiltonian 
-Fidelity of Trotter-evolved state with exact state: 0.9928527668210639
+Fidelity of Trotter-evolved state with exact state: 0.9928527668210921

The fidelity of the final result can be improved by increasing the number of Trotter steps.

@@ -558,7 +558,7 @@

Implementing Trotter simulation of the double-factorized Hamiltonian 
-Fidelity of Trotter-evolved state with exact state: 0.9999320851284026
+Fidelity of Trotter-evolved state with exact state: 0.999932085128466

As mentioned above, ffsim already includes functionality for Trotter simulation of double-factorized Hamiltonians. The implementation in ffsim includes higher-order Trotter-Suzuki formulas. The first-order asymmetric formula that we just implemented corresponds to order=0 in ffsim’s implementation. order=1 corresponds to the first-order symmetric (commonly known as the second-order) formula, order=2 corresponds to the second-order symmetric (fourth-order) formula, and so on.

@@ -587,7 +587,7 @@

Implementing Trotter simulation of the double-factorized Hamiltonian 
-Fidelity of Trotter-evolved state with exact state: 0.9999320851284026
+Fidelity of Trotter-evolved state with exact state: 0.999932085128466

A higher order formula achieves a higher fidelity with fewer Trotter steps:

@@ -615,7 +615,7 @@

Implementing Trotter simulation of the double-factorized Hamiltonian 
-Fidelity of Trotter-evolved state with exact state: 0.9999913261305021
+Fidelity of Trotter-evolved state with exact state: 0.9999913261305375

diff --git a/tutorials/double-factorized.ipynb b/tutorials/double-factorized.ipynb index 3d7877558..468fa1869 100644 --- a/tutorials/double-factorized.ipynb +++ b/tutorials/double-factorized.ipynb @@ -43,10 +43,10 @@ "execution_count": 1, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:40:14.711120Z", - "iopub.status.busy": "2024-04-24T06:40:14.710911Z", - "iopub.status.idle": "2024-04-24T06:40:15.491103Z", - "shell.execute_reply": "2024-04-24T06:40:15.490495Z" + "iopub.execute_input": "2024-04-24T07:13:33.304169Z", + "iopub.status.busy": "2024-04-24T07:13:33.303980Z", + "iopub.status.idle": "2024-04-24T07:13:34.095562Z", + "shell.execute_reply": "2024-04-24T07:13:34.094949Z" } }, "outputs": [ @@ -109,10 +109,10 @@ "execution_count": 2, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:40:15.495265Z", - "iopub.status.busy": "2024-04-24T06:40:15.494140Z", - "iopub.status.idle": "2024-04-24T06:40:15.500225Z", - "shell.execute_reply": "2024-04-24T06:40:15.499720Z" + "iopub.execute_input": "2024-04-24T07:13:34.099267Z", + "iopub.status.busy": "2024-04-24T07:13:34.098155Z", + "iopub.status.idle": "2024-04-24T07:13:34.104600Z", + "shell.execute_reply": "2024-04-24T07:13:34.103979Z" } }, "outputs": [ @@ -215,10 +215,10 @@ "execution_count": 3, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:40:15.502694Z", - "iopub.status.busy": "2024-04-24T06:40:15.502334Z", - "iopub.status.idle": "2024-04-24T06:40:15.506441Z", - "shell.execute_reply": "2024-04-24T06:40:15.505961Z" + "iopub.execute_input": "2024-04-24T07:13:34.106916Z", + "iopub.status.busy": "2024-04-24T07:13:34.106566Z", + "iopub.status.idle": "2024-04-24T07:13:34.110796Z", + "shell.execute_reply": "2024-04-24T07:13:34.110309Z" } }, "outputs": [], @@ -273,10 +273,10 @@ "execution_count": 4, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:40:15.508756Z", - "iopub.status.busy": "2024-04-24T06:40:15.508392Z", - "iopub.status.idle": "2024-04-24T06:40:15.511821Z", - "shell.execute_reply": "2024-04-24T06:40:15.511322Z" + "iopub.execute_input": "2024-04-24T07:13:34.113143Z", + "iopub.status.busy": "2024-04-24T07:13:34.112766Z", + "iopub.status.idle": "2024-04-24T07:13:34.116180Z", + "shell.execute_reply": "2024-04-24T07:13:34.115721Z" } }, "outputs": [], @@ -313,10 +313,10 @@ "execution_count": 5, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:40:15.514296Z", - "iopub.status.busy": "2024-04-24T06:40:15.513863Z", - "iopub.status.idle": "2024-04-24T06:40:15.519131Z", - "shell.execute_reply": "2024-04-24T06:40:15.518570Z" + "iopub.execute_input": "2024-04-24T07:13:34.118250Z", + "iopub.status.busy": "2024-04-24T07:13:34.118060Z", + "iopub.status.idle": "2024-04-24T07:13:34.124011Z", + "shell.execute_reply": "2024-04-24T07:13:34.123434Z" } }, "outputs": [ @@ -352,10 +352,10 @@ "execution_count": 6, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:40:15.522284Z", - "iopub.status.busy": "2024-04-24T06:40:15.521599Z", - "iopub.status.idle": "2024-04-24T06:40:16.342247Z", - "shell.execute_reply": "2024-04-24T06:40:16.341647Z" + "iopub.execute_input": "2024-04-24T07:13:34.127621Z", + "iopub.status.busy": "2024-04-24T07:13:34.126498Z", + "iopub.status.idle": "2024-04-24T07:13:34.959622Z", + "shell.execute_reply": "2024-04-24T07:13:34.959044Z" } }, "outputs": [ @@ -363,7 +363,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Fidelity of evolved state w.r.t. initial state: 0.9315062301396794\n" + "Fidelity of evolved state w.r.t. initial state: 0.931506230139706\n" ] } ], @@ -394,10 +394,10 @@ "execution_count": 7, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:40:16.346624Z", - "iopub.status.busy": "2024-04-24T06:40:16.345427Z", - "iopub.status.idle": "2024-04-24T06:40:16.374825Z", - "shell.execute_reply": "2024-04-24T06:40:16.374271Z" + "iopub.execute_input": "2024-04-24T07:13:34.963777Z", + "iopub.status.busy": "2024-04-24T07:13:34.962800Z", + "iopub.status.idle": "2024-04-24T07:13:34.994265Z", + "shell.execute_reply": "2024-04-24T07:13:34.993645Z" } }, "outputs": [ @@ -405,7 +405,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Fidelity of Trotter-evolved state with exact state: 0.9928527668210639\n" + "Fidelity of Trotter-evolved state with exact state: 0.9928527668210921\n" ] } ], @@ -435,10 +435,10 @@ "execution_count": 8, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:40:16.378941Z", - "iopub.status.busy": "2024-04-24T06:40:16.377960Z", - "iopub.status.idle": "2024-04-24T06:40:16.489574Z", - "shell.execute_reply": "2024-04-24T06:40:16.488995Z" + "iopub.execute_input": "2024-04-24T07:13:34.997907Z", + "iopub.status.busy": "2024-04-24T07:13:34.996932Z", + "iopub.status.idle": "2024-04-24T07:13:35.144603Z", + "shell.execute_reply": "2024-04-24T07:13:35.144044Z" } }, "outputs": [ @@ -446,7 +446,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Fidelity of Trotter-evolved state with exact state: 0.9999320851284026\n" + "Fidelity of Trotter-evolved state with exact state: 0.999932085128466\n" ] } ], @@ -478,10 +478,10 @@ "execution_count": 9, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:40:16.494464Z", - "iopub.status.busy": "2024-04-24T06:40:16.492729Z", - "iopub.status.idle": "2024-04-24T06:40:16.583248Z", - "shell.execute_reply": "2024-04-24T06:40:16.582651Z" + "iopub.execute_input": "2024-04-24T07:13:35.148420Z", + "iopub.status.busy": "2024-04-24T07:13:35.147427Z", + "iopub.status.idle": "2024-04-24T07:13:35.294623Z", + "shell.execute_reply": "2024-04-24T07:13:35.294088Z" } }, "outputs": [ @@ -489,7 +489,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Fidelity of Trotter-evolved state with exact state: 0.9999320851284026\n" + "Fidelity of Trotter-evolved state with exact state: 0.999932085128466\n" ] } ], @@ -520,10 +520,10 @@ "execution_count": 10, "metadata": { "execution": { - "iopub.execute_input": "2024-04-24T06:40:16.587653Z", - "iopub.status.busy": "2024-04-24T06:40:16.586620Z", - "iopub.status.idle": "2024-04-24T06:40:16.652763Z", - "shell.execute_reply": "2024-04-24T06:40:16.652137Z" + "iopub.execute_input": "2024-04-24T07:13:35.298402Z", + "iopub.status.busy": "2024-04-24T07:13:35.297445Z", + "iopub.status.idle": "2024-04-24T07:13:35.384316Z", + "shell.execute_reply": "2024-04-24T07:13:35.383754Z" } }, "outputs": [ @@ -531,7 +531,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Fidelity of Trotter-evolved state with exact state: 0.9999913261305021\n" + "Fidelity of Trotter-evolved state with exact state: 0.9999913261305375\n" ] } ],