diff --git a/.buildinfo b/.buildinfo index bc2e2c87c..100ebe1e2 100644 --- a/.buildinfo +++ b/.buildinfo @@ -1,4 +1,4 @@ # Sphinx build info version 1 # This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done. -config: 48b5fef6269d8e3c7b52f0c1d13b2ff8 +config: 7c4d33a9a9509ef2f8cc758bfeb1c27b tags: 645f666f9bcd5a90fca523b33c5a78b7 diff --git a/.doctrees/environment.pickle b/.doctrees/environment.pickle index 2881bbe13..1f3658693 100644 Binary files a/.doctrees/environment.pickle and b/.doctrees/environment.pickle differ diff --git a/.doctrees/nbsphinx/tutorials/01-introduction.ipynb b/.doctrees/nbsphinx/tutorials/01-introduction.ipynb index 5b15bd022..ef181695a 100644 --- a/.doctrees/nbsphinx/tutorials/01-introduction.ipynb +++ b/.doctrees/nbsphinx/tutorials/01-introduction.ipynb @@ -16,10 +16,10 @@ "execution_count": 1, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:33.517901Z", - "iopub.status.busy": "2024-04-14T06:23:33.517701Z", - "iopub.status.idle": "2024-04-14T06:23:34.163809Z", - "shell.execute_reply": "2024-04-14T06:23:34.163266Z" + "iopub.execute_input": "2024-04-14T06:27:19.630729Z", + "iopub.status.busy": "2024-04-14T06:27:19.630222Z", + "iopub.status.idle": "2024-04-14T06:27:20.381697Z", + "shell.execute_reply": "2024-04-14T06:27:20.381044Z" } }, "outputs": [], @@ -71,10 +71,10 @@ "execution_count": 2, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:34.166803Z", - "iopub.status.busy": "2024-04-14T06:23:34.166266Z", - "iopub.status.idle": "2024-04-14T06:23:34.171295Z", - "shell.execute_reply": "2024-04-14T06:23:34.170681Z" + "iopub.execute_input": "2024-04-14T06:27:20.384791Z", + "iopub.status.busy": "2024-04-14T06:27:20.384280Z", + "iopub.status.idle": "2024-04-14T06:27:20.388761Z", + "shell.execute_reply": "2024-04-14T06:27:20.388185Z" } }, "outputs": [ @@ -113,10 +113,10 @@ "execution_count": 3, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:34.208944Z", - "iopub.status.busy": "2024-04-14T06:23:34.208492Z", - "iopub.status.idle": "2024-04-14T06:23:34.211405Z", - "shell.execute_reply": "2024-04-14T06:23:34.210948Z" + "iopub.execute_input": "2024-04-14T06:27:20.427061Z", + "iopub.status.busy": "2024-04-14T06:27:20.426626Z", + "iopub.status.idle": "2024-04-14T06:27:20.429502Z", + "shell.execute_reply": "2024-04-14T06:27:20.429016Z" } }, "outputs": [], @@ -136,10 +136,10 @@ "execution_count": 4, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:34.213651Z", - "iopub.status.busy": "2024-04-14T06:23:34.213289Z", - "iopub.status.idle": "2024-04-14T06:23:34.219908Z", - "shell.execute_reply": "2024-04-14T06:23:34.219331Z" + "iopub.execute_input": "2024-04-14T06:27:20.431772Z", + "iopub.status.busy": "2024-04-14T06:27:20.431418Z", + "iopub.status.idle": "2024-04-14T06:27:20.438271Z", + "shell.execute_reply": "2024-04-14T06:27:20.437686Z" } }, "outputs": [ diff --git a/.doctrees/nbsphinx/tutorials/02-orbital-rotation.ipynb b/.doctrees/nbsphinx/tutorials/02-orbital-rotation.ipynb index fcd1c3200..9fb1e9d31 100644 --- a/.doctrees/nbsphinx/tutorials/02-orbital-rotation.ipynb +++ b/.doctrees/nbsphinx/tutorials/02-orbital-rotation.ipynb @@ -43,10 +43,10 @@ "execution_count": 1, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:35.946325Z", - "iopub.status.busy": "2024-04-14T06:23:35.946123Z", - "iopub.status.idle": "2024-04-14T06:23:36.566488Z", - "shell.execute_reply": "2024-04-14T06:23:36.565844Z" + "iopub.execute_input": "2024-04-14T06:27:22.043347Z", + "iopub.status.busy": "2024-04-14T06:27:22.042873Z", + "iopub.status.idle": "2024-04-14T06:27:22.654360Z", + "shell.execute_reply": "2024-04-14T06:27:22.653814Z" } }, "outputs": [], @@ -107,10 +107,10 @@ "execution_count": 2, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:36.569582Z", - "iopub.status.busy": "2024-04-14T06:23:36.569027Z", - "iopub.status.idle": "2024-04-14T06:23:36.599753Z", - "shell.execute_reply": "2024-04-14T06:23:36.599192Z" + "iopub.execute_input": "2024-04-14T06:27:22.657228Z", + "iopub.status.busy": "2024-04-14T06:27:22.656760Z", + "iopub.status.idle": "2024-04-14T06:27:22.687683Z", + "shell.execute_reply": "2024-04-14T06:27:22.687120Z" } }, "outputs": [], @@ -155,10 +155,10 @@ "execution_count": 3, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:36.603387Z", - "iopub.status.busy": "2024-04-14T06:23:36.602283Z", - "iopub.status.idle": "2024-04-14T06:23:36.621799Z", - "shell.execute_reply": "2024-04-14T06:23:36.621262Z" + "iopub.execute_input": "2024-04-14T06:27:22.690791Z", + "iopub.status.busy": "2024-04-14T06:27:22.690298Z", + "iopub.status.idle": "2024-04-14T06:27:22.708684Z", + "shell.execute_reply": "2024-04-14T06:27:22.708111Z" } }, "outputs": [], @@ -198,10 +198,10 @@ "execution_count": 4, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:36.625155Z", - "iopub.status.busy": "2024-04-14T06:23:36.624895Z", - "iopub.status.idle": "2024-04-14T06:23:36.637057Z", - "shell.execute_reply": "2024-04-14T06:23:36.636537Z" + "iopub.execute_input": "2024-04-14T06:27:22.711505Z", + "iopub.status.busy": "2024-04-14T06:27:22.711264Z", + "iopub.status.idle": "2024-04-14T06:27:22.717189Z", + "shell.execute_reply": "2024-04-14T06:27:22.716676Z" } }, "outputs": [], diff --git a/.doctrees/nbsphinx/tutorials/03-double-factorized.ipynb b/.doctrees/nbsphinx/tutorials/03-double-factorized.ipynb index 701527a80..3a12b2ab3 100644 --- a/.doctrees/nbsphinx/tutorials/03-double-factorized.ipynb +++ b/.doctrees/nbsphinx/tutorials/03-double-factorized.ipynb @@ -43,10 +43,10 @@ "execution_count": 1, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:38.368788Z", - "iopub.status.busy": "2024-04-14T06:23:38.368595Z", - "iopub.status.idle": "2024-04-14T06:23:39.181651Z", - "shell.execute_reply": "2024-04-14T06:23:39.181037Z" + "iopub.execute_input": "2024-04-14T06:27:24.434926Z", + "iopub.status.busy": "2024-04-14T06:27:24.434728Z", + "iopub.status.idle": "2024-04-14T06:27:25.244420Z", + "shell.execute_reply": "2024-04-14T06:27:25.243814Z" } }, "outputs": [ @@ -54,7 +54,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "converged SCF energy = -77.4456267643962\n" + "converged SCF energy = -77.4456267643963\n" ] } ], @@ -109,10 +109,10 @@ "execution_count": 2, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:39.184928Z", - "iopub.status.busy": "2024-04-14T06:23:39.184369Z", - "iopub.status.idle": "2024-04-14T06:23:39.189762Z", - "shell.execute_reply": "2024-04-14T06:23:39.189154Z" + "iopub.execute_input": "2024-04-14T06:27:25.248359Z", + "iopub.status.busy": "2024-04-14T06:27:25.247272Z", + "iopub.status.idle": "2024-04-14T06:27:25.253468Z", + "shell.execute_reply": "2024-04-14T06:27:25.252994Z" } }, "outputs": [ @@ -215,10 +215,10 @@ "execution_count": 3, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:39.192145Z", - "iopub.status.busy": "2024-04-14T06:23:39.191753Z", - "iopub.status.idle": "2024-04-14T06:23:39.196247Z", - "shell.execute_reply": "2024-04-14T06:23:39.195688Z" + "iopub.execute_input": "2024-04-14T06:27:25.255753Z", + "iopub.status.busy": "2024-04-14T06:27:25.255475Z", + "iopub.status.idle": "2024-04-14T06:27:25.259947Z", + "shell.execute_reply": "2024-04-14T06:27:25.259441Z" } }, "outputs": [], @@ -273,10 +273,10 @@ "execution_count": 4, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:39.198602Z", - "iopub.status.busy": "2024-04-14T06:23:39.198251Z", - "iopub.status.idle": "2024-04-14T06:23:39.201631Z", - "shell.execute_reply": "2024-04-14T06:23:39.201163Z" + "iopub.execute_input": "2024-04-14T06:27:25.262285Z", + "iopub.status.busy": "2024-04-14T06:27:25.261914Z", + "iopub.status.idle": "2024-04-14T06:27:25.265628Z", + "shell.execute_reply": "2024-04-14T06:27:25.265013Z" } }, "outputs": [], @@ -313,10 +313,10 @@ "execution_count": 5, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:39.203878Z", - "iopub.status.busy": "2024-04-14T06:23:39.203527Z", - "iopub.status.idle": "2024-04-14T06:23:39.209342Z", - "shell.execute_reply": "2024-04-14T06:23:39.208787Z" + "iopub.execute_input": "2024-04-14T06:27:25.268378Z", + "iopub.status.busy": "2024-04-14T06:27:25.267938Z", + "iopub.status.idle": "2024-04-14T06:27:25.276477Z", + "shell.execute_reply": "2024-04-14T06:27:25.275917Z" } }, "outputs": [ @@ -324,7 +324,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Hartree Fock energy: -77.44562676439627\n" + "Hartree Fock energy: -77.44562676439624\n" ] } ], @@ -352,10 +352,10 @@ "execution_count": 6, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:39.212390Z", - "iopub.status.busy": "2024-04-14T06:23:39.211987Z", - "iopub.status.idle": "2024-04-14T06:23:40.068753Z", - "shell.execute_reply": "2024-04-14T06:23:40.068172Z" + "iopub.execute_input": "2024-04-14T06:27:25.279340Z", + "iopub.status.busy": "2024-04-14T06:27:25.278826Z", + "iopub.status.idle": "2024-04-14T06:27:26.219158Z", + "shell.execute_reply": "2024-04-14T06:27:26.218538Z" } }, "outputs": [ @@ -363,7 +363,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Fidelity of evolved state w.r.t. initial state: 0.9315062301398414\n" + "Fidelity of evolved state w.r.t. initial state: 0.9315062301402518\n" ] } ], @@ -394,10 +394,10 @@ "execution_count": 7, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:40.072575Z", - "iopub.status.busy": "2024-04-14T06:23:40.071590Z", - "iopub.status.idle": "2024-04-14T06:23:40.105131Z", - "shell.execute_reply": "2024-04-14T06:23:40.104373Z" + "iopub.execute_input": "2024-04-14T06:27:26.223390Z", + "iopub.status.busy": "2024-04-14T06:27:26.222418Z", + "iopub.status.idle": "2024-04-14T06:27:26.247798Z", + "shell.execute_reply": "2024-04-14T06:27:26.247269Z" } }, "outputs": [ @@ -405,7 +405,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Fidelity of Trotter-evolved state with exact state: 0.9928527668211954\n" + "Fidelity of Trotter-evolved state with exact state: 0.9928527668216444\n" ] } ], @@ -435,10 +435,10 @@ "execution_count": 8, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:40.109263Z", - "iopub.status.busy": "2024-04-14T06:23:40.108279Z", - "iopub.status.idle": "2024-04-14T06:23:40.256687Z", - "shell.execute_reply": "2024-04-14T06:23:40.256063Z" + "iopub.execute_input": "2024-04-14T06:27:26.251615Z", + "iopub.status.busy": "2024-04-14T06:27:26.250679Z", + "iopub.status.idle": "2024-04-14T06:27:26.410011Z", + "shell.execute_reply": "2024-04-14T06:27:26.409423Z" } }, "outputs": [ @@ -446,7 +446,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Fidelity of Trotter-evolved state with exact state: 0.999932085128549\n" + "Fidelity of Trotter-evolved state with exact state: 0.9999320851289344\n" ] } ], @@ -478,10 +478,10 @@ "execution_count": 9, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:40.260898Z", - "iopub.status.busy": "2024-04-14T06:23:40.259912Z", - "iopub.status.idle": "2024-04-14T06:23:40.407165Z", - "shell.execute_reply": "2024-04-14T06:23:40.406555Z" + "iopub.execute_input": "2024-04-14T06:27:26.414147Z", + "iopub.status.busy": "2024-04-14T06:27:26.413128Z", + "iopub.status.idle": "2024-04-14T06:27:26.558989Z", + "shell.execute_reply": "2024-04-14T06:27:26.558428Z" } }, "outputs": [ @@ -489,7 +489,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Fidelity of Trotter-evolved state with exact state: 0.999932085128549\n" + "Fidelity of Trotter-evolved state with exact state: 0.9999320851289344\n" ] } ], @@ -520,10 +520,10 @@ "execution_count": 10, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:40.411437Z", - "iopub.status.busy": "2024-04-14T06:23:40.410447Z", - "iopub.status.idle": "2024-04-14T06:23:40.497475Z", - "shell.execute_reply": "2024-04-14T06:23:40.496842Z" + "iopub.execute_input": "2024-04-14T06:27:26.563103Z", + "iopub.status.busy": "2024-04-14T06:27:26.561971Z", + "iopub.status.idle": "2024-04-14T06:27:26.649056Z", + "shell.execute_reply": "2024-04-14T06:27:26.648488Z" } }, "outputs": [ @@ -531,7 +531,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Fidelity of Trotter-evolved state with exact state: 0.9999913261306368\n" + "Fidelity of Trotter-evolved state with exact state: 0.9999913261310335\n" ] } ], diff --git a/.doctrees/nbsphinx/tutorials/04-lucj.ipynb b/.doctrees/nbsphinx/tutorials/04-lucj.ipynb index ac35c10bd..fcac0f4cd 100644 --- a/.doctrees/nbsphinx/tutorials/04-lucj.ipynb +++ b/.doctrees/nbsphinx/tutorials/04-lucj.ipynb @@ -14,10 +14,10 @@ "execution_count": 1, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:42.561669Z", - "iopub.status.busy": "2024-04-14T06:23:42.561070Z", - "iopub.status.idle": "2024-04-14T06:23:43.518718Z", - "shell.execute_reply": "2024-04-14T06:23:43.518059Z" + "iopub.execute_input": "2024-04-14T06:27:28.647511Z", + "iopub.status.busy": "2024-04-14T06:27:28.647313Z", + "iopub.status.idle": "2024-04-14T06:27:29.572991Z", + "shell.execute_reply": "2024-04-14T06:27:29.572422Z" } }, "outputs": [ @@ -32,7 +32,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "CASCI E = -77.6290254326718 E(CI) = -3.57322412553863 S^2 = 0.0000000\n" + "CASCI E = -77.6290254326717 E(CI) = -3.57322412553863 S^2 = 0.0000000\n" ] } ], @@ -107,10 +107,10 @@ "execution_count": 2, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:43.522329Z", - "iopub.status.busy": "2024-04-14T06:23:43.521663Z", - "iopub.status.idle": "2024-04-14T06:23:43.723028Z", - "shell.execute_reply": "2024-04-14T06:23:43.722407Z" + "iopub.execute_input": "2024-04-14T06:27:29.577316Z", + "iopub.status.busy": "2024-04-14T06:27:29.576043Z", + "iopub.status.idle": "2024-04-14T06:27:29.815475Z", + "shell.execute_reply": "2024-04-14T06:27:29.814836Z" } }, "outputs": [ @@ -118,14 +118,14 @@ "name": "stdout", "output_type": "stream", "text": [ - "E(CCSD) = -77.49387212754473 E_corr = -0.04824536314851524\n" + "E(CCSD) = -77.49387212754468 E_corr = -0.04824536314851467\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Energy at initialization: -77.46975600021636\n" + "Energy at initialization: -77.4697560002172\n" ] } ], @@ -170,10 +170,10 @@ "execution_count": 3, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:23:43.726174Z", - "iopub.status.busy": "2024-04-14T06:23:43.725798Z", - "iopub.status.idle": "2024-04-14T06:24:30.247340Z", - "shell.execute_reply": "2024-04-14T06:24:30.246755Z" + "iopub.execute_input": "2024-04-14T06:27:29.818363Z", + "iopub.status.busy": "2024-04-14T06:27:29.817790Z", + "iopub.status.idle": "2024-04-14T06:28:14.708130Z", + "shell.execute_reply": "2024-04-14T06:28:14.707582Z" } }, "outputs": [ @@ -185,10 +185,10 @@ " message: STOP: TOTAL NO. of ITERATIONS REACHED LIMIT\n", " success: False\n", " status: 1\n", - " fun: -77.50959946166863\n", - " x: [ 1.252e+00 6.199e-01 ... 8.726e-02 1.371e-01]\n", + " fun: -77.50964086775728\n", + " x: [ 1.376e+00 -5.946e-01 ... 8.755e-02 1.397e-01]\n", " nit: 5\n", - " jac: [ 1.521e-04 -5.684e-04 ... -3.674e-03 -4.748e-03]\n", + " jac: [-8.527e-05 8.370e-04 ... -4.155e-03 -3.068e-03]\n", " nfev: 584\n", " njev: 8\n", " hess_inv: <72x72 LbfgsInvHessProduct with dtype=float64>\n" @@ -243,10 +243,10 @@ "execution_count": 4, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:24:30.251635Z", - "iopub.status.busy": "2024-04-14T06:24:30.250629Z", - "iopub.status.idle": "2024-04-14T06:25:03.357300Z", - "shell.execute_reply": "2024-04-14T06:25:03.356700Z" + "iopub.execute_input": "2024-04-14T06:28:14.711540Z", + "iopub.status.busy": "2024-04-14T06:28:14.710803Z", + "iopub.status.idle": "2024-04-14T06:28:46.037755Z", + "shell.execute_reply": "2024-04-14T06:28:46.037213Z" } }, "outputs": [ @@ -258,10 +258,10 @@ " message: STOP: TOTAL NO. of ITERATIONS REACHED LIMIT\n", " success: False\n", " status: 1\n", - " fun: -77.45740904439641\n", - " x: [ 1.249e+00 6.259e-01 ... -7.862e-04 1.414e-02]\n", + " fun: -77.45740457955615\n", + " x: [ 1.378e+00 -5.850e-01 ... 1.693e-03 1.235e-02]\n", " nit: 5\n", - " jac: [-4.263e-05 1.194e-04 ... -2.522e-03 -2.574e-03]\n", + " jac: [-3.084e-04 8.811e-04 ... -1.472e-03 -3.141e-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": "2024-04-14T06:25:03.361547Z", - "iopub.status.busy": "2024-04-14T06:25:03.360602Z", - "iopub.status.idle": "2024-04-14T06:25:24.281671Z", - "shell.execute_reply": "2024-04-14T06:25:24.278046Z" + "iopub.execute_input": "2024-04-14T06:28:46.040994Z", + "iopub.status.busy": "2024-04-14T06:28:46.040770Z", + "iopub.status.idle": "2024-04-14T06:29:01.779034Z", + "shell.execute_reply": "2024-04-14T06:29:01.778391Z" } }, "outputs": [ @@ -325,34 +325,34 @@ "Number of parameters: 46\n", " message: Stop: Total number of iterations reached limit.\n", " success: False\n", - " fun: -77.47279176917233\n", - " x: [ 1.459e+00 6.405e-01 ... 8.381e-02 -5.099e-01]\n", + " fun: -77.47670094008087\n", + " x: [ 1.203e+00 -7.290e-01 ... -6.892e-01 3.889e-01]\n", " nit: 5\n", - " jac: [-3.537e-03 2.956e-03 ... 2.880e-03 2.257e-03]\n", - " nfev: 700\n", + " jac: [ 4.514e-03 3.034e-03 ... 3.112e-03 -8.923e-03]\n", + " nfev: 643\n", " njev: 5\n", - " nlinop: 470\n", + " nlinop: 413\n", "\n", "Iteration 1\n", - " Energy: -77.457090689731\n", - " Norm of gradient: 0.012327117956958011\n", - " Regularization hyperparameter: 0.025872901351309822\n", - " Variation hyperparameter: 0.7774219914937698\n", + " Energy: -77.4561701428464\n", + " Norm of gradient: 0.020178708617073507\n", + " Regularization hyperparameter: 0.0017211959563998682\n", + " Variation hyperparameter: 0.23863811965379816\n", "Iteration 2\n", - " Energy: -77.45810206331204\n", - " Norm of gradient: 0.007842750623987389\n", - " Regularization hyperparameter: 0.0004598183587570829\n", - " Variation hyperparameter: 0.7793387589027908\n", + " Energy: -77.45818252794383\n", + " Norm of gradient: 0.016170711610366127\n", + " Regularization hyperparameter: 0.0022737732303941527\n", + " Variation hyperparameter: 0.23864126894267035\n", "Iteration 3\n", - " Energy: -77.45812603123487\n", - " Norm of gradient: 0.007633002626591864\n", - " Regularization hyperparameter: 1.0432606313329105\n", - " Variation hyperparameter: 0.7761677959408932\n", + " Energy: -77.45824776871719\n", + " Norm of gradient: 0.01565468785413535\n", + " Regularization hyperparameter: 1.0972265700421266\n", + " Variation hyperparameter: 0.23438337504497775\n", "Iteration 4\n", - " Energy: -77.4632114975537\n", - " Norm of gradient: 0.023790911850183817\n", - " Regularization hyperparameter: 0.002246858928021853\n", - " Variation hyperparameter: 0.7281885451930055\n" + " Energy: -77.46309898169862\n", + " Norm of gradient: 0.028846990789665336\n", + " Regularization hyperparameter: 0.0047852775445017025\n", + " Variation hyperparameter: 0.2647444833148662\n" ] } ], diff --git a/.doctrees/nbsphinx/tutorials/05-entanglement-forging.ipynb b/.doctrees/nbsphinx/tutorials/05-entanglement-forging.ipynb index e7fd55243..afd40c059 100644 --- a/.doctrees/nbsphinx/tutorials/05-entanglement-forging.ipynb +++ b/.doctrees/nbsphinx/tutorials/05-entanglement-forging.ipynb @@ -14,10 +14,10 @@ "execution_count": 1, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:25:25.923587Z", - "iopub.status.busy": "2024-04-14T06:25:25.923115Z", - "iopub.status.idle": "2024-04-14T06:25:26.765742Z", - "shell.execute_reply": "2024-04-14T06:25:26.765148Z" + "iopub.execute_input": "2024-04-14T06:29:03.444034Z", + "iopub.status.busy": "2024-04-14T06:29:03.443505Z", + "iopub.status.idle": "2024-04-14T06:29:04.258040Z", + "shell.execute_reply": "2024-04-14T06:29:04.257440Z" } }, "outputs": [ @@ -82,10 +82,10 @@ "execution_count": 2, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:25:26.769959Z", - "iopub.status.busy": "2024-04-14T06:25:26.768864Z", - "iopub.status.idle": "2024-04-14T06:25:26.784667Z", - "shell.execute_reply": "2024-04-14T06:25:26.784110Z" + "iopub.execute_input": "2024-04-14T06:29:04.262337Z", + "iopub.status.busy": "2024-04-14T06:29:04.261221Z", + "iopub.status.idle": "2024-04-14T06:29:04.277837Z", + "shell.execute_reply": "2024-04-14T06:29:04.277287Z" } }, "outputs": [ @@ -131,10 +131,10 @@ "execution_count": 3, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:25:26.788536Z", - "iopub.status.busy": "2024-04-14T06:25:26.787535Z", - "iopub.status.idle": "2024-04-14T06:25:27.724792Z", - "shell.execute_reply": "2024-04-14T06:25:27.724231Z" + "iopub.execute_input": "2024-04-14T06:29:04.281570Z", + "iopub.status.busy": "2024-04-14T06:29:04.280616Z", + "iopub.status.idle": "2024-04-14T06:29:05.201327Z", + "shell.execute_reply": "2024-04-14T06:29:05.200756Z" } }, "outputs": [ @@ -146,10 +146,10 @@ " message: STOP: TOTAL NO. of f AND g EVALUATIONS EXCEEDS LIMIT\n", " success: False\n", " status: 1\n", - " fun: -75.68085222771742\n", + " fun: -75.68085255364095\n", " x: [ 2.996e+00 -7.549e-01 ... 2.650e+00 8.012e-01]\n", " nit: 6\n", - " jac: [ 1.758e-03 9.115e-03 ... -1.192e-02 9.663e-04]\n", + " jac: [ 1.756e-03 9.115e-03 ... -1.192e-02 9.479e-04]\n", " nfev: 112\n", " njev: 7\n", " hess_inv: <15x15 LbfgsInvHessProduct with dtype=float64>\n" diff --git a/.doctrees/nbsphinx/tutorials/06-fermion-operator.ipynb b/.doctrees/nbsphinx/tutorials/06-fermion-operator.ipynb index 2738d1284..51a6e639d 100644 --- a/.doctrees/nbsphinx/tutorials/06-fermion-operator.ipynb +++ b/.doctrees/nbsphinx/tutorials/06-fermion-operator.ipynb @@ -29,10 +29,10 @@ "execution_count": 1, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:25:29.122567Z", - "iopub.status.busy": "2024-04-14T06:25:29.122048Z", - "iopub.status.idle": "2024-04-14T06:25:29.752389Z", - "shell.execute_reply": "2024-04-14T06:25:29.751743Z" + "iopub.execute_input": "2024-04-14T06:29:06.646093Z", + "iopub.status.busy": "2024-04-14T06:29:06.645872Z", + "iopub.status.idle": "2024-04-14T06:29:07.253582Z", + "shell.execute_reply": "2024-04-14T06:29:07.253079Z" } }, "outputs": [ @@ -40,9 +40,9 @@ "data": { "text/plain": [ "FermionOperator({\n", - " (cre_a(0), des_a(3)): 0.5,\n", " (cre_a(3), des_a(0)): -0.25,\n", - " (cre_b(1), des_b(5), cre_a(4)): 1+1j\n", + " (cre_b(1), des_b(5), cre_a(4)): 1+1j,\n", + " (cre_a(0), des_a(3)): 0.5\n", "})" ] }, @@ -76,17 +76,17 @@ "execution_count": 2, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:25:29.754898Z", - "iopub.status.busy": "2024-04-14T06:25:29.754478Z", - "iopub.status.idle": "2024-04-14T06:25:29.758557Z", - "shell.execute_reply": "2024-04-14T06:25:29.757962Z" + "iopub.execute_input": "2024-04-14T06:29:07.256242Z", + "iopub.status.busy": "2024-04-14T06:29:07.255765Z", + "iopub.status.idle": "2024-04-14T06:29:07.259799Z", + "shell.execute_reply": "2024-04-14T06:29:07.259292Z" } }, "outputs": [ { "data": { "text/plain": [ - "'FermionOperator({((True, False, 0), (False, False, 3)): 0.5+0j, ((True, False, 3), (False, False, 0)): -0.25+0j, ((True, True, 1), (False, True, 5), (True, False, 4)): 1+1j})'" + "'FermionOperator({((True, False, 3), (False, False, 0)): -0.25+0j, ((True, True, 1), (False, True, 5), (True, False, 4)): 1+1j, ((True, False, 0), (False, False, 3)): 0.5+0j})'" ] }, "execution_count": 2, @@ -110,10 +110,10 @@ "execution_count": 3, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:25:29.760749Z", - "iopub.status.busy": "2024-04-14T06:25:29.760556Z", - "iopub.status.idle": "2024-04-14T06:25:29.764773Z", - "shell.execute_reply": "2024-04-14T06:25:29.764220Z" + "iopub.execute_input": "2024-04-14T06:29:07.262013Z", + "iopub.status.busy": "2024-04-14T06:29:07.261663Z", + "iopub.status.idle": "2024-04-14T06:29:07.265904Z", + "shell.execute_reply": "2024-04-14T06:29:07.265398Z" } }, "outputs": [ @@ -121,17 +121,17 @@ "data": { "text/plain": [ "FermionOperator({\n", - " (cre_a(0), des_a(3), des_a(3), des_b(3)): -0.125,\n", - " (cre_b(2)): 0-0.25j,\n", - " (des_a(3), des_b(3)): 0.0625,\n", " (cre_a(3), des_a(0), cre_b(2)): 0-0.25j,\n", - " (cre_a(3), des_a(0)): -0.5,\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), cre_b(2)): -1+1j,\n", + " (des_a(3), des_b(3)): 0.0625,\n", + " (cre_b(1), des_b(5), cre_a(4)): 2+2j,\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_b(2)): 0-0.25j,\n", + " (cre_a(0), des_a(3), des_a(3), des_b(3)): -0.125,\n", " (cre_a(0), des_a(3)): 1,\n", - " (cre_b(1), des_b(5), cre_a(4), cre_b(2)): -1+1j,\n", - " (cre_b(1), des_b(5), cre_a(4)): 2+2j\n", + " (cre_a(0), des_a(3), cre_b(2)): 0+0.5j,\n", + " (cre_a(3), des_a(0)): -0.5,\n", + " (cre_a(3), des_a(0), des_a(3), des_b(3)): 0.0625\n", "})" ] }, @@ -169,10 +169,10 @@ "execution_count": 4, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:25:29.767042Z", - "iopub.status.busy": "2024-04-14T06:25:29.766693Z", - "iopub.status.idle": "2024-04-14T06:25:29.770655Z", - "shell.execute_reply": "2024-04-14T06:25:29.770180Z" + "iopub.execute_input": "2024-04-14T06:29:07.268169Z", + "iopub.status.busy": "2024-04-14T06:29:07.267805Z", + "iopub.status.idle": "2024-04-14T06:29:07.271743Z", + "shell.execute_reply": "2024-04-14T06:29:07.271288Z" } }, "outputs": [ @@ -180,17 +180,17 @@ "data": { "text/plain": [ "FermionOperator({\n", - " (cre_a(0), des_a(3), des_a(3), des_b(3)): 0+0.5j,\n", - " (cre_b(2)): -5,\n", - " (des_a(3), des_b(3)): 0-1.25j,\n", " (cre_a(3), des_a(0), cre_b(2)): -1,\n", - " (cre_a(3), des_a(0)): 0+3j,\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), cre_b(2)): 4+4j,\n", + " (des_a(3), des_b(3)): 0-1.25j,\n", + " (cre_b(1), des_b(5), cre_a(4)): 12-12j,\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_b(2)): -5,\n", + " (cre_a(0), des_a(3), des_a(3), des_b(3)): 0+0.5j,\n", " (cre_a(0), des_a(3)): 0-6j,\n", - " (cre_b(1), des_b(5), cre_a(4), cre_b(2)): 4+4j,\n", - " (cre_b(1), des_b(5), cre_a(4)): 12-12j\n", + " (cre_a(0), des_a(3), cre_b(2)): 2,\n", + " (cre_a(3), des_a(0)): 0+3j,\n", + " (cre_a(3), des_a(0), des_a(3), des_b(3)): 0-0.25j\n", "})" ] }, @@ -219,10 +219,10 @@ "execution_count": 5, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:25:29.772661Z", - "iopub.status.busy": "2024-04-14T06:25:29.772469Z", - "iopub.status.idle": "2024-04-14T06:25:29.776167Z", - "shell.execute_reply": "2024-04-14T06:25:29.775672Z" + "iopub.execute_input": "2024-04-14T06:29:07.273751Z", + "iopub.status.busy": "2024-04-14T06:29:07.273547Z", + "iopub.status.idle": "2024-04-14T06:29:07.277286Z", + "shell.execute_reply": "2024-04-14T06:29:07.276728Z" } }, "outputs": [ @@ -231,15 +231,15 @@ "text/plain": [ "FermionOperator({\n", " (cre_b(2)): -5,\n", - " (cre_b(1), cre_a(4), des_b(5)): -12+12j,\n", - " (cre_b(1), cre_a(4), des_b(5), des_b(3), des_a(3)): -1+1j,\n", - " (cre_a(3), des_a(0)): 0+3j,\n", + " (des_b(3), des_a(3)): 0+1.25j,\n", " (cre_b(2), cre_a(3), des_a(0)): -1,\n", - " (cre_a(0), des_a(3)): 0-6j,\n", " (cre_b(2), cre_a(0), des_a(3)): 2,\n", - " (des_b(3), des_a(3)): 0+1.25j,\n", " (cre_a(3), des_b(3), des_a(3), des_a(0)): 0+0.25j,\n", - " (cre_b(2), cre_b(1), cre_a(4), des_b(5)): 4+4j\n", + " (cre_b(2), cre_b(1), cre_a(4), des_b(5)): 4+4j,\n", + " (cre_a(3), des_a(0)): 0+3j,\n", + " (cre_a(0), des_a(3)): 0-6j,\n", + " (cre_b(1), cre_a(4), des_b(5)): -12+12j,\n", + " (cre_b(1), cre_a(4), des_b(5), des_b(3), des_a(3)): -1+1j\n", "})" ] }, @@ -264,10 +264,10 @@ "execution_count": 6, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:25:29.778514Z", - "iopub.status.busy": "2024-04-14T06:25:29.778083Z", - "iopub.status.idle": "2024-04-14T06:25:29.781183Z", - "shell.execute_reply": "2024-04-14T06:25:29.780636Z" + "iopub.execute_input": "2024-04-14T06:29:07.279747Z", + "iopub.status.busy": "2024-04-14T06:29:07.279404Z", + "iopub.status.idle": "2024-04-14T06:29:07.282495Z", + "shell.execute_reply": "2024-04-14T06:29:07.281876Z" } }, "outputs": [ @@ -297,10 +297,10 @@ "execution_count": 7, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:25:29.783603Z", - "iopub.status.busy": "2024-04-14T06:25:29.783173Z", - "iopub.status.idle": "2024-04-14T06:25:29.787510Z", - "shell.execute_reply": "2024-04-14T06:25:29.787016Z" + "iopub.execute_input": "2024-04-14T06:29:07.284671Z", + "iopub.status.busy": "2024-04-14T06:29:07.284321Z", + "iopub.status.idle": "2024-04-14T06:29:07.288464Z", + "shell.execute_reply": "2024-04-14T06:29:07.287911Z" } }, "outputs": [ @@ -340,21 +340,21 @@ "execution_count": 8, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:25:29.789683Z", - "iopub.status.busy": "2024-04-14T06:25:29.789341Z", - "iopub.status.idle": "2024-04-14T06:25:29.794947Z", - "shell.execute_reply": "2024-04-14T06:25:29.794450Z" + "iopub.execute_input": "2024-04-14T06:29:07.290775Z", + "iopub.status.busy": "2024-04-14T06:29:07.290429Z", + "iopub.status.idle": "2024-04-14T06:29:07.296079Z", + "shell.execute_reply": "2024-04-14T06:29:07.295520Z" } }, "outputs": [ { "data": { "text/plain": [ - "array([ 0. +0.j , 0. +0.j ,\n", - " 0. +0.j , 0. +0.j ,\n", - " -0.08127924-0.08210612j, 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", + " 0.03031105-0.0725495j, 0. +0.j ,\n", + " 0. +0.j , 0. +0.j ,\n", + " 0. +0.j ])" ] }, "execution_count": 8, @@ -379,10 +379,10 @@ "execution_count": 9, "metadata": { "execution": { - "iopub.execute_input": "2024-04-14T06:25:29.797217Z", - "iopub.status.busy": "2024-04-14T06:25:29.796840Z", - "iopub.status.idle": "2024-04-14T06:25:29.808137Z", - "shell.execute_reply": "2024-04-14T06:25:29.807618Z" + "iopub.execute_input": "2024-04-14T06:29:07.298277Z", + "iopub.status.busy": "2024-04-14T06:29:07.298096Z", + "iopub.status.idle": "2024-04-14T06:29:07.308835Z", + "shell.execute_reply": "2024-04-14T06:29:07.308376Z" } }, "outputs": [ diff --git a/.doctrees/tutorials/03-double-factorized.doctree b/.doctrees/tutorials/03-double-factorized.doctree index cf8f521b3..e1f066c2d 100644 Binary files a/.doctrees/tutorials/03-double-factorized.doctree and b/.doctrees/tutorials/03-double-factorized.doctree differ diff --git a/.doctrees/tutorials/04-lucj.doctree b/.doctrees/tutorials/04-lucj.doctree index 30f92f936..6a327cc83 100644 Binary files a/.doctrees/tutorials/04-lucj.doctree and b/.doctrees/tutorials/04-lucj.doctree differ diff --git a/.doctrees/tutorials/05-entanglement-forging.doctree b/.doctrees/tutorials/05-entanglement-forging.doctree index 38ebf6fc1..77f622c12 100644 Binary files a/.doctrees/tutorials/05-entanglement-forging.doctree and b/.doctrees/tutorials/05-entanglement-forging.doctree differ diff --git a/.doctrees/tutorials/06-fermion-operator.doctree b/.doctrees/tutorials/06-fermion-operator.doctree index 49274afe1..fa2138a48 100644 Binary files a/.doctrees/tutorials/06-fermion-operator.doctree and b/.doctrees/tutorials/06-fermion-operator.doctree differ diff --git a/_modules/ffsim/cistring.html b/_modules/ffsim/cistring.html index 8e7943a91..a5d914436 100644 --- a/_modules/ffsim/cistring.html +++ b/_modules/ffsim/cistring.html @@ -5,7 +5,7 @@ -
Here, mol_hamiltonian
is an instance of MolecularHamiltonian
, a dataclass that stores the one- and two-body tensors, and df_hamiltonian
is an instance of DoubleFactorizedHamiltonian
, a dataclass that stores the updated one-body-tensor, diagonal Coulomb matrices, and orbital rotations. In the cell below, we print out the shapes of the tensors describing the original and double-factorized representations.
-Hartree Fock energy: -77.44562676439627
+Hartree Fock energy: -77.44562676439624
Now, we set the evolution time and calculate the exact result of time evolution by directly exponentiating the Hamiltonian using SciPy. Later, we will compare the result of our approximate time evolution with this exact result.
@@ -496,7 +496,7 @@
-Fidelity of evolved state w.r.t. initial state: 0.9315062301398414
+Fidelity of evolved state w.r.t. initial state: 0.9315062301402518
Now, let’s test our implementation.
@@ -523,7 +523,7 @@
-Fidelity of Trotter-evolved state with exact state: 0.9928527668211954
+Fidelity of Trotter-evolved state with exact state: 0.9928527668216444
The fidelity of the final result can be improved by increasing the number of Trotter steps.
@@ -550,7 +550,7 @@
-Fidelity of Trotter-evolved state with exact state: 0.999932085128549
+Fidelity of Trotter-evolved state with exact state: 0.9999320851289344
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.
-Fidelity of Trotter-evolved state with exact state: 0.999932085128549
+Fidelity of Trotter-evolved state with exact state: 0.9999320851289344
A higher order formula achieves a higher fidelity with fewer Trotter steps:
@@ -607,7 +607,7 @@
-Fidelity of Trotter-evolved state with exact state: 0.9999913261306368
+Fidelity of Trotter-evolved state with exact state: 0.9999913261310335
-E(CCSD) = -77.49387212754473 E_corr = -0.04824536314851524
+E(CCSD) = -77.49387212754468 E_corr = -0.04824536314851467
-Energy at initialization: -77.46975600021636
+Energy at initialization: -77.4697560002172
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.)
-'FermionOperator({((True, False, 0), (False, False, 3)): 0.5+0j, ((True, False, 3), (False, False, 0)): -0.25+0j, ((True, True, 1), (False, True, 5), (True, False, 4)): 1+1j})'
+'FermionOperator({((True, False, 3), (False, False, 0)): -0.25+0j, ((True, True, 1), (False, True, 5), (True, False, 4)): 1+1j, ((True, False, 0), (False, False, 3)): 0.5+0j})'
FermionOperators support arithmetic operations. Note that when multiplying a FermionOperator by a scalar, the scalar must go on the left, i.e. 2 * op
and not op * 2
.
FermionOperator({
- (cre_a(0), des_a(3), des_a(3), des_b(3)): -0.125,
- (cre_b(2)): 0-0.25j,
- (des_a(3), des_b(3)): 0.0625,
(cre_a(3), des_a(0), cre_b(2)): 0-0.25j,
- (cre_a(3), des_a(0)): -0.5,
- (cre_a(3), des_a(0), des_a(3), des_b(3)): 0.0625,
+ (cre_b(1), des_b(5), cre_a(4), cre_b(2)): -1+1j,
+ (des_a(3), des_b(3)): 0.0625,
+ (cre_b(1), des_b(5), cre_a(4)): 2+2j,
(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_b(2)): 0-0.25j,
+ (cre_a(0), des_a(3), des_a(3), des_b(3)): -0.125,
(cre_a(0), des_a(3)): 1,
- (cre_b(1), des_b(5), cre_a(4), cre_b(2)): -1+1j,
- (cre_b(1), des_b(5), cre_a(4)): 2+2j
+ (cre_a(0), des_a(3), cre_b(2)): 0+0.5j,
+ (cre_a(3), des_a(0)): -0.5,
+ (cre_a(3), des_a(0), des_a(3), des_b(3)): 0.0625
})
FermionOperator({
- (cre_a(0), des_a(3), des_a(3), des_b(3)): 0+0.5j,
- (cre_b(2)): -5,
- (des_a(3), des_b(3)): 0-1.25j,
(cre_a(3), des_a(0), cre_b(2)): -1,
- (cre_a(3), des_a(0)): 0+3j,
- (cre_a(3), des_a(0), des_a(3), des_b(3)): 0-0.25j,
+ (cre_b(1), des_b(5), cre_a(4), cre_b(2)): 4+4j,
+ (des_a(3), des_b(3)): 0-1.25j,
+ (cre_b(1), des_b(5), cre_a(4)): 12-12j,
(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_b(2)): -5,
+ (cre_a(0), des_a(3), des_a(3), des_b(3)): 0+0.5j,
(cre_a(0), des_a(3)): 0-6j,
- (cre_b(1), des_b(5), cre_a(4), cre_b(2)): 4+4j,
- (cre_b(1), des_b(5), cre_a(4)): 12-12j
+ (cre_a(0), des_a(3), cre_b(2)): 2,
+ (cre_a(3), des_a(0)): 0+3j,
+ (cre_a(3), des_a(0), des_a(3), des_b(3)): 0-0.25j
})
FermionOperator({ (cre_b(2)): -5, - (cre_b(1), cre_a(4), des_b(5)): -12+12j, - (cre_b(1), cre_a(4), des_b(5), des_b(3), des_a(3)): -1+1j, - (cre_a(3), des_a(0)): 0+3j, + (des_b(3), des_a(3)): 0+1.25j, (cre_b(2), cre_a(3), des_a(0)): -1, - (cre_a(0), des_a(3)): 0-6j, (cre_b(2), cre_a(0), des_a(3)): 2, - (des_b(3), des_a(3)): 0+1.25j, (cre_a(3), des_b(3), des_a(3), des_a(0)): 0+0.25j, - (cre_b(2), cre_b(1), cre_a(4), des_b(5)): 4+4j + (cre_b(2), cre_b(1), cre_a(4), des_b(5)): 4+4j, + (cre_a(3), des_a(0)): 0+3j, + (cre_a(0), des_a(3)): 0-6j, + (cre_b(1), cre_a(4), des_b(5)): -12+12j, + (cre_b(1), cre_a(4), des_b(5), des_b(3), des_a(3)): -1+1j })@@ -443,11 +443,11 @@
-array([ 0. +0.j , 0. +0.j ,
- 0. +0.j , 0. +0.j ,
- -0.08127924-0.08210612j, 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 ,
+ 0.03031105-0.0725495j, 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
.