{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Copyright 2020 The OpenFermion Developers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:44:39.470500Z",
     "iopub.status.busy": "2024-06-30T09:44:39.470248Z",
     "iopub.status.idle": "2024-06-30T09:44:39.474442Z",
     "shell.execute_reply": "2024-06-30T09:44:39.473804Z"
    }
   },
   "outputs": [],
   "source": [
    "#@title Licensed under the Apache License, Version 2.0 (the \"License\");\n",
    "# you may not use this file except in compliance with the License.\n",
    "# You may obtain a copy of the License at\n",
    "#\n",
    "# https://www.apache.org/licenses/LICENSE-2.0\n",
    "#\n",
    "# Unless required by applicable law or agreed to in writing, software\n",
    "# distributed under the License is distributed on an \"AS IS\" BASIS,\n",
    "# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
    "# See the License for the specific language governing permissions and\n",
    "# limitations under the License."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Hamiltonian Time Evolution and Expectation Value Computation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<table class=\"tfo-notebook-buttons\" align=\"left\">\n",
    "  <td>\n",
    "    <a target=\"_blank\" href=\"https://quantumai.google/openfermion/fqe/tutorials/hamiltonian_time_evolution_and_expectation_estimation\"><img src=\"https://quantumai.google/site-assets/images/buttons/quantumai_logo_1x.png\" />View on QuantumAI</a>\n",
    "  </td>\n",
    "  <td>\n",
    "    <a target=\"_blank\" href=\"https://colab.research.google.com/github/quantumlib/OpenFermion/blob/master/docs/fqe/tutorials/hamiltonian_time_evolution_and_expectation_estimation.ipynb\"><img src=\"https://quantumai.google/site-assets/images/buttons/colab_logo_1x.png\" />Run in Google Colab</a>\n",
    "  </td>\n",
    "  <td>\n",
    "    <a target=\"_blank\" href=\"https://github.com/quantumlib/OpenFermion/blob/master/docs/fqe/tutorials/hamiltonian_time_evolution_and_expectation_estimation.ipynb\"><img src=\"https://quantumai.google/site-assets/images/buttons/github_logo_1x.png\" />View source on GitHub</a>\n",
    "  </td>\n",
    "  <td>\n",
    "    <a href=\"https://storage.googleapis.com/tensorflow_docs/OpenFermion/docs/fqe/tutorials/hamiltonian_time_evolution_and_expectation_estimation.ipynb\"><img src=\"https://quantumai.google/site-assets/images/buttons/download_icon_1x.png\" />Download notebook</a>\n",
    "  </td>\n",
    "</table>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This tutorial describes the FQE's capabilities for Hamiltonian time-evolution and expectation value estimation\n",
    "\n",
    "Where possible, LiH will be used as an example molecule for the API."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:44:39.478109Z",
     "iopub.status.busy": "2024-06-30T09:44:39.477572Z",
     "iopub.status.idle": "2024-06-30T09:44:53.944692Z",
     "shell.execute_reply": "2024-06-30T09:44:53.943709Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\r\n",
      "tensorflow-metadata 1.15.0 requires protobuf<4.21,>=3.20.3; python_version < \"3.11\", but you have protobuf 4.25.3 which is incompatible.\u001b[0m\u001b[31m\r\n",
      "\u001b[0m"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    import fqe\n",
    "except ImportError:\n",
    "    !pip install fqe --quiet"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:44:53.949832Z",
     "iopub.status.busy": "2024-06-30T09:44:53.948994Z",
     "iopub.status.idle": "2024-06-30T09:44:55.748670Z",
     "shell.execute_reply": "2024-06-30T09:44:55.747696Z"
    },
    "id": "3i55Vs4SYsm5",
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "Print = True\n",
    "from openfermion import FermionOperator, MolecularData\n",
    "from openfermion.utils import hermitian_conjugated\n",
    "import numpy\n",
    "import fqe\n",
    "\n",
    "numpy.set_printoptions(floatmode='fixed', precision=6, linewidth=80, suppress=True)\n",
    "numpy.random.seed(seed=409)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:44:55.752899Z",
     "iopub.status.busy": "2024-06-30T09:44:55.752417Z",
     "iopub.status.idle": "2024-06-30T09:44:56.091500Z",
     "shell.execute_reply": "2024-06-30T09:44:56.090260Z"
    },
    "id": "obExleOZZSAq"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  % Total    % Received % Xferd  Average Speed   Time    Time     Time  Current\r\n",
      "                                 Dload  Upload   Total   Spent    Left  Speed\r\n",
      "\r",
      "  0     0    0     0    0     0      0      0 --:--:-- --:--:-- --:--:--     0"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\r",
      "100 62086  100 62086    0     0   306k      0 --:--:-- --:--:-- --:--:--  306k\r\n"
     ]
    }
   ],
   "source": [
    "!curl -O https://raw.githubusercontent.com/quantumlib/OpenFermion-FQE/master/tests/unittest_data/build_lih_data.py"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:44:56.096152Z",
     "iopub.status.busy": "2024-06-30T09:44:56.095812Z",
     "iopub.status.idle": "2024-06-30T09:44:56.107847Z",
     "shell.execute_reply": "2024-06-30T09:44:56.106970Z"
    },
    "id": "E3RaAyBkZuov"
   },
   "outputs": [],
   "source": [
    "import build_lih_data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:44:56.111584Z",
     "iopub.status.busy": "2024-06-30T09:44:56.111013Z",
     "iopub.status.idle": "2024-06-30T09:44:56.128233Z",
     "shell.execute_reply": "2024-06-30T09:44:56.127485Z"
    },
    "id": "g8XMRUw6Zb8W"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sector N = 4 : S_z = 0\n",
      "a'000011'b'000011' (-0.9870890035778126+0j)\n",
      "a'000011'b'000101' (-0.0384854879340124+0j)\n",
      "a'000011'b'100001' (-0.0036477466809967+0j)\n",
      "a'000101'b'000011' (-0.0384854879340073+0j)\n",
      "a'000101'b'000101' (0.0344508540105218+0j)\n",
      "a'000101'b'100001' (-0.0592953695921639+0j)\n",
      "a'000101'b'100010' (-0.0010777697106179+0j)\n",
      "a'001001'b'001001' (0.0271656962509733+0j)\n",
      "a'001001'b'001010' (-0.0031209939957822+0j)\n",
      "a'010001'b'010001' (0.0271656962509733+0j)\n",
      "a'010001'b'010010' (-0.0031209939957822+0j)\n",
      "a'100001'b'000011' (-0.003647746680997+0j)\n",
      "a'100001'b'000101' (-0.0592953695921649+0j)\n",
      "a'100001'b'100001' (0.1135892009408387+0j)\n",
      "a'000110'b'000110' (0.0037623492033852+0j)\n",
      "a'001010'b'001001' (-0.0031209939957822+0j)\n",
      "a'001010'b'001010' (0.001794202149606+0j)\n",
      "a'010010'b'010001' (-0.0031209939957822+0j)\n",
      "a'010010'b'010010' (0.001794202149606+0j)\n",
      "a'100010'b'000101' (-0.001077769710618+0j)\n",
      "a'100010'b'100010' (0.0010515758855392+0j)\n"
     ]
    }
   ],
   "source": [
    "h1e, h2e, wfn = build_lih_data.build_lih_data('energy')\n",
    "lih_hamiltonian = fqe.get_restricted_hamiltonian(([h1e, h2e]))\n",
    "lihwfn = fqe.Wavefunction([[4, 0, 6]])\n",
    "lihwfn.set_wfn(strategy='from_data', raw_data={(4, 0): wfn})\n",
    "if Print:\n",
    "    lihwfn.print_wfn()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Application of one- and two-body fermionic gates\n",
    "\n",
    "The API for time propagation can be invoked through the fqe namespace or the wavefunction object"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:44:56.134170Z",
     "iopub.status.busy": "2024-06-30T09:44:56.132696Z",
     "iopub.status.idle": "2024-06-30T09:44:58.982883Z",
     "shell.execute_reply": "2024-06-30T09:44:58.982138Z"
    }
   },
   "outputs": [],
   "source": [
    "# dummy geometry\n",
    "from openfermion.chem.molecular_data import spinorb_from_spatial\n",
    "from openfermion import jordan_wigner, get_sparse_operator, InteractionOperator, get_fermion_operator\n",
    "\n",
    "h1s, h2s = spinorb_from_spatial(h1e, numpy.einsum(\"ijlk\", -2 * h2e) * 0.5)\n",
    "mol = InteractionOperator(0, h1s, h2s)\n",
    "ham_fop = get_fermion_operator(mol)\n",
    "ham_mat = get_sparse_operator(jordan_wigner(ham_fop)).toarray()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:44:58.987251Z",
     "iopub.status.busy": "2024-06-30T09:44:58.986702Z",
     "iopub.status.idle": "2024-06-30T09:45:06.824924Z",
     "shell.execute_reply": "2024-06-30T09:45:06.823783Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sector N = 4 : S_z = 0\n",
      "a'000011'b'000011' (-0.9832017407384288-0.08751592993809965j)\n",
      "a'000011'b'000101' (-0.038333927909412015-0.003412147794935153j)\n",
      "a'000011'b'100001' (-0.003633381455202132-0.00032341158146809005j)\n",
      "a'000101'b'000011' (-0.038333927909406935-0.0034121477949347074j)\n",
      "a'000101'b'000101' (0.03431518282571416+0.003054434433325243j)\n",
      "a'000101'b'100001' (-0.05906185803441181-0.005257164690403508j)\n",
      "a'000101'b'100010' (-0.001073525344578889-9.555564554363788e-05j)\n",
      "a'001001'b'001001' (0.02705871481472957+0.0024085276354275367j)\n",
      "a'001001'b'001010' (-0.003108703183821363-0.00027670930255657j)\n",
      "a'010001'b'010001' (0.02705871481472957+0.0024085276354275367j)\n",
      "a'010001'b'010010' (-0.003108703183821363-0.00027670930255657j)\n",
      "a'100001'b'000011' (-0.0036333814552024306-0.0003234115814681176j)\n",
      "a'100001'b'000101' (-0.0590618580344128-0.0052571646904036005j)\n",
      "a'100001'b'100001' (0.11314187441703756+0.010070889930469404j)\n",
      "a'000110'b'000110' (0.003747532670094563+0.0003335721898296519j)\n",
      "a'001010'b'001001' (-0.003108703183821363-0.0002767093025565706j)\n",
      "a'001010'b'001010' (0.0017871363875789756+0.00015907509845870593j)\n",
      "a'010010'b'010001' (-0.003108703183821363-0.0002767093025565706j)\n",
      "a'010010'b'010010' (0.0017871363875789756+0.00015907509845870593j)\n",
      "a'100010'b'000101' (-0.0010735253445789887-9.555564554364526e-05j)\n",
      "a'100010'b'100010' (0.0010474346607506372+9.323348063182962e-05j)\n",
      "Sector N = 4 : S_z = 0\n",
      "a'000011'b'000011' (-0.9832017407384288-0.08751592993809965j)\n",
      "a'000011'b'000101' (-0.038333927909412015-0.003412147794935153j)\n",
      "a'000011'b'100001' (-0.003633381455202132-0.00032341158146809005j)\n",
      "a'000101'b'000011' (-0.038333927909406935-0.0034121477949347074j)\n",
      "a'000101'b'000101' (0.03431518282571416+0.003054434433325243j)\n",
      "a'000101'b'100001' (-0.05906185803441181-0.005257164690403508j)\n",
      "a'000101'b'100010' (-0.001073525344578889-9.555564554363788e-05j)\n",
      "a'001001'b'001001' (0.02705871481472957+0.0024085276354275367j)\n",
      "a'001001'b'001010' (-0.003108703183821363-0.00027670930255657j)\n",
      "a'010001'b'010001' (0.02705871481472957+0.0024085276354275367j)\n",
      "a'010001'b'010010' (-0.003108703183821363-0.00027670930255657j)\n",
      "a'100001'b'000011' (-0.0036333814552024306-0.0003234115814681176j)\n",
      "a'100001'b'000101' (-0.0590618580344128-0.0052571646904036005j)\n",
      "a'100001'b'100001' (0.11314187441703756+0.010070889930469404j)\n",
      "a'000110'b'000110' (0.003747532670094563+0.0003335721898296519j)\n",
      "a'001010'b'001001' (-0.003108703183821363-0.0002767093025565706j)\n",
      "a'001010'b'001010' (0.0017871363875789756+0.00015907509845870593j)\n",
      "a'010010'b'010001' (-0.003108703183821363-0.0002767093025565706j)\n",
      "a'010010'b'010010' (0.0017871363875789756+0.00015907509845870593j)\n",
      "a'100010'b'000101' (-0.0010735253445789887-9.555564554364526e-05j)\n",
      "a'100010'b'100010' (0.0010474346607506372+9.323348063182962e-05j)\n"
     ]
    }
   ],
   "source": [
    "from scipy.linalg import expm\n",
    "time = 0.01\n",
    "evolved1 = lihwfn.time_evolve(time, lih_hamiltonian)\n",
    "if Print:\n",
    "    evolved1.print_wfn()\n",
    "evolved2 = fqe.time_evolve(lihwfn, time, lih_hamiltonian)\n",
    "if Print:\n",
    "    evolved2.print_wfn()\n",
    "assert numpy.isclose(fqe.vdot(evolved1, evolved2), 1)\n",
    "cirq_wf = fqe.to_cirq(lihwfn)\n",
    "evolve_cirq = expm(-1j * time * ham_mat) @ cirq_wf\n",
    "test_evolve = fqe.from_cirq(evolve_cirq, thresh=1.0E-12)\n",
    "assert numpy.isclose(fqe.vdot(test_evolve, evolved1), 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exact evolution implementation of quadratic Hamiltonians\n",
    "\n",
    "Listed here are examples of evolving the special Hamiltonians.\n",
    "\n",
    "Diagonal Hamiltonian evolution is supported."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:45:06.829922Z",
     "iopub.status.busy": "2024-06-30T09:45:06.829255Z",
     "iopub.status.idle": "2024-06-30T09:45:06.862750Z",
     "shell.execute_reply": "2024-06-30T09:45:06.861889Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sector N = 4 : S_z = 2\n",
      "a'0111'b'0001' (0.06410384332680892+0.006457824553220213j)\n",
      "a'0111'b'0010' (-0.04750282105641742-0.004806746363757904j)\n",
      "a'0111'b'0100' (-0.13678251578172124-0.14669354294155937j)\n",
      "a'0111'b'1000' (-0.17590908446219902+0.18037032855388507j)\n",
      "a'1011'b'0001' (-0.024476509055323645-0.35002973813257837j)\n",
      "a'1011'b'0010' (-0.29306549119285774-0.24993699345042755j)\n",
      "a'1011'b'0100' (-0.24463823979331925+0.216230018507552j)\n",
      "a'1011'b'1000' (0.2630370061724331-0.3219176890144332j)\n",
      "a'1101'b'0001' (0.06881598452335648-0.11820307831930973j)\n",
      "a'1101'b'0010' (0.07453400242973146-0.06864240203689581j)\n",
      "a'1101'b'0100' (-0.10280462830183762-0.20883766019199357j)\n",
      "a'1101'b'1000' (0.18607595463278417-0.08025697543327445j)\n",
      "a'1110'b'0001' (0.11456775354687777+0.3103763043669054j)\n",
      "a'1110'b'0010' (-0.06508766236659613-0.19892441460633234j)\n",
      "a'1110'b'0100' (0.10203424947892725-0.20758467025020982j)\n",
      "a'1110'b'1000' (0.07609612213727357+0.04672521448420223j)\n",
      "-2.0 [0^ 0] +\n",
      "-1.7 [1^ 1] +\n",
      "-0.7 [2^ 2] +\n",
      "-0.55 [3^ 3] +\n",
      "-0.1 [4^ 4] +\n",
      "-0.06 [5^ 5] +\n",
      "0.5 [6^ 6] +\n",
      "0.3 [7^ 7]\n",
      "Sector N = 4 : S_z = 2\n",
      "a'0111'b'0001' (0.0637484451008294+0.009334986580321275j)\n",
      "a'0111'b'0010' (-0.047315172642354755-0.006395096304800911j)\n",
      "a'0111'b'0100' (-0.132531714879653-0.15054499897167728j)\n",
      "a'0111'b'1000' (-0.18036290425047674+0.17591669673256966j)\n",
      "a'1011'b'0001' (-0.010810197548109522-0.35071791614011905j)\n",
      "a'1011'b'0010' (-0.2860822817529272-0.25790077721904764j)\n",
      "a'1011'b'0100' (-0.24946214917080445+0.21064644645479583j)\n",
      "a'1011'b'1000' (0.26910559751344887-0.31686218219200535j)\n",
      "a'1101'b'0001' (0.07267851126583812-0.11586820724539217j)\n",
      "a'1101'b'0010' (0.07599247436942777-0.06702418007556614j)\n",
      "a'1101'b'0100' (-0.09932391825824108-0.21051536565432893j)\n",
      "a'1101'b'1000' (0.1871035427295063-0.07783127453836906j)\n",
      "a'1110'b'0001' (0.108337728499401+0.31260543349216774j)\n",
      "a'1110'b'0010' (-0.06339447392546516-0.19947046697326787j)\n",
      "a'1110'b'0100' (0.1027808914964281-0.207216002598293j)\n",
      "a'1110'b'1000' (0.07609612213727357+0.04672521448420223j)\n"
     ]
    }
   ],
   "source": [
    "wfn = fqe.Wavefunction([[4, 2, 4]])\n",
    "wfn.set_wfn(strategy='random')\n",
    "if Print:\n",
    "    wfn.print_wfn()\n",
    "\n",
    "diagonal = FermionOperator('0^ 0', -2.0) + \\\n",
    "           FermionOperator('1^ 1', -1.7) + \\\n",
    "           FermionOperator('2^ 2', -0.7) + \\\n",
    "           FermionOperator('3^ 3', -0.55) + \\\n",
    "           FermionOperator('4^ 4', -0.1) + \\\n",
    "           FermionOperator('5^ 5', -0.06) + \\\n",
    "           FermionOperator('6^ 6', 0.5) + \\\n",
    "           FermionOperator('7^ 7', 0.3)\n",
    "if Print:\n",
    "    print(diagonal)\n",
    "    \n",
    "evolved = wfn.time_evolve(time, diagonal)\n",
    "if Print:\n",
    "    evolved.print_wfn()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exact evolution of dense quadratic hamiltonians is supported.  Here is an evolution example using a spin restricted Hamiltonian on a number and spin conserving wavefunction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:45:06.868970Z",
     "iopub.status.busy": "2024-06-30T09:45:06.868495Z",
     "iopub.status.idle": "2024-06-30T09:45:07.064821Z",
     "shell.execute_reply": "2024-06-30T09:45:07.064062Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial Energy: (12.222344460445168+5.551115123125783e-17j)\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Final Energy:   (12.222344460445179+1.9168694409543718e-16j)\n"
     ]
    }
   ],
   "source": [
    "norb = 4 \n",
    "h1e = numpy.zeros((norb, norb), dtype=numpy.complex128) \n",
    "for i in range(norb): \n",
    "    for j in range(norb): \n",
    "        h1e[i, j] += (i+j) * 0.02 \n",
    "    h1e[i, i] += i * 2.0 \n",
    "\n",
    "hamil = fqe.get_restricted_hamiltonian((h1e,)) \n",
    "wfn = fqe.Wavefunction([[4, 0, norb]]) \n",
    "wfn.set_wfn(strategy='random') \n",
    "initial_energy = wfn.expectationValue(hamil) \n",
    "print('Initial Energy: {}'.format(initial_energy))\n",
    "evolved = wfn.time_evolve(time, hamil) \n",
    "final_energy = evolved.expectationValue(hamil)\n",
    "print('Final Energy:   {}'.format(final_energy))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The GSO Hamiltonian is for evolution of quadratic hamiltonians that are spin broken and number conserving."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:45:07.077955Z",
     "iopub.status.busy": "2024-06-30T09:45:07.068588Z",
     "iopub.status.idle": "2024-06-30T09:45:07.530067Z",
     "shell.execute_reply": "2024-06-30T09:45:07.529258Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial Energy: (28.720309503101298-2.393918396847994e-16j)\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Final Energy:   (28.72030950310126+9.020562075079397e-17j)\n"
     ]
    }
   ],
   "source": [
    "norb = 4 \n",
    "h1e = numpy.zeros((2*norb, 2*norb), dtype=numpy.complex128) \n",
    "for i in range(2*norb): \n",
    "    for j in range(2*norb): \n",
    "        h1e[i, j] += (i+j) * 0.02 \n",
    "    h1e[i, i] += i * 2.0 \n",
    "\n",
    "hamil = fqe.get_gso_hamiltonian((h1e,)) \n",
    "wfn = fqe.get_number_conserving_wavefunction(4, norb) \n",
    "wfn.set_wfn(strategy='random') \n",
    "initial_energy = wfn.expectationValue(hamil) \n",
    "print('Initial Energy: {}'.format(initial_energy))\n",
    "evolved = wfn.time_evolve(time, hamil) \n",
    "final_energy = evolved.expectationValue(hamil)\n",
    "print('Final Energy:   {}'.format(final_energy))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The BCS hamiltonian evovles spin conserved and number broken wavefunctions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:45:07.534438Z",
     "iopub.status.busy": "2024-06-30T09:45:07.533502Z",
     "iopub.status.idle": "2024-06-30T09:45:07.868580Z",
     "shell.execute_reply": "2024-06-30T09:45:07.867792Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6.0 [] +\n",
      "(0.05-0.15000000000000002j) [0 1] +\n",
      "(0.25-0.35000000000000003j) [0 3] +\n",
      "(0.45-0.55j) [0 5] +\n",
      "(0.65-0.75j) [0 7] +\n",
      "(0.05+0.05j) [0^ 1^] +\n",
      "(0.15000000000000002+0.15000000000000002j) [0^ 3^] +\n",
      "(0.25+0.25j) [0^ 5^] +\n",
      "(0.35000000000000003+0.35000000000000003j) [0^ 7^] +\n",
      "(0.05-0.05j) [1 0] +\n",
      "(0.15000000000000002-0.15000000000000002j) [1 2] +\n",
      "(0.25-0.25j) [1 4] +\n",
      "(0.35000000000000003-0.35000000000000003j) [1 6] +\n",
      "(0.05+0.15000000000000002j) [1^ 0^] +\n",
      "(0.15000000000000002+0.25j) [1^ 2^] +\n",
      "(0.25+0.35000000000000003j) [1^ 4^] +\n",
      "(0.35000000000000003+0.45j) [1^ 6^] +\n",
      "(0.15000000000000002-0.25j) [2 1] +\n",
      "(0.35000000000000003-0.45j) [2 3] +\n",
      "(0.55-0.65j) [2 5] +\n",
      "(0.75-0.8500000000000001j) [2 7] +\n",
      "(0.15000000000000002+0.15000000000000002j) [2^ 1^] +\n",
      "(0.25+0.25j) [2^ 3^] +\n",
      "(0.35000000000000003+0.35000000000000003j) [2^ 5^] +\n",
      "(0.45+0.45j) [2^ 7^] +\n",
      "(0.15000000000000002-0.15000000000000002j) [3 0] +\n",
      "(0.25-0.25j) [3 2] +\n",
      "(0.35000000000000003-0.35000000000000003j) [3 4] +\n",
      "(0.45-0.45j) [3 6] +\n",
      "(0.25+0.35000000000000003j) [3^ 0^] +\n",
      "(0.35000000000000003+0.45j) [3^ 2^] +\n",
      "(0.45+0.55j) [3^ 4^] +\n",
      "(0.55+0.65j) [3^ 6^] +\n",
      "(0.25-0.35000000000000003j) [4 1] +\n",
      "(0.45-0.55j) [4 3] +\n",
      "(0.65-0.75j) [4 5] +\n",
      "(0.8500000000000001-0.9500000000000001j) [4 7] +\n",
      "(0.25+0.25j) [4^ 1^] +\n",
      "(0.35000000000000003+0.35000000000000003j) [4^ 3^] +\n",
      "(0.45+0.45j) [4^ 5^] +\n",
      "(0.55+0.55j) [4^ 7^] +\n",
      "(0.25-0.25j) [5 0] +\n",
      "(0.35000000000000003-0.35000000000000003j) [5 2] +\n",
      "(0.45-0.45j) [5 4] +\n",
      "(0.55-0.55j) [5 6] +\n",
      "(0.45+0.55j) [5^ 0^] +\n",
      "(0.55+0.65j) [5^ 2^] +\n",
      "(0.65+0.75j) [5^ 4^] +\n",
      "(0.75+0.8500000000000001j) [5^ 6^] +\n",
      "(0.35000000000000003-0.45j) [6 1] +\n",
      "(0.55-0.65j) [6 3] +\n",
      "(0.75-0.8500000000000001j) [6 5] +\n",
      "(0.9500000000000001-1.05j) [6 7] +\n",
      "(0.35000000000000003+0.35000000000000003j) [6^ 1^] +\n",
      "(0.45+0.45j) [6^ 3^] +\n",
      "(0.55+0.55j) [6^ 5^] +\n",
      "(0.65+0.65j) [6^ 7^] +\n",
      "(0.35000000000000003-0.35000000000000003j) [7 0] +\n",
      "(0.45-0.45j) [7 2] +\n",
      "(0.55-0.55j) [7 4] +\n",
      "(0.65-0.65j) [7 6] +\n",
      "(0.65+0.75j) [7^ 0^] +\n",
      "(0.75+0.8500000000000001j) [7^ 2^] +\n",
      "(0.8500000000000001+0.9500000000000001j) [7^ 4^] +\n",
      "(0.9500000000000001+1.05j) [7^ 6^]\n",
      "Sector N = 2 : S_z = 2\n",
      "a'0011'b'0000' (-0.03822050376543366+0.04722267594475418j)\n",
      "a'0101'b'0000' (0.11266351755026058-0.08045492107765523j)\n",
      "a'1001'b'0000' (0.20320077458016653-0.12569862804377432j)\n",
      "a'0110'b'0000' (0.05619850454412105+0.09496478341558393j)\n",
      "a'1010'b'0000' (0.10559885537015444+0.11842688052122244j)\n",
      "a'1100'b'0000' (0.0077637482095351895+0.31531590367980844j)\n",
      "Sector N = 4 : S_z = 2\n",
      "a'0111'b'0001' (0.2824942893468279+0.03355206579818916j)\n",
      "a'0111'b'0010' (-0.05772107497136629-0.02636869741981419j)\n",
      "a'0111'b'0100' (0.023785796648682776-0.0020224394959580906j)\n",
      "a'0111'b'1000' (0.21640450174424197-0.04378508800670816j)\n",
      "a'1011'b'0001' (-0.05498328060358216+0.02734692187255261j)\n",
      "a'1011'b'0010' (-0.04928412842784853+0.14503876601698823j)\n",
      "a'1011'b'0100' (0.15795758958015643-0.4182365189352067j)\n",
      "a'1011'b'1000' (-0.17447882444539087-0.009486157624653761j)\n",
      "a'1101'b'0001' (-0.1445095969304484-0.18205850652200903j)\n",
      "a'1101'b'0010' (-0.03991499779909757-0.15433749648217981j)\n",
      "a'1101'b'0100' (0.06570900360151104+0.004102951665073998j)\n",
      "a'1101'b'1000' (0.059818195036244165-0.12107474028451727j)\n",
      "a'1110'b'0001' (-0.06872552283670944-0.03502572909303779j)\n",
      "a'1110'b'0010' (0.01187150936783494-0.08220106790262457j)\n",
      "a'1110'b'0100' (0.245935671612729+0.2003604108656353j)\n",
      "a'1110'b'1000' (-0.08678168538325304-0.15114749448953507j)\n",
      "Sector N = 6 : S_z = 2\n",
      "a'1111'b'0011' (-0.07458559356076411-0.07500964684800192j)\n",
      "a'1111'b'0101' (0.00026534389687275984-0.07669551428142163j)\n",
      "a'1111'b'1001' (-0.09607654962925911-0.03066759719778399j)\n",
      "a'1111'b'0110' (0.09851785462192013+0.04062010213534433j)\n",
      "a'1111'b'1010' (-0.06147228452062744+0.19634181999024788j)\n",
      "a'1111'b'1100' (0.203465187540792+0.15201301478908172j)\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sector N = 2 : S_z = 2\n",
      "a'0011'b'0000' (-0.038097743113607495+0.047539945454499716j)\n",
      "a'0101'b'0000' (0.11218647680909512-0.0812198942753196j)\n",
      "a'1001'b'0000' (0.2025964775787488-0.1269139163437523j)\n",
      "a'0110'b'0000' (0.056815624811621314+0.09444679519610732j)\n",
      "a'1010'b'0000' (0.1060979033896154+0.1180590472361861j)\n",
      "a'1100'b'0000' (0.009711552915153309+0.3149941116139492j)\n",
      "Sector N = 4 : S_z = 2\n",
      "a'0111'b'0001' (0.2827334081828877+0.03191913437012366j)\n",
      "a'0111'b'0010' (-0.05776957775932644-0.026097211137678548j)\n",
      "a'0111'b'0100' (0.02376911509399819-0.002175581749758095j)\n",
      "a'0111'b'1000' (0.21605500834702818-0.045249841427133605j)\n",
      "a'1011'b'0001' (-0.054883877134095975+0.02766525346243461j)\n",
      "a'1011'b'0010' (-0.04852768913042599+0.14528128840122675j)\n",
      "a'1011'b'0100' (0.15548424682043674-0.4189660547258269j)\n",
      "a'1011'b'1000' (-0.17451327690229249-0.008546537415431717j)\n",
      "a'1101'b'0001' (-0.14555517224730116-0.18111178278510304j)\n",
      "a'1101'b'0010' (-0.04074794646804691-0.15419679959996305j)\n",
      "a'1101'b'0100' (0.06575967845578282+0.003736909200068359j)\n",
      "a'1101'b'1000' (0.0589613844419772-0.12164982883882651j)\n",
      "a'1110'b'0001' (-0.0689906450290815-0.03462388136753807j)\n",
      "a'1110'b'0010' (0.011244555293778628-0.08231180714509588j)\n",
      "a'1110'b'0100' (0.24720056742353744+0.19907277517865116j)\n",
      "a'1110'b'1000' (-0.08770534327870647-0.15070288403482224j)\n",
      "Sector N = 6 : S_z = 2\n",
      "a'1111'b'0011' (-0.07500249390617937-0.07463966565297969j)\n",
      "a'1111'b'0101' (-0.00027596423073685146-0.07657867707171727j)\n",
      "a'1111'b'1001' (-0.09614340563343148-0.030226554766595443j)\n",
      "a'1111'b'0110' (0.0988673405671825+0.040013301020912787j)\n",
      "a'1111'b'1010' (-0.06049739731068076+0.1966835109908999j)\n",
      "a'1111'b'1100' (0.20447453839378954+0.15086235436317388j)\n"
     ]
    }
   ],
   "source": [
    "norb = 4\n",
    "time = 0.001\n",
    "wfn_spin = fqe.get_spin_conserving_wavefunction(2, norb)\n",
    "hamil = FermionOperator('', 6.0)\n",
    "for i in range(0, 2*norb, 2):\n",
    "    for j in range(0, 2*norb, 2):\n",
    "        opstring = str(i) + ' ' + str(j + 1)\n",
    "        hamil += FermionOperator(opstring, (i+1 + j*2)*0.1 - (i+1 + 2*(j + 1))*0.1j)\n",
    "        opstring = str(i) + '^ ' + str(j + 1) + '^ '\n",
    "        hamil += FermionOperator(opstring, (i+1 + j)*0.1 + (i+1 + j)*0.1j)\n",
    "h_noncon = (hamil + hermitian_conjugated(hamil))/2.0\n",
    "if Print:\n",
    "    print(h_noncon)\n",
    "\n",
    "wfn_spin.set_wfn(strategy='random')\n",
    "if Print:\n",
    "    wfn_spin.print_wfn()\n",
    "\n",
    "spin_evolved = wfn_spin.time_evolve(time, h_noncon)\n",
    "if Print:\n",
    "    spin_evolved.print_wfn()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exact Evolution Implementation of Diagonal Coulomb terms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:45:07.874080Z",
     "iopub.status.busy": "2024-06-30T09:45:07.872603Z",
     "iopub.status.idle": "2024-06-30T09:45:07.886777Z",
     "shell.execute_reply": "2024-06-30T09:45:07.886052Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sector N = 5 : S_z = 1\n",
      "a'0111'b'0011' (0.1767197633238888+0.07705027591386174j)\n",
      "a'0111'b'0101' (0.26511616271206534+0.008961114965964636j)\n",
      "a'0111'b'1001' (-0.21500788967807205+0.18883222357115242j)\n",
      "a'0111'b'0110' (0.05899888340530983+0.24718508330513655j)\n",
      "a'0111'b'1010' (0.026779941886877087+0.178826903379466j)\n",
      "a'0111'b'1100' (0.040105194632471065-0.13765308148589672j)\n",
      "a'1011'b'0011' (-0.07737195675705617-0.13361350504300393j)\n",
      "a'1011'b'0101' (-0.13687313057404554+0.29187074296331617j)\n",
      "a'1011'b'1001' (-0.06315193751752252-0.015963783432630548j)\n",
      "a'1011'b'0110' (0.1412070788641405+0.1547599681431957j)\n",
      "a'1011'b'1010' (0.19284938811527336+0.05815106367478927j)\n",
      "a'1011'b'1100' (0.0601422593196422+0.28502769589072846j)\n",
      "a'1101'b'0011' (-0.18150775903742453+0.016874070450244708j)\n",
      "a'1101'b'0101' (-0.0348624766019713-0.13176446392846256j)\n",
      "a'1101'b'1001' (-0.0025269874019722425+0.15813527253622475j)\n",
      "a'1101'b'0110' (0.2350325470053182+0.00014186481294887606j)\n",
      "a'1101'b'1010' (-0.05882009410190486-0.17131076761878405j)\n",
      "a'1101'b'1100' (-0.0805012226704995-0.22440941132365497j)\n",
      "a'1110'b'0011' (0.03604887823935152+0.03924039987192205j)\n",
      "a'1110'b'0101' (0.01791557236805855+0.049178708081171994j)\n",
      "a'1110'b'1001' (0.043394070527428906+0.04684086505377531j)\n",
      "a'1110'b'0110' (-0.06429592223560363-0.0008430408719242828j)\n",
      "a'1110'b'1010' (0.2544814342392375-0.29724856959461543j)\n",
      "a'1110'b'1100' (0.030682862125531125+0.04316975493381036j)\n",
      "Sector N = 5 : S_z = 1\n",
      "a'0111'b'0011' (0.17246919202766037+0.08614521210711293j)\n",
      "a'0111'b'0101' (0.2640044105946365+0.02585638217504639j)\n",
      "a'0111'b'1001' (-0.22865588002522289+0.1720537410593985j)\n",
      "a'0111'b'0110' (0.039317599959618595+0.25106863600429263j)\n",
      "a'0111'b'1010' (0.01016559936487448+0.18053500283686325j)\n",
      "a'0111'b'1100' (0.05486817182308945-0.13246237654467485j)\n",
      "a'1011'b'0011' (-0.06892245945470493-0.1381618000932499j)\n",
      "a'1011'b'0101' (-0.15852659275216138+0.2806993121856342j)\n",
      "a'1011'b'1001' (-0.06148222691741648-0.021516165247931745j)\n",
      "a'1011'b'0110' (0.12632522987699016+0.16712876220911863j)\n",
      "a'1011'b'1010' (0.18549537660753673+0.0785104958599347j)\n",
      "a'1011'b'1100' (0.023846943350919615+0.2903260272067709j)\n",
      "a'1101'b'0011' (-0.18226377985147815+0.00311695341631818j)\n",
      "a'1101'b'0101' (-0.022781831587704093-0.13438100453072824j)\n",
      "a'1101'b'1001' (-0.019218653219683423+0.15698341776715583j)\n",
      "a'1101'b'0110' (0.2336171388678771+0.025755595598224452j)\n",
      "a'1101'b'1010' (-0.03682570987971141-0.1773444379284259j)\n",
      "a'1101'b'1100' (-0.04695986639822977-0.23374088579008845j)\n",
      "a'1110'b'0011' (0.03227444359792561+0.042399185073162045j)\n",
      "a'1110'b'0101' (0.012449211992330995+0.05083827477971962j)\n",
      "a'1110'b'1001' (0.037163719046413746+0.05192273088508236j)\n",
      "a'1110'b'0110' (-0.06364995585406888-0.00913013985728614j)\n",
      "a'1110'b'1010' (0.29549075882045067-0.25652041624443533j)\n",
      "a'1110'b'1100' (0.02295227609762028+0.047731109260085336j)\n"
     ]
    }
   ],
   "source": [
    "norb = 4\n",
    "wfn = fqe.Wavefunction([[5, 1, norb]])\n",
    "vij = numpy.zeros((norb, norb, norb, norb), dtype=numpy.complex128)\n",
    "for i in range(norb):\n",
    "            for j in range(norb):\n",
    "                vij[i, j] += 4*(i % norb + 1)*(j % norb + 1)*0.21\n",
    "                \n",
    "wfn.set_wfn(strategy='random')\n",
    "\n",
    "if Print:\n",
    "    wfn.print_wfn()\n",
    "    \n",
    "hamil = fqe.get_diagonalcoulomb_hamiltonian(vij)\n",
    "    \n",
    "evolved = wfn.time_evolve(time, hamil)\n",
    "if Print:\n",
    "    evolved.print_wfn()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exact evolution of individual n-body anti-Hermitian gnerators"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:45:07.892430Z",
     "iopub.status.busy": "2024-06-30T09:45:07.890989Z",
     "iopub.status.idle": "2024-06-30T09:45:07.909286Z",
     "shell.execute_reply": "2024-06-30T09:45:07.908565Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sector N = 4 : S_z = -2\n",
      "a'001'b'111' (0.15192470437087405+0.002030736551023001j)\n",
      "a'010'b'111' (-0.03463482137049871+0.27009718718635434j)\n",
      "a'100'b'111' (-0.227167326143701-0.042680013855770464j)\n",
      "Sector N = 4 : S_z = 0\n",
      "a'011'b'011' (-0.3135681471723931+0.02343272674191714j)\n",
      "a'011'b'101' (-0.02499985483324474+0.08816424695516287j)\n",
      "a'011'b'110' (0.08135663091791555-0.2354257740286914j)\n",
      "a'101'b'011' (0.10687726150559686-0.2956005054158534j)\n",
      "a'101'b'101' (0.17170942273731685+0.06147474695535962j)\n",
      "a'101'b'110' (-0.06659232012041506-0.17366091158923022j)\n",
      "a'110'b'011' (0.10656876871906572+0.05034757634263565j)\n",
      "a'110'b'101' (-0.5439254382876493-0.12805668259580452j)\n",
      "a'110'b'110' (0.05259450329119359+0.030994001966578325j)\n",
      "Sector N = 4 : S_z = 2\n",
      "a'111'b'001' (-0.27915567439519695-0.16997476921880741j)\n",
      "a'111'b'010' (0.21779510862047607-0.13520849680378372j)\n",
      "a'111'b'100' (-0.062444490395427814+0.08399365005639217j)\n",
      "Sector N = 4 : S_z = -2\n",
      "a'001'b'111' (0.15192470437087405+0.002030736551023001j)\n",
      "a'010'b'111' (-0.03463482137049871+0.27009718718635434j)\n",
      "a'100'b'111' (-0.2277896098680148-0.04319797642130905j)\n",
      "Sector N = 4 : S_z = 0\n",
      "a'011'b'011' (-0.3135681471723931+0.02343272674191714j)\n",
      "a'011'b'101' (-0.02499985483324474+0.08816424695516287j)\n",
      "a'011'b'110' (0.08135663091791555-0.2354257740286914j)\n",
      "a'101'b'011' (0.10687726150559686-0.2956005054158534j)\n",
      "a'101'b'101' (0.17170942273731685+0.06147474695535962j)\n",
      "a'101'b'110' (-0.06659232012041506-0.17366091158923022j)\n",
      "a'110'b'011' (0.10656876871906572+0.05034757634263565j)\n",
      "a'110'b'101' (-0.5439254382876493-0.12805668259580452j)\n",
      "a'110'b'110' (0.05259450329119359+0.030994001966578325j)\n",
      "Sector N = 4 : S_z = 2\n",
      "a'111'b'001' (-0.27915567439519695-0.16997476921880741j)\n",
      "a'111'b'010' (0.2174388128701414-0.13456899986199047j)\n",
      "a'111'b'100' (-0.062444490395427814+0.08399365005639217j)\n"
     ]
    }
   ],
   "source": [
    "norb = 3\n",
    "nele = 4\n",
    "ops = FermionOperator('5^ 1^ 2 0', 3.0 - 1.j)\n",
    "ops += FermionOperator('0^ 2^ 1 5', 3.0 + 1.j)\n",
    "wfn = fqe.get_number_conserving_wavefunction(nele, norb)\n",
    "wfn.set_wfn(strategy='random')\n",
    "wfn.normalize()\n",
    "if Print:\n",
    "    wfn.print_wfn()\n",
    "evolved = wfn.time_evolve(time, ops)\n",
    "if Print:\n",
    "    evolved.print_wfn()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " Approximate evolution of sums of n-body generators\n",
    "\n",
    "Approximate evolution can be done for dense operators."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:45:07.914920Z",
     "iopub.status.busy": "2024-06-30T09:45:07.913484Z",
     "iopub.status.idle": "2024-06-30T09:45:07.926822Z",
     "shell.execute_reply": "2024-06-30T09:45:07.926101Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sector N = 4 : S_z = 0\n",
      "a'000011'b'000011' (-0.9870501056639676-0.008762984250848186j)\n",
      "a'000011'b'000101' (-0.0384839713479148-0.00034165891188627537j)\n",
      "a'000011'b'100001' (-0.003647602935297334-3.238325409440231e-05j)\n",
      "a'000101'b'000011' (-0.0384839713479097-0.00034165891188623075j)\n",
      "a'000101'b'000101' (0.03444949641617426+0.0003058410149547989j)\n",
      "a'000101'b'100001' (-0.059293032957665366-0.000526400752638063j)\n",
      "a'000101'b'100010' (-0.0010777272393492372-9.568002262034463e-06j)\n",
      "a'001001'b'001001' (0.02716462574072948+0.0002411662625463919j)\n",
      "a'001001'b'001010' (-0.003120871007714629-2.770694732178501e-05j)\n",
      "a'010001'b'010001' (0.02716462574072948+0.0002411662625463919j)\n",
      "a'010001'b'010010' (-0.003120871007714629-2.770694732178501e-05j)\n",
      "a'100001'b'000011' (-0.0036476029352976343-3.238325409440512e-05j)\n",
      "a'100001'b'000101' (-0.059293032957666365-0.0005264007526380719j)\n",
      "a'100001'b'100001' (0.11358472476587893+0.0010083998412408197j)\n",
      "a'000110'b'000110' (0.0037622009416752988+3.340063743457015e-05j)\n",
      "a'001010'b'001001' (-0.003120871007714629-2.7706947321785068e-05j)\n",
      "a'001010'b'001010' (0.0017941314460251166+1.5928215395129844e-05j)\n",
      "a'010010'b'010001' (-0.003120871007714629-2.7706947321785068e-05j)\n",
      "a'010010'b'010010' (0.0017941314460251166+1.5928215395129844e-05j)\n",
      "a'100010'b'000101' (-0.0010777272393493371-9.568002262035205e-06j)\n",
      "a'100010'b'100010' (0.0010515344463539172+9.335483532366947e-06j)\n"
     ]
    }
   ],
   "source": [
    "lih_evolved = lihwfn.apply_generated_unitary(time, 'taylor', lih_hamiltonian, accuracy=1.e-8)\n",
    "if Print:\n",
    "    lih_evolved.print_wfn()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:45:07.932303Z",
     "iopub.status.busy": "2024-06-30T09:45:07.930893Z",
     "iopub.status.idle": "2024-06-30T09:45:07.952291Z",
     "shell.execute_reply": "2024-06-30T09:45:07.951570Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sector N = 2 : S_z = 0\n",
      "a'01'b'01' (0.3495794429313121+0.022158760002413482j)\n",
      "a'01'b'10' (0.5664453364412798-0.3945979322462498j)\n",
      "a'10'b'01' (-0.39816876837018494+0.14573910133744658j)\n",
      "a'10'b'10' (-0.4662060110250305-0.06007542464228622j)\n",
      "Sector N = 2 : S_z = 0\n",
      "a'01'b'01' (0.34443641067646297-0.051014551799515924j)\n",
      "a'01'b'10' (0.42102942409764665-0.5481625677445289j)\n",
      "a'10'b'01' (-0.33530047561075+0.26085960635871003j)\n",
      "a'10'b'10' (-0.4509597513637166+0.13105059079720657j)\n"
     ]
    }
   ],
   "source": [
    "norb = 2\n",
    "nalpha = 1\n",
    "nbeta = 1\n",
    "nele = nalpha + nbeta\n",
    "time = 0.05\n",
    "h1e = numpy.zeros((norb*2, norb*2), dtype=numpy.complex128)\n",
    "for i in range(2*norb):\n",
    "    for j in range(2*norb):\n",
    "        h1e[i, j] += (i+j) * 0.02\n",
    "    h1e[i, i] += i * 2.0\n",
    "hamil = fqe.get_general_hamiltonian((h1e,))\n",
    "spec_lim = [-1.13199078e-03, 6.12720338e+00]\n",
    "wfn = fqe.Wavefunction([[nele, nalpha - nbeta, norb]])\n",
    "wfn.set_wfn(strategy='random')\n",
    "if Print:\n",
    "    wfn.print_wfn()\n",
    "evol_wfn = wfn.apply_generated_unitary(time, 'chebyshev', hamil, spec_lim=spec_lim)\n",
    "if Print:\n",
    "    evol_wfn.print_wfn()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "API for determining desired expectation values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:45:07.957926Z",
     "iopub.status.busy": "2024-06-30T09:45:07.956500Z",
     "iopub.status.idle": "2024-06-30T09:45:08.002576Z",
     "shell.execute_reply": "2024-06-30T09:45:08.001815Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.999908+0.000000j -0.000284+0.000000j  0.000441+0.000000j\n",
      "  -0.000000+0.000000j -0.000000+0.000000j -0.001285+0.000000j]\n",
      " [-0.000284+0.000000j  1.951766+0.000000j  0.073757+0.000000j\n",
      "   0.000000+0.000000j  0.000000+0.000000j  0.010948+0.000000j]\n",
      " [ 0.000441+0.000000j  0.073757+0.000000j  0.012402+0.000000j\n",
      "   0.000000+0.000000j  0.000000+0.000000j -0.017277+0.000000j]\n",
      " [-0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "   0.001525+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      " [-0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "   0.000000+0.000000j  0.001525+0.000000j -0.000000+0.000000j]\n",
      " [-0.001285+0.000000j  0.010948+0.000000j -0.017277+0.000000j\n",
      "   0.000000+0.000000j -0.000000+0.000000j  0.032874+0.000000j]]\n",
      "(0.0737569627134323+0j)\n",
      "[[ 0.000170-0.017754j  0.000284+0.000003j -0.000441-0.000004j\n",
      "   0.000000+0.000000j  0.000000+0.000000j  0.001285+0.000011j]\n",
      " [ 0.000284+0.000003j  0.048311-0.017327j -0.073754-0.000655j\n",
      "  -0.000000-0.000000j -0.000000-0.000000j -0.010948-0.000097j]\n",
      " [-0.000441-0.000004j -0.073754-0.000655j  1.987599-0.000110j\n",
      "  -0.000000-0.000000j -0.000000-0.000000j  0.017276+0.000153j]\n",
      " [ 0.000000+0.000000j -0.000000-0.000000j -0.000000-0.000000j\n",
      "   1.998475-0.000014j -0.000000-0.000000j -0.000000-0.000000j]\n",
      " [ 0.000000+0.000000j -0.000000-0.000000j -0.000000-0.000000j\n",
      "  -0.000000-0.000000j  1.998475-0.000014j  0.000000+0.000000j]\n",
      " [ 0.001285+0.000011j -0.010948-0.000097j  0.017276+0.000153j\n",
      "  -0.000000-0.000000j  0.000000+0.000000j  1.967127-0.000292j]]\n",
      "(-0.07375405619488316-0.0006547850894203785j)\n"
     ]
    }
   ],
   "source": [
    "rdm1 = lihwfn.expectationValue('i^ j')\n",
    "if Print:\n",
    "    print(rdm1)\n",
    "val = lihwfn.expectationValue('5^ 3')\n",
    "if Print:\n",
    "    print(2.*val)\n",
    "trdm1 = fqe.expectationValue(lih_evolved, 'i j^', lihwfn)\n",
    "if Print:\n",
    "    print(trdm1)\n",
    "val = fqe.expectationValue(lih_evolved, '5 3^', lihwfn)\n",
    "if Print:\n",
    "    print(2*val)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2.B.1 RDMs \n",
    "In addition to the above API higher order density matrices in addition to hole densities can be calculated."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:45:08.008418Z",
     "iopub.status.busy": "2024-06-30T09:45:08.006986Z",
     "iopub.status.idle": "2024-06-30T09:45:08.362886Z",
     "shell.execute_reply": "2024-06-30T09:45:08.361735Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[[[-0.000047+0.000000j -0.000260+0.000000j  0.000445+0.000000j\n",
      "    -0.000000+0.000000j -0.000000+0.000000j -0.001287+0.000000j]\n",
      "   [ 0.000308+0.000000j  0.048269+0.000000j -0.073761+0.000000j\n",
      "    -0.000000+0.000000j -0.000000+0.000000j -0.010927+0.000000j]\n",
      "   [-0.000437+0.000000j -0.073761+0.000000j  1.987537+0.000000j\n",
      "    -0.000000+0.000000j -0.000000+0.000000j  0.017274+0.000000j]\n",
      "   [ 0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j\n",
      "     1.998412+0.000000j -0.000000+0.000000j -0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j\n",
      "    -0.000000+0.000000j  1.998412+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.001283+0.000000j -0.010927+0.000000j  0.017274+0.000000j\n",
      "    -0.000000+0.000000j  0.000000+0.000000j  1.967051+0.000000j]]\n",
      "\n",
      "  [[-0.000544+0.000000j -0.096416+0.000000j  0.147514+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.021885+0.000000j]\n",
      "   [ 0.000733+0.000000j  0.000242+0.000000j  0.000079+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j -0.000157+0.000000j]\n",
      "   [ 0.000282+0.000000j  0.000213+0.000000j -0.000180+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j -0.000110+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "    -0.000103+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j -0.000103+0.000000j -0.000000+0.000000j]\n",
      "   [-0.000012+0.000000j  0.000449+0.000000j  0.000021+0.000000j\n",
      "    -0.000000+0.000000j -0.000000+0.000000j -0.000447+0.000000j]]\n",
      "\n",
      "  [[ 0.000886+0.000000j  0.147514+0.000000j -3.975076+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j -0.034551+0.000000j]\n",
      "   [ 0.000282+0.000000j -0.000073+0.000000j  0.000684+0.000000j\n",
      "    -0.000000+0.000000j -0.000000+0.000000j -0.000288+0.000000j]\n",
      "   [-0.007548+0.000000j -0.000414+0.000000j -0.000511+0.000000j\n",
      "    -0.000000+0.000000j -0.000000+0.000000j  0.000102+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000433+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000433+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000293+0.000000j -0.003503+0.000000j  0.002338+0.000000j\n",
      "    -0.000000+0.000000j -0.000000+0.000000j  0.000595+0.000000j]]\n",
      "\n",
      "  [[-0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "    -3.996823+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000207+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j\n",
      "    -0.000866+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [-0.003666+0.000000j -0.006155+0.000000j -0.000351+0.000000j\n",
      "     0.000000+0.000000j -0.000000+0.000000j  0.000189+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j -0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.002480+0.000000j  0.000000+0.000000j -0.000000+0.000000j]]\n",
      "\n",
      "  [[-0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j -3.996823+0.000000j -0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000207+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j -0.000866+0.000000j -0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "    -0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [-0.003666+0.000000j -0.006155+0.000000j -0.000351+0.000000j\n",
      "    -0.000000+0.000000j  0.000000+0.000000j  0.000189+0.000000j]\n",
      "   [-0.000000+0.000000j  0.000000+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.002480+0.000000j -0.000000+0.000000j]]\n",
      "\n",
      "  [[-0.002572+0.000000j  0.021885+0.000000j -0.034551+0.000000j\n",
      "     0.000000+0.000000j -0.000000+0.000000j -3.934082+0.000000j]\n",
      "   [-0.000012+0.000000j -0.000182+0.000000j -0.000002+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000625+0.000000j]\n",
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      "     0.000000+0.000000j  0.000000+0.000000j  0.000091+0.000000j]\n",
      "   [ 0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j -0.000045+0.000000j  0.000000+0.000000j]]\n",
      "\n",
      "  [[-0.000000+0.000000j  0.000000+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j -0.000181+0.000000j  0.000000+0.000000j]\n",
      "   [-0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000026+0.000000j -0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j -0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j -0.000001+0.000000j -0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j -0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000362+0.000000j -0.000052+0.000000j  0.000001+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j -0.000011+0.000000j]\n",
      "   [-0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000005+0.000000j  0.000000+0.000000j]]\n",
      "\n",
      "  [[-0.000000+0.000000j -0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000008+0.000000j -0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j -0.000001+0.000000j  0.000000+0.000000j]\n",
      "   [-0.000000+0.000000j  0.000000+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "    -0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [-0.000015+0.000000j  0.000001+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j -0.000000+0.000000j  0.000001+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j]]\n",
      "\n",
      "  [[ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "    -0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j]]\n",
      "\n",
      "  [[ 0.003666+0.000000j  0.006155+0.000000j  0.000351+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j -0.000189+0.000000j]\n",
      "   [ 0.006155+0.000000j  0.053634+0.000000j  0.002096+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000196+0.000000j]\n",
      "   [ 0.000351+0.000000j  0.002096+0.000000j -0.001890+0.000000j\n",
      "     0.000000+0.000000j -0.000000+0.000000j  0.003214+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "    -0.001525+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j -0.001525+0.000000j  0.000000+0.000000j]\n",
      "   [-0.000189+0.000000j  0.000196+0.000000j  0.003214+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j -0.006178+0.000000j]]\n",
      "\n",
      "  [[ 0.000000+0.000000j  0.000000+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j -0.000045+0.000000j  0.000000+0.000000j]\n",
      "   [-0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000005+0.000000j -0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j -0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j -0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000091+0.000000j -0.000011+0.000000j  0.000001+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j -0.000003+0.000000j]\n",
      "   [-0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000002+0.000000j  0.000000+0.000000j]]]\n",
      "\n",
      "\n",
      " [[[ 0.001287+0.000000j  0.010927+0.000000j -0.017274+0.000000j\n",
      "     0.000000+0.000000j -0.000000+0.000000j  0.032857+0.000000j]\n",
      "   [-0.021885+0.000000j  0.000157+0.000000j  0.000110+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000163+0.000000j]\n",
      "   [ 0.034551+0.000000j  0.000288+0.000000j -0.000102+0.000000j\n",
      "     0.000000+0.000000j -0.000000+0.000000j -0.000154+0.000000j]\n",
      "   [-0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "    -0.000189+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j -0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j -0.000189+0.000000j -0.000000+0.000000j]\n",
      "   [-0.065735+0.000000j -0.000057+0.000000j  0.000178+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000033+0.000000j]]\n",
      "\n",
      "  [[ 0.000012+0.000000j  0.002122+0.000000j -0.000021+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000163+0.000000j]\n",
      "   [-0.001103+0.000000j -0.007202+0.000000j -0.000282+0.000000j\n",
      "    -0.000000+0.000000j -0.000000+0.000000j -0.000020+0.000000j]\n",
      "   [ 0.000002+0.000000j -0.000278+0.000000j  0.000251+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j -0.000433+0.000000j]\n",
      "   [-0.000000+0.000000j -0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000196+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [-0.000000+0.000000j -0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000196+0.000000j -0.000000+0.000000j]\n",
      "   [-0.000057+0.000000j -0.000036+0.000000j -0.000433+0.000000j\n",
      "     0.000000+0.000000j -0.000000+0.000000j  0.000828+0.000000j]]\n",
      "\n",
      "  [[-0.000293+0.000000j  0.003503+0.000000j  0.000232+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j -0.000154+0.000000j]\n",
      "   [-0.000715+0.000000j -0.117061+0.000000j -0.004564+0.000000j\n",
      "    -0.000000+0.000000j -0.000000+0.000000j -0.000433+0.000000j]\n",
      "   [-0.000126+0.000000j -0.004564+0.000000j  0.004087+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j -0.007032+0.000000j]\n",
      "   [-0.000000+0.000000j -0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.003214+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [-0.000000+0.000000j -0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.003214+0.000000j -0.000000+0.000000j]\n",
      "   [ 0.000178+0.000000j -0.000433+0.000000j -0.007038+0.000000j\n",
      "     0.000000+0.000000j -0.000000+0.000000j  0.013471+0.000000j]]\n",
      "\n",
      "  [[-0.000000+0.000000j -0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000091+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "    -0.000011+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000001+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [-0.000045+0.000000j  0.000005+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000002+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "    -0.000003+0.000000j  0.000000+0.000000j  0.000000+0.000000j]]\n",
      "\n",
      "  [[ 0.000000+0.000000j -0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000091+0.000000j -0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j\n",
      "    -0.000000+0.000000j -0.000011+0.000000j -0.000000+0.000000j]\n",
      "   [-0.000000+0.000000j -0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000001+0.000000j -0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [-0.000045+0.000000j  0.000005+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000002+0.000000j]\n",
      "   [ 0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j -0.000003+0.000000j  0.000000+0.000000j]]\n",
      "\n",
      "  [[ 0.002124+0.000000j  0.000521+0.000000j  0.000001+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000033+0.000000j]\n",
      "   [ 0.000521+0.000000j  0.224250+0.000000j  0.008743+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j  0.000828+0.000000j]\n",
      "   [ 0.000001+0.000000j  0.008743+0.000000j -0.007835+0.000000j\n",
      "     0.000000+0.000000j -0.000000+0.000000j  0.013471+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j  0.000000+0.000000j\n",
      "    -0.006178+0.000000j  0.000000+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000000+0.000000j  0.000000+0.000000j -0.000000+0.000000j\n",
      "     0.000000+0.000000j -0.006178+0.000000j  0.000000+0.000000j]\n",
      "   [ 0.000033+0.000000j  0.000828+0.000000j  0.013471+0.000000j\n",
      "     0.000000+0.000000j  0.000000+0.000000j -0.025808+0.000000j]]]]\n"
     ]
    }
   ],
   "source": [
    "rdm2 = lihwfn.expectationValue('i^ j k l^')\n",
    "if Print:\n",
    "    print(rdm2)\n",
    "rdm2 = fqe.expectationValue(lihwfn, 'i^ j^ k l', lihwfn)\n",
    "if Print:\n",
    "    print(rdm2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2.B.2 Hamiltonian expectations (or any expectation values)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:45:08.367495Z",
     "iopub.status.busy": "2024-06-30T09:45:08.366868Z",
     "iopub.status.idle": "2024-06-30T09:45:08.532435Z",
     "shell.execute_reply": "2024-06-30T09:45:08.531588Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(-8.87771957038547+0j)\n",
      "(-8.87771957038547+0j)\n"
     ]
    }
   ],
   "source": [
    "li_h_energy = lihwfn.expectationValue(lih_hamiltonian)\n",
    "if Print:\n",
    "    print(li_h_energy)\n",
    "li_h_energy = fqe.expectationValue(lihwfn, lih_hamiltonian, lihwfn)\n",
    "if Print:\n",
    "    print(li_h_energy)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2.B.3 Symmetry operations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-06-30T09:45:08.538494Z",
     "iopub.status.busy": "2024-06-30T09:45:08.537006Z",
     "iopub.status.idle": "2024-06-30T09:45:08.548241Z",
     "shell.execute_reply": "2024-06-30T09:45:08.547580Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(-1.8969111426479763e-23+0j)\n",
      "0j\n",
      "(1.0000000000000018+0j)\n",
      "(4.000000000000007+0j)\n"
     ]
    }
   ],
   "source": [
    "op = fqe.get_s2_operator()\n",
    "print(lihwfn.expectationValue(op))\n",
    "op = fqe.get_sz_operator()\n",
    "print(lihwfn.expectationValue(op))\n",
    "op = fqe.get_time_reversal_operator()\n",
    "print(lihwfn.expectationValue(op))\n",
    "op = fqe.get_number_operator()\n",
    "print(lihwfn.expectationValue(op))\n"
   ]
  }
 ],
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