This code tutorial shows how to estimate a 1-RDM and perform variational optimization
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try:
import recirq
except ImportError:
!pip install --quiet git+https://github.com/quantumlib/ReCirq
import numpy as np
import cirq
from recirq.hfvqe.gradient_hf import rhf_func_generator
from recirq.hfvqe.opdm_functionals import OpdmFunctional
from recirq.hfvqe.analysis import (
compute_opdm, mcweeny_purification,
resample_opdm, fidelity_witness,
fidelity)
from recirq.hfvqe.third_party.higham import fixed_trace_positive_projection
from recirq.hfvqe.molecular_example import make_h6_1_3
Set up the experiment
Generate the input files, set up quantum resources, and set up the OpdmFunctional to make measurements.
rhf_objective, molecule, parameters, obi, tbi = make_h6_1_3()
ansatz, energy, gradient = rhf_func_generator(rhf_objective)
# settings for quantum resources
qubits = [cirq.GridQubit(0, x) for x in range(molecule.n_orbitals)]
sampler = cirq.Simulator(dtype=np.complex128) # this can be a QuantumEngine
# OpdmFunctional contains an interface for running experiments
opdm_func = OpdmFunctional(qubits=qubits,
sampler=sampler,
constant=molecule.nuclear_repulsion,
one_body_integrals=obi,
two_body_integrals=tbi,
# only simulate spin-up electrons:
num_electrons=molecule.n_electrons // 2,
clean_xxyy=True,
purification=True
)
Optimization terminated successfully.
Current function value: -2.924060
Iterations: 7
Function evaluations: 15
Gradient evaluations: 15
The displayed text is the output of the gradient based restricted Hartree-Fock. We define the gradient in rhf_objective and use the conjugate-gradient optimizer to optimize the basis rotation parameters. This is equivalent to doing Hartree-Fock theory from the canonical transformation perspective.
Estimate Quantities
Next, we will do the following:
Do measurements for a given set of parameters
Compute 1-RDM, variances, and purification
Compute energy, fidelities, and errorbars
# 1.
# default to 250_000 shots for each circuit.
# 7 circuits total, printed for your viewing pleasure
# return value is a dictionary with circuit results for each permutation
measurement_data = opdm_func.calculate_data(parameters)
# 2.
opdm, var_dict = compute_opdm(measurement_data, return_variance=True)
opdm_pure = mcweeny_purification(opdm)
# 3.
raw_energies = []
raw_fidelity_witness = []
purified_eneriges = []
purified_fidelity_witness = []
purified_fidelity = []
true_unitary = ansatz(parameters)
nocc = molecule.n_electrons // 2
nvirt = molecule.n_orbitals - nocc
initial_fock_state = [1] * nocc + [0] * nvirt
# 1000 repetitions of the measurement
for _ in range(1000):
new_opdm = resample_opdm(opdm, var_dict)
raw_energies.append(opdm_func.energy_from_opdm(new_opdm))
raw_fidelity_witness.append(
fidelity_witness(target_unitary=true_unitary,
omega=initial_fock_state,
measured_opdm=new_opdm)
)
# fix positivity and trace of sampled 1-RDM if strictly outside
# feasible set
w, v = np.linalg.eigh(new_opdm)
if len(np.where(w < 0)[0]) > 0:
new_opdm = fixed_trace_positive_projection(new_opdm, nocc)
new_opdm_pure = mcweeny_purification(new_opdm)
purified_eneriges.append(opdm_func.energy_from_opdm(new_opdm_pure))
purified_fidelity_witness.append(
fidelity_witness(target_unitary=true_unitary,
omega=initial_fock_state,
measured_opdm=new_opdm_pure)
)
purified_fidelity.append(
fidelity(target_unitary=true_unitary,
measured_opdm=new_opdm_pure)
)
print("Canonical Hartree-Fock energy ", molecule.hf_energy)
print("True energy ", energy(parameters))
print("Raw energy ", opdm_func.energy_from_opdm(opdm),
"+- ", np.std(raw_energies))
print("Raw fidelity witness ", np.mean(raw_fidelity_witness).real,
"+- ", np.std(raw_fidelity_witness))
print("purified energy ", opdm_func.energy_from_opdm(opdm_pure),
"+- ", np.std(purified_eneriges))
print("Purified fidelity witness ", np.mean(purified_fidelity_witness).real,
"+- ", np.std(purified_fidelity_witness))
print("Purified fidelity ", np.mean(purified_fidelity).real,
"+- ", np.std(purified_fidelity))
/tmpfs/tmp/ipykernel_8722/2348574811.py:8: DeprecationWarning: `alltrue` is deprecated as of NumPy 1.25.0, and will be removed in NumPy 2.0. Please use `all` instead. opdm, var_dict = compute_opdm(measurement_data, return_variance=True) /tmpfs/tmp/ipykernel_8722/2348574811.py:8: DeprecationWarning: `alltrue` is deprecated as of NumPy 1.25.0, and will be removed in NumPy 2.0. Please use `all` instead. opdm, var_dict = compute_opdm(measurement_data, return_variance=True) /tmpfs/tmp/ipykernel_8722/2348574811.py:8: DeprecationWarning: `alltrue` is deprecated as of NumPy 1.25.0, and will be removed in NumPy 2.0. Please use `all` instead. opdm, var_dict = compute_opdm(measurement_data, return_variance=True) /tmpfs/tmp/ipykernel_8722/2348574811.py:8: DeprecationWarning: `alltrue` is deprecated as of NumPy 1.25.0, and will be removed in NumPy 2.0. Please use `all` instead. opdm, var_dict = compute_opdm(measurement_data, return_variance=True) /tmpfs/tmp/ipykernel_8722/2348574811.py:8: DeprecationWarning: `alltrue` is deprecated as of NumPy 1.25.0, and will be removed in NumPy 2.0. Please use `all` instead. opdm, var_dict = compute_opdm(measurement_data, return_variance=True) /tmpfs/tmp/ipykernel_8722/2348574811.py:8: DeprecationWarning: `alltrue` is deprecated as of NumPy 1.25.0, and will be removed in NumPy 2.0. Please use `all` instead. opdm, var_dict = compute_opdm(measurement_data, return_variance=True) Canonical Hartree-Fock energy -2.9240604849733085 True energy -2.924060484972228 Raw energy -2.9213044931760517 +- 0.0015177972177468448 Raw fidelity witness 0.997033277160794 +- 0.0020815856454034325 purified energy -2.9240559215724597 +- 5.5249068450963004e-06 Purified fidelity witness 0.999978892040198 +- 8.959420956931912e-06 Purified fidelity 0.999989446541326 +- 4.479680869499371e-06
This prints out the various energies estimated from the 1-RDM along with error bars. Generated from resampling the 1-RDM based on the estimated covariance.
Optimization
We use the sampling functionality to variationally relax the parameters of my ansatz such that the energy is decreased.
For this we will need the augmented Hessian optimizer
The optimizerer code we have takes: rhf_objective object, initial parameters, a function that takes a n x n unitary and returns an opdm maximum iterations, hassian_update which indicates how much of the hessian to use rtol which is the gradient stopping condition.
A natural thing that we will want to save is the variance dictionary of the non-purified 1-RDM. This is accomplished by wrapping the 1-RDM estimation code in another object that keeps track of the variance dictionaries.
from recirq.hfvqe.mfopt import moving_frame_augmented_hessian_optimizer
from recirq.hfvqe.opdm_functionals import RDMGenerator
rdm_generator = RDMGenerator(opdm_func, purification=True)
opdm_generator = rdm_generator.opdm_generator
result = moving_frame_augmented_hessian_optimizer(
rhf_objective=rhf_objective,
initial_parameters=parameters + 1.0E-1,
opdm_aa_measurement_func=opdm_generator,
verbose=True, delta=0.03,
max_iter=20,
hessian_update='diagonal',
rtol=0.50E-2)
ITERATION NUMBER : 0 unitary [[1. 0. 0. 0. 0. 0.] [0. 1. 0. 0. 0. 0.] [0. 0. 1. 0. 0. 0.] [0. 0. 0. 1. 0. 0.] [0. 0. 0. 0. 1. 0.] [0. 0. 0. 0. 0. 1.]] Current Energy: -2.826944816125242 true energy -2.826271292768161 dvec [((0.1299870221271307+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.14785875594365422+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.10661179110340353+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((0.3291181404785592+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.14195780888672413+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.26123255023886893+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((0.2049363275749181+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.21880083571875025+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.23952163627369352+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.05253760808447634 ITERATION NUMBER : 1 unitary [[ 0.98389642 0.01872322 -0.03839118 -0.09592923 -0.10702297 -0.09729877] [ 0.01872322 0.92317564 -0.00126495 0.34627269 -0.09792622 0.13379184] [-0.03839118 -0.00126495 0.85968895 -0.09412702 -0.4916874 -0.09403736] [ 0.09592923 -0.34627269 0.09412702 0.92774284 -0.01313115 -0.03396633] [ 0.10702297 0.09792622 0.4916874 -0.01313115 0.85816974 -0.02392281] [ 0.09729877 -0.13379184 0.09403736 -0.03396633 -0.02392281 0.98084844]] Current Energy: -2.856222940083195 true energy -2.8559010881780926 dvec [((0.12488407592470022+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.13514380712847476+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.0928458882062743+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((0.25154696956895983+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.12835758815319542+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.19546294860716798+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((0.17380260578353235+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.18455779836519476+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.18496326996440982+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.05629823462148727 ITERATION NUMBER : 2 unitary [[ 0.98880318 0.01613381 -0.02186015 -0.08789021 -0.08139252 -0.08473884] [ 0.02247946 0.91785167 0.00758451 0.35712186 -0.0867756 0.14805214] [-0.03343422 -0.00691063 0.87259735 -0.08269081 -0.47423973 -0.07528062] [ 0.08665783 -0.35787127 0.0827736 0.92542384 -0.00460151 -0.03371423] [ 0.07789632 0.08521881 0.4746078 -0.01322966 0.87223159 -0.02131639] [ 0.08406949 -0.14800734 0.07703534 -0.03629333 -0.01152461 0.98165253]] Current Energy: -2.8809521867113626 true energy -2.8808066255875593 dvec [((0.12277840454655436+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.11261923813504812+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.07029691288548916+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((0.16299967970151746+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.11133776730769589+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.1267692491358285+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((0.13688735744277908+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.15293025487822626+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.1409990538949392+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.06674752088983352 ITERATION NUMBER : 3 unitary [[ 9.92897134e-01 1.14708767e-02 -5.87132795e-03 -7.73607512e-02 -5.51144722e-02 -7.04765060e-02] [ 2.60723325e-02 9.10785024e-01 1.75064563e-02 3.70478768e-01 -7.25889396e-02 1.64197217e-01] [-2.93383420e-02 -1.42441206e-02 8.84573501e-01 -6.92757090e-02 -4.56626961e-01 -5.62030492e-02] [ 7.43882045e-02 -3.71979072e-01 6.89839555e-02 9.22036930e-01 4.77509757e-03 -3.41215130e-02] [ 4.84030407e-02 7.01558609e-02 4.57095462e-01 -1.51310369e-02 8.84964453e-01 -2.01991868e-02] [ 6.88354423e-02 -1.63847153e-01 5.91077422e-02 -3.97570692e-02 4.17009561e-04 9.81499500e-01]] Current Energy: -2.9018341899705713 true energy -2.901989720679283 dvec [((0.10823962290260936+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.07877886595937186+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.052326697973545604+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((0.07155652133150209+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.08810064764737702+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.0676933910532364+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((0.09645938650964846+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.10937796775160928+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.08697546770177016+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.07444466243528106 ITERATION NUMBER : 4 unitary [[ 0.99631107 0.0024377 0.00967737 -0.06067026 -0.0280509 -0.05288601] [ 0.02950918 0.9003074 0.02991369 0.38804047 -0.05219794 0.18542097] [-0.02600898 -0.02444426 0.89590486 -0.05330925 -0.43830834 -0.03353238] [ 0.05481648 -0.39040119 0.05165075 0.91673742 0.01532443 -0.03566417] [ 0.01848879 0.0499805 0.43855608 -0.02034989 0.89663691 -0.02137324] [ 0.04970227 -0.18420635 0.03696997 -0.04560952 0.01348072 0.97978006]] Current Energy: -2.916350091963808 true energy -2.916294431421 dvec [((0.05547658628322175+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.04768171050447512+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.04090104889062215+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((0.0036330103322619756+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.061088133844132475+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.021908877058534698+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((0.06256944537920127+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.04866892612100086+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.0352340320654215+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.05950386717666047 ITERATION NUMBER : 5 unitary [[ 0.99850536 -0.01284269 0.02063662 -0.03573815 -0.00766817 -0.03256109] [ 0.03177609 0.88621633 0.04449454 0.40823477 -0.02409787 0.21070072] [-0.02397584 -0.03814092 0.90506224 -0.03328546 -0.42143598 -0.01079631] [ 0.02560615 -0.41135435 0.02889711 0.90943257 0.02699468 -0.03873723] [-0.00338342 0.02383547 0.42123655 -0.03050134 0.90574463 -0.02601022] [ 0.02711757 -0.2078952 0.01346961 -0.05441792 0.02549832 0.97583365]] Current Energy: -2.9220503327626277 true energy -2.9220593319024903 dvec [((0.03169861974595674+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.01226242399530797+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.02649034661623485+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((-0.015886888428376496+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.031038505180421373+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.012123410003152812+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((0.021814630200472522+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.011648215480956944+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.017311740944372024+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.0314367978236929 ITERATION NUMBER : 6 unitary [[ 9.99161355e-01 -2.62376019e-02 2.17451708e-02 -1.76168094e-02 -4.99399605e-03 -1.34175502e-02] [ 3.22551566e-02 8.73733167e-01 5.74556381e-02 4.25533374e-01 4.34211221e-03 2.26166535e-01] [-2.33761239e-02 -5.23760674e-02 9.08937967e-01 -1.00499666e-02 -4.12841990e-01 1.60191396e-03] [ 4.19615182e-03 -4.28589205e-01 2.96808819e-03 9.01646261e-01 3.85557903e-02 -4.28059183e-02] [-5.67712120e-03 -2.17383857e-03 4.12369417e-01 -4.27947303e-02 9.09410455e-01 -3.24922207e-02] [ 6.31752633e-03 -2.22407861e-01 -6.50857543e-04 -6.09313131e-02 3.16823819e-02 9.72511188e-01]] Current Energy: -2.9235191222993095 true energy -2.923523540416502 dvec [((0.010470453868872218+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.006654678834670941+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.015406083888045022+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((-0.006161242409113891+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.016739936458211514+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((-0.0006175128415948361+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((0.014443853801779698+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.0068722360919028225+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.00934817472494195+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.016440600286779992 ITERATION NUMBER : 7 unitary [[ 9.99171401e-01 -3.33334717e-02 1.88278996e-02 -6.27296257e-03 -1.04553939e-02 -6.49892292e-03] [ 3.22155394e-02 8.70070997e-01 6.39639958e-02 4.29782935e-01 1.93073272e-02 2.29697805e-01] [-2.34065991e-02 -6.19620849e-02 9.10697245e-01 5.89293874e-03 -4.07615661e-01 7.62270620e-03] [-8.30652195e-03 -4.32253859e-01 -1.44183921e-02 8.99336180e-01 4.61478442e-02 -4.40967730e-02] [ 7.74008526e-04 -1.60701098e-02 4.07331432e-01 -4.97124101e-02 9.11064378e-01 -3.62302859e-02] [-1.09736318e-03 -2.25664337e-01 -7.60659340e-03 -6.27348664e-02 3.46332434e-02 9.71535598e-01]] Current Energy: -2.9238977258522887 true energy -2.9238985467981786 dvec [((0.0058177015338317685+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.00022287486490180086+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.005788189203595353+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((-0.003928357424124193+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.008270309225101262+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.003279243317207664+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((0.006193137507152789+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.00028777942950965196+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.011944724050746515+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.008557842065769396 ITERATION NUMBER : 8 unitary [[ 0.99910509 -0.03622306 0.01761532 -0.00266714 -0.01250181 -0.00178883] [ 0.03219019 0.86792442 0.0673857 0.43202715 0.02740345 0.2318011 ] [-0.02353155 -0.06747426 0.91035666 0.01528878 -0.40717511 0.01090527] [-0.01236178 -0.43417585 -0.02445977 0.89787205 0.05048702 -0.04491194] [ 0.00315061 -0.02366583 0.40702787 -0.05343483 0.9107369 -0.03822745] [-0.00602797 -0.22753808 -0.01138637 -0.06389193 0.03620154 0.97091076]] Current Energy: -2.9239992303712206 true energy -2.9240004898549405 dvec [((0.004493105551805139+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.005310059574327508+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.005828402878573054+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((0.0029030352679713465+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.006780753516660748+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.0027794006823822492+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((-0.00016796679356435092+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((-0.0014325599294213731+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((-0.0008584509999620546+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.006435670841346409 ITERATION NUMBER : 9 unitary [[ 9.99052004e-01 -3.76758170e-02 1.68360087e-02 -6.11585502e-04 -1.38473740e-02 2.27080694e-04] [ 3.22047921e-02 8.67744106e-01 6.91141391e-02 4.31884087e-01 3.12976015e-02 2.31738751e-01] [-2.36157511e-02 -7.03804624e-02 9.10689804e-01 1.85300826e-02 -4.05648111e-01 1.54665336e-02] [-1.46055432e-02 -4.33903863e-01 -2.81833503e-02 8.97761490e-01 5.21554756e-02 -4.49881008e-02] [ 4.73649086e-03 -2.73433225e-02 4.05630375e-01 -5.53007421e-02 9.11096296e-01 -3.92335905e-02] [-8.03026593e-03 -2.27215911e-01 -1.59236675e-02 -6.40197902e-02 3.82317413e-02 9.70821653e-01]] Current Energy: -2.9240309078794757 true energy -2.924038119315364 dvec [((0.00015690520220090396+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((-0.004966975009435104+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.0032159958543552603+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((-0.003721992361756998+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.0037069721834967324+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.0011342841449119567+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((0.00402905366738221+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((-0.0014770239849206807+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.0015238891270694986+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.00444774326842547 Finished Optimization
Each iteration prints out a variety of information that the user might find useful. Watching energies go down is known to be one of the best forms of entertainment during a shelter-in-place order.
After the optimization we can print the energy as a function of iteration number to see close the energy gets to the true minium.
import matplotlib.pyplot as plt
plt.semilogy(range(len(result.func_vals)),
np.abs(np.array(result.func_vals) - energy(parameters)),
'C0o-')
plt.xlabel("Optimization Iterations", fontsize=18)
plt.ylabel(r"$|E - E^{*}|$", fontsize=18)
plt.tight_layout()
plt.show()
