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))
Canonical Hartree-Fock energy -2.9240604849733085 True energy -2.9240604849722245 Raw energy -2.923310113426198 +- 0.0015387966847919925 Raw fidelity witness 0.9998962380218406 +- 0.0020851195410641725 purified energy -2.924053293202899 +- 6.782494183238227e-06 Purified fidelity witness 0.9999777468102539 +- 9.408923684089392e-06 Purified fidelity 0.9999888737162103 +- 4.704445649606818e-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: -2.7806001654427446 true energy -2.78086438438673 dvec [((0.2911168206360509+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.2925072879800671+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.3276477356656424+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((0.22723047667465024+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.2358797536275172+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.3286782257732593+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((0.47016052534132213+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.31157452190474044+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.28897191928863986+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.06521539918959801 ITERATION NUMBER : 1 unitary [[ 0.98631229 -0.00332115 0.00365105 -0.10006404 -0.08539822 -0.09928762] [-0.00332115 0.87717394 0.01627076 0.34088804 -0.09402936 -0.3244132 ] [ 0.00365105 0.01627076 0.94020018 -0.096165 0.31064861 -0.09997639] [ 0.10006404 -0.34088804 0.096165 0.92807028 0.02868246 0.04896628] [ 0.08539822 0.09402936 -0.31064861 0.02868246 0.94154959 -0.00497452] [ 0.09928762 0.3244132 0.09997639 0.04896628 -0.00497452 0.93406655]] Current Energy: -2.835028807279577 true energy -2.8350117950387927 dvec [((0.23365247173156004+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.2300811849902434+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.23957524121500343+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((0.19743096867270954+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.18843250872462958+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.26358064489649924+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((0.34335680776514943+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.25136757751079997+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.2241578324498773+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.06671397112242869 ITERATION NUMBER : 2 unitary [[ 9.91754397e-01 1.49912987e-04 2.79681237e-03 -7.99460714e-02 -7.02205932e-02 -7.13657141e-02] [ 1.52286835e-03 8.77928747e-01 1.36883859e-02 3.60559141e-01 -7.62447176e-02 -3.05344515e-01] [ 9.60437622e-03 1.37508107e-02 9.37982973e-01 -6.68677122e-02 3.30255587e-01 -7.97907827e-02] [ 8.03227992e-02 -3.58595449e-01 6.76250383e-02 9.25387047e-01 2.22821361e-02 5.95543972e-02] [ 6.96504132e-02 7.75009388e-02 -3.29929403e-01 2.52553246e-02 9.37888602e-01 4.01892303e-03] [ 7.08923163e-02 3.07335185e-01 8.10320382e-02 4.64265347e-02 -7.45212519e-03 9.44321034e-01]] Current Energy: -2.8764859703482992 true energy -2.8764618025436715 dvec [((0.17683705746535822+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.16967720016336202+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.15313562820949236+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((0.1624347501703286+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.14315767509065064+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.19819114480277386+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((0.22295128108154222+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.18777245009067767+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.14927245326897776+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.07156426974656199 ITERATION NUMBER : 3 unitary [[ 9.95814979e-01 1.77168397e-03 -3.25347965e-04 -5.93830522e-02 -5.32271291e-02 -4.46072629e-02] [ 5.35749376e-03 8.77679206e-01 9.52251665e-03 3.81210199e-01 -5.74948562e-02 -2.84487181e-01] [ 1.41341060e-02 1.01287182e-02 9.33816667e-01 -3.78044424e-02 3.50439818e-01 -5.87096841e-02] [ 5.94620987e-02 -3.77328551e-01 3.92710456e-02 9.20634691e-01 1.47357861e-02 6.89915315e-02] [ 5.14180963e-02 5.98214849e-02 -3.50006086e-01 2.02639448e-02 9.33118116e-01 1.23799508e-02] [ 4.40963637e-02 2.89167834e-01 6.20417076e-02 4.17660142e-02 -1.12510553e-02 9.53266654e-01]] Current Energy: -2.9059250981058153 true energy -2.905895665531206 dvec [((0.10712028116499778+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.10513303841986035+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.07085875963239022+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((0.11200883886140417+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.08737314114932304+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.12075922829429009+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((0.10678803862894143+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.12596682277908816+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.09044822158791423+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.08277819841336062 ITERATION NUMBER : 4 unitary [[ 9.98591343e-01 9.86400043e-04 -6.41256944e-03 -3.69353164e-02 -3.27280746e-02 -1.83818291e-02] [ 8.12140016e-03 8.75984796e-01 3.05541484e-03 4.04218236e-01 -3.62607992e-02 -2.60515100e-01] [ 1.71365894e-02 5.18838423e-03 9.27033915e-01 -9.13950489e-03 3.72586973e-01 -3.71881093e-02] [ 3.58653603e-02 -3.98397564e-01 1.12761182e-02 9.13129126e-01 5.98857558e-03 7.76224322e-02] [ 2.87912934e-02 3.95210117e-02 -3.72197627e-01 1.28924555e-02 9.26549089e-01 2.04608790e-02] [ 1.84512237e-02 2.68961365e-01 4.35825608e-02 3.44797472e-02 -1.62449206e-02 9.61232098e-01]] Current Energy: -2.9217148543517415 true energy -2.921758008688324 dvec [((0.022270467490081373+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.02870213638285865+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((-0.005212514590905883+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((0.04950006790751265+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.025221759828651956+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.02920109464386811+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((-0.006656462628961187+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.04651434937892024+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.03462350037627562+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.03932840510746992 ITERATION NUMBER : 5 unitary [[ 9.99739762e-01 -3.36920924e-03 -1.71758084e-02 -1.15830027e-02 -5.68179393e-03 6.89930605e-03] [ 9.34448431e-03 8.72221637e-01 -6.65551302e-03 4.32180784e-01 -1.13384255e-02 -2.28449141e-01] [ 1.83003594e-02 1.93772940e-04 9.16858975e-01 1.76339549e-02 3.98231732e-01 -1.16280184e-02] [ 8.14664068e-03 -4.24364978e-01 -1.46732042e-02 9.01260018e-01 -3.79067850e-03 8.57245937e-02] [-1.74488485e-03 1.53021802e-02 -3.97887127e-01 1.92798101e-03 9.16873017e-01 2.80853342e-02] [-5.36415907e-03 2.42689586e-01 2.23796074e-02 2.23899650e-02 -2.40862288e-02 9.69273278e-01]] Current Energy: -2.9236925009781163 true energy -2.923680902873468 dvec [((-0.0008831076418034247+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((-0.001417174180218167+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((-0.015429286227810035+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((-0.0024663081276074927+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.007661199505215627+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((-0.008090714558493084+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((-0.013348230219161834+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((-0.002010545482956594+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.013811467195305696+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.014037184113811685 ITERATION NUMBER : 6 unitary [[ 9.99517350e-01 -7.55504407e-03 -2.58652027e-02 -3.59224194e-03 1.37458626e-02 6.09310291e-03] [ 9.15214549e-03 8.70717965e-01 -1.17717755e-02 4.44663281e-01 9.14168456e-06 -2.09529129e-01] [ 1.80854748e-02 5.20622664e-03 9.12680862e-01 1.46771373e-02 4.07968970e-01 2.31430968e-03] [-8.73541912e-04 -4.36108894e-01 -1.10871822e-02 8.95448096e-01 -2.30987970e-03 8.86155500e-02] [-2.32618259e-02 4.07840751e-03 -4.07425084e-01 -3.77949400e-03 9.12404296e-01 3.08410693e-02] [-3.51346841e-03 2.27064839e-01 9.37744029e-03 1.43061975e-02 -2.97562749e-02 9.73268295e-01]] Current Energy: -2.9239124746768237 true energy -2.923923275884299 dvec [((0.009588389811279796+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((-0.00042682481350120316+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((-0.007988698496559227+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((-0.0012235326722757883+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.009014465452053857+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((7.232024338735266e-05+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((6.320446913860456e-05+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((-0.0017907260065313513+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.006850554560939836+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.008730083325537267 ITERATION NUMBER : 7 unitary [[ 9.99547615e-01 -8.44055434e-03 -2.54610175e-02 -3.97084926e-03 1.29140063e-02 1.58734465e-03] [ 9.06379060e-03 8.70847279e-01 -1.33766895e-02 4.44025170e-01 3.64004219e-03 -2.10219711e-01] [ 1.81545531e-02 1.11946831e-02 9.14036122e-01 5.11842720e-03 4.04981926e-01 6.81902169e-03] [-3.17517002e-04 -4.35431648e-01 -1.25848513e-03 8.95855379e-01 2.07804354e-03 8.85235369e-02] [-2.21769952e-02 6.55157362e-04 -4.04589782e-01 -5.50840016e-03 9.13663159e-01 3.16883243e-02] [ 9.51420034e-04 2.27656109e-01 4.03630530e-03 1.45766624e-02 -3.20145637e-02 9.73097163e-01]] Current Energy: -2.924000323250029 true energy -2.924011580243367 dvec [((-0.002506486236702607+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.000970222916061636+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((-0.0009972475304001272+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((0.0007410392613995068+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.006419835105327357+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.0010387340913406706+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((-0.0038948199868477553+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.003030044782189878+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.008797993873664919+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.005407411310355213 ITERATION NUMBER : 8 unitary [[ 9.99550786e-01 -1.01115340e-02 -2.52420351e-02 -5.16232450e-04 1.24427757e-02 1.93233999e-03] [ 8.96175359e-03 8.70772598e-01 -1.52575094e-02 4.43801445e-01 7.90442346e-03 -2.10759993e-01] [ 1.81687496e-02 1.42636280e-02 9.14025309e-01 1.69440189e-04 4.04897214e-01 9.07727969e-03] [-4.06453662e-03 -4.35146954e-01 3.92759243e-03 8.95969518e-01 4.28599430e-03 8.85179911e-02] [-2.16911717e-02 -3.39371787e-03 -4.04561828e-01 -7.57097885e-03 9.13631920e-01 3.26664495e-02] [ 8.84691687e-04 2.28221381e-01 1.44324578e-03 1.48756369e-02 -3.31555119e-02 9.72929372e-01]] Current Energy: -2.9240143866114994 true energy -2.924044410391702 dvec [((0.00589705645570891+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.00404162819963827+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((-0.006644391930941089+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((0.00017639805829225072+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.003111598236476687+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.0047583482414612355+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((-0.00012995239687356067+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((-0.00233567632163785+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((-4.8472672176985004e-05+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.006109753576019443 ITERATION NUMBER : 9 unitary [[ 9.99545355e-01 -9.97828078e-03 -2.53754603e-02 -1.44023218e-03 1.27757045e-02 5.56824670e-04] [ 8.88953193e-03 8.70820365e-01 -1.65944528e-02 4.44207076e-01 1.08885603e-02 -2.09473160e-01] [ 1.81680785e-02 1.53909500e-02 9.13861755e-01 -4.02985386e-04 4.05131773e-01 1.25625422e-02] [-3.04982425e-03 -4.35551155e-01 4.54429259e-03 8.95762262e-01 4.57562845e-03 8.86243347e-02] [-2.20476040e-02 -6.27665741e-03 -4.04861923e-01 -9.03923469e-03 9.13436888e-01 3.33567733e-02] [ 2.14053604e-03 2.27137048e-01 -1.89103574e-03 1.43561567e-02 -3.46205637e-02 9.73137154e-01]] Current Energy: -2.9240355355406593 true energy -2.9240425939686174 dvec [((-0.005190058381876124+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((-0.005883606666771651+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.0019075442067241885+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((0.005729838893404778+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((-0.0013449985210631638+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((-0.0033416372143696693+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((-0.005230313644305731+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((-0.0005180992693374492+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.002032508436276999+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.005124913817028045 ITERATION NUMBER : 10 unitary [[ 9.99533626e-01 -1.10223599e-02 -2.53904255e-02 6.78878622e-04 1.28598297e-02 7.26520586e-04] [ 8.84407915e-03 8.69761815e-01 -1.72292062e-02 4.45908089e-01 1.23918987e-02 -2.10123152e-01] [ 1.81474567e-02 1.73894657e-02 9.13097242e-01 -4.55627222e-03 4.06758244e-01 1.21932231e-02] [-5.35035662e-03 -4.37131362e-01 8.85693586e-03 8.94892170e-01 6.39807402e-03 8.90851777e-02] [-2.21708144e-02 -7.69864194e-03 -4.06484802e-01 -9.74716151e-03 9.12683189e-01 3.36701819e-02] [ 2.19335390e-03 2.27919729e-01 -1.88911427e-03 1.47564160e-02 -3.46014071e-02 9.72948718e-01]] Current Energy: -2.9240348869683186 true energy -2.9240398745392264 dvec [((0.004042167816728785+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((3.609654478309299e-05+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((0.002981406788836377+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((-0.006636552029626864+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((-0.0007365370578159158+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((-0.0024062148842306785+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((0.0066385112909700325+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.0009670130370709601+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((0.005726425249959032+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.005200993283179341 ITERATION NUMBER : 11 unitary [[ 0.99947936 -0.01053793 -0.02640569 -0.00124032 0.01516341 -0.00111345] [ 0.00887189 0.8709544 -0.01697012 0.44337877 0.01174756 -0.21059204] [ 0.01817401 0.0170084 0.91362188 -0.00333667 0.40557752 0.01308522] [-0.00326344 -0.43471497 0.00753572 0.89615454 0.00591946 0.0884729 ] [-0.02460974 -0.00704587 -0.4052724 -0.00944276 0.91317296 0.03354066] [ 0.00396503 0.22805999 -0.00270504 0.01484809 -0.03491472 0.97289573]] Current Energy: -2.9240347808531197 true energy -2.9240466574369943 dvec [((-0.0014939403782711948+0j), -1.0 [0^ 6] + -1.0 [1^ 7] + 1.0 [6^ 0] + 1.0 [7^ 1]), ((0.0008572335746679094+0j), -1.0 [2^ 6] + -1.0 [3^ 7] + 1.0 [6^ 2] + 1.0 [7^ 3]), ((-0.005459424058817685+0j), -1.0 [4^ 6] + -1.0 [5^ 7] + 1.0 [6^ 4] + 1.0 [7^ 5]), ((0.0039320590777360085+0j), -1.0 [0^ 8] + -1.0 [1^ 9] + 1.0 [8^ 0] + 1.0 [9^ 1]), ((0.0019486339517142386+0j), -1.0 [2^ 8] + -1.0 [3^ 9] + 1.0 [8^ 2] + 1.0 [9^ 3]), ((0.0010438622826220156+0j), -1.0 [4^ 8] + -1.0 [5^ 9] + 1.0 [8^ 4] + 1.0 [9^ 5]), ((-0.006087631242539529+0j), -1.0 [0^ 10] + -1.0 [1^ 11] + 1.0 [10^ 0] + 1.0 [11^ 1]), ((0.0020174858826655373+0j), -1.0 [2^ 10] + -1.0 [3^ 11] + 1.0 [10^ 2] + 1.0 [11^ 3]), ((-0.0022790059841688486+0j), -1.0 [4^ 10] + -1.0 [5^ 11] + 1.0 [10^ 4] + 1.0 [11^ 5])] New fr values norm 0.0049865164006613885 Finished Optimization
Each interation 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()
