import os, sys
import numpy as np
# path to access c++ files
installation_path = os.getenv("INSTALL_PATH")
sys.path.append(installation_path)

from cunqa import getQPUs

qpus  = getQPUs()

for q in qpus:
    print(f"QPU {q.id}, backend: {q.backend.name}, simulator: {q.backend.simulator}, version: {q.backend.version}.")

QPU 0, backend: BasicAer, simulator: AerSimulator, version: 0.0.1.
QPU 1, backend: BasicAer, simulator: AerSimulator, version: 0.0.1.
QPU 2, backend: BasicAer, simulator: AerSimulator, version: 0.0.1.
QPU 3, backend: BasicAer, simulator: AerSimulator, version: 0.0.1.
QPU 4, backend: BasicAer, simulator: AerSimulator, version: 0.0.1.

Paralelization for gradient-free optimizers: Differential Evolution

#TODO: Introduction and explanation

We recover the variational circuit used before:

from qiskit import QuantumCircuit
from qiskit.circuit import Parameter

def hardware_efficient_ansatz(num_qubits, num_layers):
    qc = QuantumCircuit(num_qubits)
    param_idx = 0
    for _ in range(num_layers):
        for qubit in range(num_qubits):
            phi = Parameter(f'phi_{param_idx}_{qubit}')
            lam = Parameter(f'lam_{param_idx}_{qubit}')
            qc.ry(phi, qubit)
            qc.rz(lam, qubit)
        param_idx += 1
        for qubit in range(num_qubits - 1):
            qc.cx(qubit, qubit + 1)
    qc.measure_all()
    return qc

def target_distribution(num_qubits):
    # Define a normal distribution over the states
    num_states = 2 ** num_qubits
    states = np.arange(num_states)
    mean = num_states / 2
    std_dev = num_states / 4
    target_probs = norm.pdf(states, mean, std_dev)
    target_probs /= target_probs.sum()  # Normalize to make it a valid probability distribution
    target_dist = {format(i, f'0{num_qubits}b'): target_probs[i] for i in range(num_states)}
    return target_dist

import pandas as pd
from scipy.stats import entropy, norm

def KL_divergence(counts, n_shots, target_dist):
    # Convert counts to probabilities
    pdf = pd.DataFrame.from_dict(counts, orient="index").reset_index()
    pdf.rename(columns={"index": "state", 0: "counts"}, inplace=True)
    pdf["probability"] = pdf["counts"] / n_shots

    # Create a dictionary for the obtained distribution
    obtained_dist = pdf.set_index("state")["probability"].to_dict()

    # Ensure all states are present in the obtained distribution
    for state in target_dist:
        if state not in obtained_dist:
            obtained_dist[state] = 0.0

    # Convert distributions to lists for KL divergence calculation
    target_probs = [target_dist[state] for state in sorted(target_dist)]
    obtained_probs = [obtained_dist[state] for state in sorted(obtained_dist)]

    # Calculate KL divergence
    kl_divergence = entropy(obtained_probs, target_probs)

    return kl_divergence

num_qubits = 6

num_layers = 3

n_shots = 999

target_dist = target_distribution(num_qubits)

def cost_function(result):

    global target_dist

    counts = result.get_counts()

    return KL_divergence(counts, n_shots, target_dist)

ansatz = hardware_efficient_ansatz(num_qubits, num_layers)

num_parameters = ansatz.num_parameters; print(num_parameters)

initial_parameters = np.zeros(num_parameters)

36

QJobMapper

init_qjobs = []
init_params = np.zeros(num_parameters)
for i in range(1*num_parameters):# we set pop=1 as the population size is pop*num_parameters
    qpu = qpus[i%len(qpus)]# we select the qpu
    init_qjobs.append(qpu.run(ansatz.assign_parameters(init_params), transpile=False, shots=n_shots))

from cunqa import QJobMapper
mapper = QJobMapper(init_qjobs)

pop=[]
total_pop=1*num_parameters
for j in range(total_pop):
    initial_point=np.random.uniform(-np.pi, np.pi, num_parameters)
    pop.append(initial_point)

bounds=[]
for i in range(0,num_parameters):
    bounds.append((-np.pi,np.pi))

print("Bounds:", len(bounds))
print("Initial population:", len(pop))

best_individual = []
energies = []

def cb(xk,convergence=1e-8):
    best_individual.append(xk)
    energy = mapper(cost_function, [xk])[0]
    energies.append(energy)

from scipy.optimize import differential_evolution
import time

tick = time.time()
result = differential_evolution(cost_function, bounds, maxiter=1000, disp=True, workers=mapper, updating='deferred',strategy='best1bin', init=pop, polish = False, callback=cb)
tack = time.time()
print(result)


print("Time:", tack-tick)

Bounds: 36
Initial population: 36
differential_evolution step 1: f(x)= 0.4204065631472025
differential_evolution step 2: f(x)= 0.4204065631472025
differential_evolution step 3: f(x)= 0.4204065631472025
differential_evolution step 4: f(x)= 0.4204065631472025
differential_evolution step 5: f(x)= 0.3765903247073238
differential_evolution step 6: f(x)= 0.3765903247073238
differential_evolution step 7: f(x)= 0.3765903247073238
differential_evolution step 8: f(x)= 0.3765903247073238
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differential_evolution step 21: f(x)= 0.3765903247073238
differential_evolution step 22: f(x)= 0.3388771721243059
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differential_evolution step 29: f(x)= 0.3114712870874271
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differential_evolution step 41: f(x)= 0.30470288752324576
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differential_evolution step 911: f(x)= 0.1946041244085572
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differential_evolution step 935: f(x)= 0.1946041244085572
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differential_evolution step 939: f(x)= 0.1946041244085572
differential_evolution step 940: f(x)= 0.1946041244085572
differential_evolution step 941: f(x)= 0.1946041244085572
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differential_evolution step 962: f(x)= 0.1946041244085572
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differential_evolution step 964: f(x)= 0.1946041244085572
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differential_evolution step 966: f(x)= 0.1946041244085572
differential_evolution step 967: f(x)= 0.1946041244085572
differential_evolution step 968: f(x)= 0.1946041244085572
differential_evolution step 969: f(x)= 0.1946041244085572
differential_evolution step 970: f(x)= 0.1946041244085572
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differential_evolution step 972: f(x)= 0.1946041244085572
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differential_evolution step 975: f(x)= 0.1946041244085572
differential_evolution step 976: f(x)= 0.1946041244085572
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differential_evolution step 978: f(x)= 0.1946041244085572
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differential_evolution step 980: f(x)= 0.1946041244085572
differential_evolution step 981: f(x)= 0.1946041244085572
differential_evolution step 982: f(x)= 0.1946041244085572
differential_evolution step 983: f(x)= 0.1946041244085572
differential_evolution step 984: f(x)= 0.1946041244085572
differential_evolution step 985: f(x)= 0.1946041244085572
differential_evolution step 986: f(x)= 0.1946041244085572
differential_evolution step 987: f(x)= 0.1946041244085572
differential_evolution step 988: f(x)= 0.1946041244085572
differential_evolution step 989: f(x)= 0.1946041244085572
differential_evolution step 990: f(x)= 0.1946041244085572
differential_evolution step 991: f(x)= 0.1946041244085572
differential_evolution step 992: f(x)= 0.1946041244085572
differential_evolution step 993: f(x)= 0.1946041244085572
differential_evolution step 994: f(x)= 0.1946041244085572
differential_evolution step 995: f(x)= 0.1946041244085572
differential_evolution step 996: f(x)= 0.1946041244085572
differential_evolution step 997: f(x)= 0.1946041244085572
differential_evolution step 998: f(x)= 0.1946041244085572
differential_evolution step 999: f(x)= 0.1946041244085572
differential_evolution step 1000: f(x)= 0.1946041244085572
             message: Maximum number of iterations has been exceeded.
             success: False
                 fun: 0.1946041244085572
                   x: [-2.130e-01 -1.847e-01 ... -1.793e+00 -9.677e-03]
                 nit: 1000
                nfev: 36036
          population: [[-2.130e-01 -1.847e-01 ... -1.793e+00 -9.677e-03]
                       [ 8.804e-01 -1.213e+00 ... -1.852e+00  2.361e+00]
                       ...
                       [-2.582e-02 -1.152e+00 ... -1.218e+00 -3.756e-01]
                       [-4.615e-01  1.583e+00 ... -2.322e+00 -2.514e-01]]
 population_energies: [ 1.946e-01  2.096e-01 ...  2.945e-01  3.171e-01]
Time: 92.09724497795105

QPUCircuitMapper

from cunqa import QJobMapper, QPUCircuitMapper
mapper = QPUCircuitMapper(qpus, ansatz, transpile=False, shots=n_shots)

pop=[]
total_pop=1*num_parameters
for j in range(total_pop):
    initial_point=np.random.uniform(-np.pi, np.pi, num_parameters)
    pop.append(initial_point)

bounds=[]
for i in range(0,num_parameters):
    bounds.append((-np.pi,np.pi))

print("Bounds:", len(bounds))
print("Initial population:", len(pop))

best_individual_ = []
energies_ = []

def cb(xk,convergence=1e-8):
    best_individual_.append(xk)
    energy = mapper(cost_function, [xk])[0]
    energies_.append(energy)

from scipy.optimize import differential_evolution
import time

tick = time.time()
result_ = differential_evolution(cost_function, bounds, maxiter=1000, disp=True, workers=mapper, updating='deferred',strategy='best1bin', init=pop, polish = False, callback=cb)
tack = time.time()
print(result_)


print("Time:", tack-tick)

Bounds: 36
Initial population: 36
differential_evolution step 1: f(x)= 0.3902868537063659
differential_evolution step 2: f(x)= 0.35153686053900457
differential_evolution step 3: f(x)= 0.35153686053900457
differential_evolution step 4: f(x)= 0.35153686053900457
differential_evolution step 5: f(x)= 0.35153686053900457
differential_evolution step 6: f(x)= 0.35153686053900457
differential_evolution step 7: f(x)= 0.35153686053900457
differential_evolution step 8: f(x)= 0.35153686053900457
differential_evolution step 9: f(x)= 0.35153686053900457
differential_evolution step 10: f(x)= 0.35153686053900457
differential_evolution step 11: f(x)= 0.35153686053900457
differential_evolution step 12: f(x)= 0.35153686053900457
differential_evolution step 13: f(x)= 0.35153686053900457
differential_evolution step 14: f(x)= 0.35153686053900457
differential_evolution step 15: f(x)= 0.35153686053900457
differential_evolution step 16: f(x)= 0.35153686053900457
differential_evolution step 17: f(x)= 0.35153686053900457
differential_evolution step 18: f(x)= 0.35153686053900457
differential_evolution step 19: f(x)= 0.35153686053900457
differential_evolution step 20: f(x)= 0.35153686053900457
differential_evolution step 21: f(x)= 0.35153686053900457
differential_evolution step 22: f(x)= 0.35153686053900457
differential_evolution step 23: f(x)= 0.35153686053900457
differential_evolution step 24: f(x)= 0.35153686053900457
differential_evolution step 25: f(x)= 0.35153686053900457
differential_evolution step 26: f(x)= 0.32071343207615494
differential_evolution step 27: f(x)= 0.32071343207615494
differential_evolution step 28: f(x)= 0.32071343207615494
differential_evolution step 29: f(x)= 0.32071343207615494
differential_evolution step 30: f(x)= 0.32071343207615494
differential_evolution step 31: f(x)= 0.32071343207615494
differential_evolution step 32: f(x)= 0.32071343207615494
differential_evolution step 33: f(x)= 0.32071343207615494
differential_evolution step 34: f(x)= 0.32071343207615494
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differential_evolution step 758: f(x)= 0.20621643229365255
differential_evolution step 759: f(x)= 0.20621643229365255
differential_evolution step 760: f(x)= 0.20621643229365255
differential_evolution step 761: f(x)= 0.20621643229365255
differential_evolution step 762: f(x)= 0.20621643229365255
differential_evolution step 763: f(x)= 0.20621643229365255
differential_evolution step 764: f(x)= 0.20621643229365255
differential_evolution step 765: f(x)= 0.20621643229365255
differential_evolution step 766: f(x)= 0.20621643229365255
differential_evolution step 767: f(x)= 0.20621643229365255
differential_evolution step 768: f(x)= 0.20621643229365255
differential_evolution step 769: f(x)= 0.20621643229365255
differential_evolution step 770: f(x)= 0.20621643229365255
differential_evolution step 771: f(x)= 0.20621643229365255
differential_evolution step 772: f(x)= 0.20621643229365255
differential_evolution step 773: f(x)= 0.20621643229365255
differential_evolution step 774: f(x)= 0.20621643229365255
differential_evolution step 775: f(x)= 0.20621643229365255
differential_evolution step 776: f(x)= 0.20621643229365255
differential_evolution step 777: f(x)= 0.20621643229365255
differential_evolution step 778: f(x)= 0.20621643229365255
differential_evolution step 779: f(x)= 0.1752673007036788
differential_evolution step 780: f(x)= 0.1752673007036788
differential_evolution step 781: f(x)= 0.1752673007036788
differential_evolution step 782: f(x)= 0.1752673007036788
differential_evolution step 783: f(x)= 0.1752673007036788
differential_evolution step 784: f(x)= 0.1752673007036788
differential_evolution step 785: f(x)= 0.1752673007036788
differential_evolution step 786: f(x)= 0.1752673007036788
differential_evolution step 787: f(x)= 0.1752673007036788
differential_evolution step 788: f(x)= 0.1752673007036788
differential_evolution step 789: f(x)= 0.1752673007036788
differential_evolution step 790: f(x)= 0.1752673007036788
differential_evolution step 791: f(x)= 0.1752673007036788
differential_evolution step 792: f(x)= 0.1752673007036788
differential_evolution step 793: f(x)= 0.1752673007036788
differential_evolution step 794: f(x)= 0.1752673007036788
differential_evolution step 795: f(x)= 0.1752673007036788
differential_evolution step 796: f(x)= 0.1752673007036788
differential_evolution step 797: f(x)= 0.1752673007036788
differential_evolution step 798: f(x)= 0.1752673007036788
differential_evolution step 799: f(x)= 0.1752673007036788
differential_evolution step 800: f(x)= 0.1752673007036788
differential_evolution step 801: f(x)= 0.1752673007036788
differential_evolution step 802: f(x)= 0.1752673007036788
differential_evolution step 803: f(x)= 0.1752673007036788
differential_evolution step 804: f(x)= 0.1752673007036788
differential_evolution step 805: f(x)= 0.1752673007036788
differential_evolution step 806: f(x)= 0.1752673007036788
differential_evolution step 807: f(x)= 0.1752673007036788
differential_evolution step 808: f(x)= 0.1752673007036788
differential_evolution step 809: f(x)= 0.1752673007036788
differential_evolution step 810: f(x)= 0.1752673007036788
differential_evolution step 811: f(x)= 0.1752673007036788
differential_evolution step 812: f(x)= 0.1752673007036788
differential_evolution step 813: f(x)= 0.1752673007036788
differential_evolution step 814: f(x)= 0.1752673007036788
differential_evolution step 815: f(x)= 0.1752673007036788
differential_evolution step 816: f(x)= 0.1752673007036788
differential_evolution step 817: f(x)= 0.1752673007036788
differential_evolution step 818: f(x)= 0.1752673007036788
differential_evolution step 819: f(x)= 0.1752673007036788
differential_evolution step 820: f(x)= 0.1752673007036788
differential_evolution step 821: f(x)= 0.1752673007036788
differential_evolution step 822: f(x)= 0.1752673007036788
differential_evolution step 823: f(x)= 0.1752673007036788
differential_evolution step 824: f(x)= 0.1752673007036788
differential_evolution step 825: f(x)= 0.1752673007036788
differential_evolution step 826: f(x)= 0.1752673007036788
differential_evolution step 827: f(x)= 0.1752673007036788
differential_evolution step 828: f(x)= 0.1752673007036788
differential_evolution step 829: f(x)= 0.1752673007036788
differential_evolution step 830: f(x)= 0.1752673007036788
differential_evolution step 831: f(x)= 0.1752673007036788
differential_evolution step 832: f(x)= 0.1752673007036788
differential_evolution step 833: f(x)= 0.1752673007036788
differential_evolution step 834: f(x)= 0.1752673007036788
differential_evolution step 835: f(x)= 0.1752673007036788
differential_evolution step 836: f(x)= 0.1752673007036788
differential_evolution step 837: f(x)= 0.1752673007036788
differential_evolution step 838: f(x)= 0.1752673007036788
differential_evolution step 839: f(x)= 0.1752673007036788
differential_evolution step 840: f(x)= 0.1752673007036788
differential_evolution step 841: f(x)= 0.1752673007036788
differential_evolution step 842: f(x)= 0.1752673007036788
differential_evolution step 843: f(x)= 0.1752673007036788
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differential_evolution step 850: f(x)= 0.1752673007036788
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differential_evolution step 857: f(x)= 0.1752673007036788
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differential_evolution step 859: f(x)= 0.1752673007036788
differential_evolution step 860: f(x)= 0.1752673007036788
differential_evolution step 861: f(x)= 0.1752673007036788
differential_evolution step 862: f(x)= 0.1752673007036788
differential_evolution step 863: f(x)= 0.1752673007036788
differential_evolution step 864: f(x)= 0.1752673007036788
differential_evolution step 865: f(x)= 0.1752673007036788
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differential_evolution step 867: f(x)= 0.1752673007036788
differential_evolution step 868: f(x)= 0.1752673007036788
differential_evolution step 869: f(x)= 0.1752673007036788
differential_evolution step 870: f(x)= 0.1752673007036788
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differential_evolution step 872: f(x)= 0.1752673007036788
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differential_evolution step 885: f(x)= 0.1752673007036788
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differential_evolution step 887: f(x)= 0.1752673007036788
differential_evolution step 888: f(x)= 0.1752673007036788
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differential_evolution step 894: f(x)= 0.1752673007036788
differential_evolution step 895: f(x)= 0.1752673007036788
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differential_evolution step 897: f(x)= 0.1752673007036788
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differential_evolution step 900: f(x)= 0.1752673007036788
differential_evolution step 901: f(x)= 0.1752673007036788
differential_evolution step 902: f(x)= 0.1752673007036788
differential_evolution step 903: f(x)= 0.1752673007036788
differential_evolution step 904: f(x)= 0.1752673007036788
differential_evolution step 905: f(x)= 0.1752673007036788
differential_evolution step 906: f(x)= 0.1752673007036788
differential_evolution step 907: f(x)= 0.1752673007036788
differential_evolution step 908: f(x)= 0.1752673007036788
differential_evolution step 909: f(x)= 0.1752673007036788
differential_evolution step 910: f(x)= 0.1752673007036788
differential_evolution step 911: f(x)= 0.1752673007036788
differential_evolution step 912: f(x)= 0.1752673007036788
differential_evolution step 913: f(x)= 0.1752673007036788
differential_evolution step 914: f(x)= 0.1752673007036788
differential_evolution step 915: f(x)= 0.1752673007036788
differential_evolution step 916: f(x)= 0.1752673007036788
differential_evolution step 917: f(x)= 0.1752673007036788
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differential_evolution step 920: f(x)= 0.1752673007036788
differential_evolution step 921: f(x)= 0.1752673007036788
differential_evolution step 922: f(x)= 0.1752673007036788
differential_evolution step 923: f(x)= 0.1752673007036788
differential_evolution step 924: f(x)= 0.1752673007036788
differential_evolution step 925: f(x)= 0.1752673007036788
differential_evolution step 926: f(x)= 0.1752673007036788
differential_evolution step 927: f(x)= 0.1752673007036788
differential_evolution step 928: f(x)= 0.1752673007036788
differential_evolution step 929: f(x)= 0.1752673007036788
differential_evolution step 930: f(x)= 0.1752673007036788
differential_evolution step 931: f(x)= 0.1752673007036788
differential_evolution step 932: f(x)= 0.1752673007036788
differential_evolution step 933: f(x)= 0.1752673007036788
differential_evolution step 934: f(x)= 0.1752673007036788
differential_evolution step 935: f(x)= 0.1752673007036788
differential_evolution step 936: f(x)= 0.1752673007036788
differential_evolution step 937: f(x)= 0.1752673007036788
differential_evolution step 938: f(x)= 0.1752673007036788
differential_evolution step 939: f(x)= 0.1752673007036788
differential_evolution step 940: f(x)= 0.1752673007036788
differential_evolution step 941: f(x)= 0.1752673007036788
differential_evolution step 942: f(x)= 0.1752673007036788
differential_evolution step 943: f(x)= 0.1752673007036788
differential_evolution step 944: f(x)= 0.1752673007036788
differential_evolution step 945: f(x)= 0.1752673007036788
differential_evolution step 946: f(x)= 0.1752673007036788
differential_evolution step 947: f(x)= 0.1752673007036788
differential_evolution step 948: f(x)= 0.1752673007036788
differential_evolution step 949: f(x)= 0.1752673007036788
differential_evolution step 950: f(x)= 0.1752673007036788
differential_evolution step 951: f(x)= 0.1752673007036788
differential_evolution step 952: f(x)= 0.1752673007036788
differential_evolution step 953: f(x)= 0.1752673007036788
differential_evolution step 954: f(x)= 0.1752673007036788
differential_evolution step 955: f(x)= 0.1752673007036788
differential_evolution step 956: f(x)= 0.1752673007036788
differential_evolution step 957: f(x)= 0.1752673007036788
differential_evolution step 958: f(x)= 0.1752673007036788
differential_evolution step 959: f(x)= 0.1752673007036788
differential_evolution step 960: f(x)= 0.1752673007036788
differential_evolution step 961: f(x)= 0.1752673007036788
differential_evolution step 962: f(x)= 0.1752673007036788
differential_evolution step 963: f(x)= 0.1752673007036788
differential_evolution step 964: f(x)= 0.1752673007036788
differential_evolution step 965: f(x)= 0.1752673007036788
differential_evolution step 966: f(x)= 0.1752673007036788
differential_evolution step 967: f(x)= 0.1752673007036788
differential_evolution step 968: f(x)= 0.1752673007036788
differential_evolution step 969: f(x)= 0.1752673007036788
differential_evolution step 970: f(x)= 0.1752673007036788
differential_evolution step 971: f(x)= 0.1752673007036788
differential_evolution step 972: f(x)= 0.1752673007036788
differential_evolution step 973: f(x)= 0.1752673007036788
differential_evolution step 974: f(x)= 0.1752673007036788
differential_evolution step 975: f(x)= 0.1752673007036788
differential_evolution step 976: f(x)= 0.1752673007036788
differential_evolution step 977: f(x)= 0.1752673007036788
differential_evolution step 978: f(x)= 0.1752673007036788
differential_evolution step 979: f(x)= 0.1752673007036788
differential_evolution step 980: f(x)= 0.1752673007036788
differential_evolution step 981: f(x)= 0.1752673007036788
differential_evolution step 982: f(x)= 0.1752673007036788
differential_evolution step 983: f(x)= 0.1752673007036788
differential_evolution step 984: f(x)= 0.1752673007036788
differential_evolution step 985: f(x)= 0.1752673007036788
differential_evolution step 986: f(x)= 0.1752673007036788
differential_evolution step 987: f(x)= 0.1752673007036788
differential_evolution step 988: f(x)= 0.1752673007036788
differential_evolution step 989: f(x)= 0.1752673007036788
differential_evolution step 990: f(x)= 0.1752673007036788
differential_evolution step 991: f(x)= 0.1752673007036788
differential_evolution step 992: f(x)= 0.1752673007036788
differential_evolution step 993: f(x)= 0.1752673007036788
differential_evolution step 994: f(x)= 0.1752673007036788
differential_evolution step 995: f(x)= 0.1752673007036788
differential_evolution step 996: f(x)= 0.1752673007036788
differential_evolution step 997: f(x)= 0.1752673007036788
differential_evolution step 998: f(x)= 0.1752673007036788
differential_evolution step 999: f(x)= 0.1752673007036788
differential_evolution step 1000: f(x)= 0.1752673007036788
             message: Maximum number of iterations has been exceeded.
             success: False
                 fun: 0.1752673007036788
                   x: [ 2.382e-01  1.605e-01 ...  2.693e-01  1.814e+00]
                 nit: 1000
                nfev: 36036
          population: [[ 2.382e-01  1.605e-01 ...  2.693e-01  1.814e+00]
                       [ 6.489e-01  2.460e+00 ...  7.980e-01  2.236e+00]
                       ...
                       [ 3.022e-01  2.399e+00 ...  2.507e+00  1.413e+00]
                       [ 2.202e+00  1.065e+00 ...  2.880e+00  1.779e+00]]
 population_energies: [ 1.753e-01  2.096e-01 ...  2.904e-01  2.062e-01]
Time: 127.24817490577698

%matplotlib inline

import matplotlib.pyplot as plt
plt.clf()
plt.plot(np.linspace(0, result.nit, result.nit), energies, label="Optimization path (QJobMapper)")
upper_bound = result.nit
plt.plot(np.linspace(0, result_.nit, result_.nit), energies_, label="Optimization path (QPUCircuitMapper)")
plt.plot(np.linspace(0, upper_bound, upper_bound), np.zeros(upper_bound), "--", label="Target cost")
plt.xlabel("Step"); plt.ylabel("Cost"); plt.legend(loc="upper right"); plt.title(f"n = {num_qubits}, l = {num_layers}, # params = {num_parameters}")
plt.grid(True)
plt.show()
# plt.savefig(f"optimization_de_n_{num_qubits}_p_{num_parameters}.png", dpi=200)

Paralelization of expectation value terms

# TODO

Paralelization for gradient optimizers

# TODO