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Esta página foi gerada a partir do tutorials/optimization/8_cvar_optimization.ipynb.

Execute interativamente no IBM Quantum lab.

Melhorando a Otimização Quântica Variacional usando o CVaR

Introdução

Este notebook mostra como utilizar a função objetiva de Valor em Risco Condicional (CVaR) introduzida em [1] dentro dos algoritmos de otimização variacional quântica fornecidos pelo Qiskit. Particularmente, é mostrado como configurar o MinimumEigenOptimizer utilizando o VQE adequadamente. Para um determinado conjunto de repetições com valores objetivos correspondentes do problema de otimização considerado, o CVaR com nível de confiança \(\alpha \in [0, 1]\) é definido como a média das \(\alpha\) melhores repetições. Assim, \(\alpha = 1\) corresponde ao valor esperado padrão, enquanto \(\alpha=0\) corresponde ao mínimo das repetições dadas, e \(\alpha \in (0, 1)\) é um tradeoff entre focar em melhores doses, mas ainda aplicando alguma média para suavizar o cenário de otimização.

Referências

[1] P. Barkoutsos et al., Improving Variational Quantum Optimization using CVaR, Quantum 4, 256 (2020).

from qiskit.circuit.library import RealAmplitudes
from qiskit.aqua.components.optimizers import COBYLA
from qiskit.aqua.algorithms import NumPyMinimumEigensolver, VQE
from qiskit.aqua.operators import PauliExpectation, CVaRExpectation
from qiskit.optimization import QuadraticProgram
from qiskit.optimization.converters import LinearEqualityToPenalty
from qiskit.optimization.algorithms import MinimumEigenOptimizer
from qiskit import execute, Aer
from qiskit.aqua import aqua_globals

import numpy as np
import matplotlib.pyplot as plt
from docplex.mp.model import Model

aqua_globals.random_seed = 123456

Otimização de Portfólio

Em seguida definimos uma instância do problema para otimização de portfólio como introduzido em [1].

# prepare problem instance
n = 6            # number of assets
q = 0.5          # risk factor
budget = n // 2  # budget
penalty = 2*n    # scaling of penalty term

# instance from [1]
mu = np.array([0.7313, 0.9893, 0.2725, 0.8750, 0.7667, 0.3622])
sigma = np.array([
    [ 0.7312, -0.6233,  0.4689, -0.5452, -0.0082, -0.3809],
    [-0.6233,  2.4732, -0.7538,  2.4659, -0.0733,  0.8945],
    [ 0.4689, -0.7538,  1.1543, -1.4095,  0.0007, -0.4301],
    [-0.5452,  2.4659, -1.4095,  3.5067,  0.2012,  1.0922],
    [-0.0082, -0.0733,  0.0007,  0.2012,  0.6231,  0.1509],
    [-0.3809,  0.8945, -0.4301,  1.0922,  0.1509,  0.8992]
])

# or create random instance
# mu, sigma = portfolio.random_model(n, seed=123)  # expected returns and covariance matrix

# create docplex model
mdl = Model('portfolio_optimization')
x = mdl.binary_var_list('x{}'.format(i) for i in range(n))
objective = mdl.sum([mu[i]*x[i] for i in range(n)])
objective -= q * mdl.sum([sigma[i,j]*x[i]*x[j] for i in range(n) for j in range(n)])
mdl.maximize(objective)
mdl.add_constraint(mdl.sum(x[i] for i in range(n)) == budget)

# case to
qp = QuadraticProgram()
qp.from_docplex(mdl)

# solve classically as reference
opt_result = MinimumEigenOptimizer(NumPyMinimumEigensolver()).solve(qp)
opt_result

optimal function value: 1.27835
optimal value: [1. 1. 0. 0. 1. 0.]
status: SUCCESS

# we convert the problem to an unconstrained problem for further analysis,
# otherwise this would not be necessary as the MinimumEigenSolver would do this
# translation automatically
linear2penalty = LinearEqualityToPenalty(penalty=penalty)
qp = linear2penalty.convert(qp)
_, offset = qp.to_ising()

Otimizador Eigen mínimo usando VQE

# set classical optimizer
maxiter = 100
optimizer = COBYLA(maxiter=maxiter)

# set variational ansatz
var_form = RealAmplitudes(n, reps=1)
m = var_form.num_parameters

# set backend
backend_name = 'qasm_simulator'  # use this for QASM simulator
# backend_name = 'statevector_simulator'  # use this for statevector simlator
backend = Aer.get_backend(backend_name)

# run variational optimization for different values of alpha
alphas = [1.0, 0.50, 0.25]  # confidence levels to be evaluated

# dictionaries to store optimization progress and results
objectives = {alpha: [] for alpha in alphas}  # set of tested objective functions w.r.t. alpha
results = {}  # results of minimum eigensolver w.r.t alpha

# callback to store intermediate results
def callback(i, params, obj, stddev, alpha):
    # we translate the objective from the internal Ising representation
    # to the original optimization problem
    objectives[alpha] += [-(obj + offset)]

# loop over all given alpha values
for alpha in alphas:

    # initialize CVaR_alpha objective
    cvar_exp = CVaRExpectation(alpha, PauliExpectation())
    cvar_exp.compute_variance = lambda x: [0]  # to be fixed in PR #1373

    # initialize VQE using CVaR
    vqe = VQE(expectation=cvar_exp, optimizer=optimizer, var_form=var_form, quantum_instance=backend,
              callback=lambda i, params, obj, stddev: callback(i, params, obj, stddev, alpha))

    # initialize optimization algorithm based on CVaR-VQE
    opt_alg = MinimumEigenOptimizer(vqe)

    # solve problem
    results[alpha] = opt_alg.solve(qp)

    # print results
    print('alpha = {}:'.format(alpha))
    print(results[alpha])
    print()

alpha = 1.0:
optimal function value: 0.7295999999999907
optimal value: [0. 1. 1. 0. 1. 0.]
status: SUCCESS

alpha = 0.5:
optimal function value: 0.7295999999999907
optimal value: [0. 1. 1. 0. 1. 0.]
status: SUCCESS

alpha = 0.25:
optimal function value: 1.2783500000000068
optimal value: [1. 1. 0. 0. 1. 0.]
status: SUCCESS

# plot resulting history of objective values
plt.figure(figsize=(10, 5))
plt.plot([0, maxiter], [opt_result.fval, opt_result.fval], 'r--', linewidth=2, label='optimum')
for alpha in alphas:
    plt.plot(objectives[alpha], label='alpha = %.2f' % alpha, linewidth=2)
plt.legend(loc='lower right', fontsize=14)
plt.xlim(0, maxiter)
plt.xticks(fontsize=14)
plt.xlabel('iterations', fontsize=14)
plt.yticks(fontsize=14)
plt.ylabel('objective value', fontsize=14)
plt.show()

# evaluate and sort all objective values
objective_values = np.zeros(2**n)
for i in range(2**n):
    x_bin = ('{0:0%sb}' % n).format(i)
    x = [0 if x_ == '0' else 1 for x_ in reversed(x_bin)]
    objective_values[i] = qp.objective.evaluate(x)
ind = np.argsort(objective_values)

# evaluate final optimal probability for each alpha
probabilities = np.zeros(len(objective_values))
for alpha in alphas:
    if backend_name == 'qasm_simulator':
        counts = results[alpha].min_eigen_solver_result.eigenstate
        shots = sum(counts.values())
        for key, val in counts.items():
            i = int(key, 2)
            probabilities[i] = val / shots
    else:
        probabilities = np.abs(results[alpha].min_eigen_solver_result.eigenstate)**2
    print('optimal probabilitiy (alpha = %.2f):  %.4f' % (alpha, probabilities[ind][-1:]))

optimal probabilitiy (alpha = 1.00):  0.0000
optimal probabilitiy (alpha = 0.50):  0.0098
optimal probabilitiy (alpha = 0.25):  0.2402

import qiskit.tools.jupyter
%qiskit_version_table
%qiskit_copyright

Version Information

Qiskit Software	Version
Qiskit	0.23.0
Terra	0.16.0
Aer	0.7.0
Ignis	0.5.0
Aqua	0.8.0
IBM Q Provider	0.11.0
System information
Python	3.7.4 (default, Aug 13 2019, 15:17:50) [Clang 4.0.1 (tags/RELEASE_401/final)]
OS	Darwin
CPUs	6
Memory (Gb)	16.0
Mon Oct 19 23:44:24 2020 CEST

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