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This page was generated from tutorials/optimization/8_cvar_optimization.ipynb.

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Improving Variational Quantum Optimization using CVaR

Introduction

This notebook shows how to use the Conditional Value at Risk (CVaR) objective function introduced in [1] within the variational quantum optimization algorithms provided by Qiskit. Particularly, it is shown how to setup the MinimumEigenOptimizer using VQE accordingly. For a given set of shots with corresponding objective values of the considered optimization problem, the CVaR with confidence level \(\alpha \in [0, 1]\) is defined as the average of the \(\alpha\) best shots. Thus, \(\alpha = 1\) corresponds to the standard expected value, while \(\alpha=0\) corresponds to the minimum of the given shots, and \(\alpha \in (0, 1)\) is a tradeoff between focusing on better shots, but still applying some averaging to smoothen the optimization landscape.

References

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

[1]:
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
[2]:
aqua_globals.random_seed = 123456

Portfolio Optimization

In the following we define a problem instance for portfolio optimization as introduced in [1].

[3]:
# prepare problem instance
n = 6            # number of assets
q = 0.5          # risk factor
budget = n // 2  # budget
penalty = 2*n    # scaling of penalty term
[4]:
# 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
[5]:
# 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)
[6]:
# solve classically as reference
opt_result = MinimumEigenOptimizer(NumPyMinimumEigensolver()).solve(qp)
opt_result
[6]:
optimal function value: 1.27835
optimal value: [1. 1. 0. 0. 1. 0.]
status: SUCCESS
[7]:
# 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()

Minimum Eigen Optimizer using VQE

[8]:
# 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
[9]:
# 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: 1.2783500000000068
optimal value: [1. 1. 0. 0. 1. 0.]
status: SUCCESS

alpha = 0.5:
optimal function value: 0.9716500000000075
optimal value: [1. 0. 0. 0. 1. 1.]
status: SUCCESS

alpha = 0.25:
optimal function value: 0.9716500000000075
optimal value: [1. 0. 0. 0. 1. 1.]
status: SUCCESS

[10]:
# 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()
../../_images/tutorials_optimization_8_cvar_optimization_13_0.png
[11]:
# 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):  0.0107
optimal probabilitiy (alpha = 0.25):  0.3516
[12]:
import qiskit.tools.jupyter
%qiskit_version_table
%qiskit_copyright

Version Information

Qiskit SoftwareVersion
Qiskit0.23.6
Terra0.16.4
Aer0.7.5
Ignis0.5.2
Aqua0.8.2
IBM Q Provider0.11.1
System information
Python3.8.7 (default, Jan 25 2021, 16:23:06) [GCC 9.3.0]
OSLinux
CPUs2
Memory (Gb)6.791378021240234
Thu Feb 18 22:54:42 2021 UTC

This code is a part of Qiskit

© Copyright IBM 2017, 2021.

This code is licensed under the Apache License, Version 2.0. You may
obtain a copy of this license in the LICENSE.txt file in the root directory
of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.

Any modifications or derivative works of this code must retain this
copyright notice, and modified files need to carry a notice indicating
that they have been altered from the originals.

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