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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]:

from qiskit.circuit.library import RealAmplitudes
from qiskit.algorithms.optimizers import COBYLA
from qiskit.algorithms import NumPyMinimumEigensolver, VQE
from qiskit.opflow import PauliExpectation, CVaRExpectation
from qiskit_optimization.converters import LinearEqualityToPenalty
from qiskit_optimization.algorithms import MinimumEigenOptimizer
from qiskit_optimization.translators import from_docplex_mp
from qiskit import execute, Aer
from qiskit.utils import algorithm_globals

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

[2]:

algorithm_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(range(n), name="x")
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 = from_docplex_mp(mdl)

[6]:

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

objective function value: 1.27835
variable values: x_0=1.0, x_1=1.0, x_2=0.0, x_3=0.0, x_4=1.0, x_5=0.0
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
ansatz = RealAmplitudes(n, reps=1)
m = ansatz.num_parameters

# set backend
backend_name = "qasm_simulator"  # use this for QASM simulator
# backend_name = 'aer_simulator_statevector'  # 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,
ansatz=ansatz,
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].prettyprint())
print()

alpha = 1.0:
objective function value: 0.9716499999999968
variable values: x_0=1.0, x_1=0.0, x_2=0.0, x_3=0.0, x_4=1.0, x_5=1.0
status: SUCCESS

alpha = 0.5:
objective function value: 1.2783500000000174
variable values: x_0=1.0, x_1=1.0, x_2=0.0, x_3=0.0, x_4=1.0, x_5=0.0
status: SUCCESS

alpha = 0.25:
objective function value: 1.2783500000000174
variable values: x_0=1.0, x_1=1.0, x_2=0.0, x_3=0.0, x_4=1.0, x_5=0.0
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()

[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):  0.0264
optimal probabilitiy (alpha = 0.25):  0.1123

[12]:

import qiskit.tools.jupyter

%qiskit_version_table


### Version Information

Qiskit SoftwareVersion
qiskit-terra0.21.0
qiskit-aer0.10.4
qiskit-optimization0.4.0
System information
Python version3.8.13
Python compilerGCC 9.4.0
Python builddefault, Jun 20 2022 14:28:56
OSLinux
CPUs2
Memory (Gb)6.783607482910156
Wed Jul 06 17:29:42 2022 UTC

### This code is a part of Qiskit

obtain a copy of this license in the LICENSE.txt file in the root directory
[ ]: