Improving Variational Quantum Optimization using CVaR

Hybrid quantum/classical variational algorithms can be implemented on noisy intermediate-scale quantum computers and can be used to find solutions for combinatorial optimization problems. Approaches discussed in the literature minimize the expectation of the problem Hamiltonian for a parameterized trial quantum state. The expectation is estimated as the sample mean of a set of measurement outcomes, while the parameters of the trial state are optimized classically. This procedure is fully justified for quantum mechanical observables such as molecular energies. In the case of classical optimization problems, which yield diagonal Hamiltonians, we argue that aggregating the samples in a different way than the expected value is more natural. In this paper we propose the Conditional Value-at-Risk as an aggregation function. We empirically show -- using classical simulation as well as quantum hardware -- that this leads to faster convergence to better solutions for all combinatorial optimization problems tested in our study. We also provide analytical results to explain the observed difference in performance between different variational algorithms.

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Solving Combinatorial Optimization Problems on Quantum Computers

Cybernetics and Computer Technologies ◽

10.34229/2707-451x.20.2.1 ◽

2020 ◽

pp. 5-13

Author(s):

Vyacheslav Korolyov ◽

Oleksandr Khodzinskyi

Keyword(s):

Combinatorial Optimization ◽

Optimization Problems ◽

Quantum Computer ◽

Independent Set ◽

Cloud Services ◽

Quantum Computers ◽

Maximal Independent Set ◽

Maximum Independent Set ◽

Combinatorial Optimization Problems ◽

D Wave

Introduction. Quantum computers provide several times faster solutions to several NP-hard combinatorial optimization problems in comparison with computing clusters. The trend of doubling the number of qubits of quantum computers every year suggests the existence of an analog of Moore's law for quantum computers, which means that soon they will also be able to get a significant acceleration of solving many applied large-scale problems. The purpose of the article is to review methods for creating algorithms of quantum computer mathematics for combinatorial optimization problems and to analyze the influence of the qubit-to-qubit coupling and connections strength on the performance of quantum data processing. Results. The article offers approaches to the classification of algorithms for solving these problems from the perspective of quantum computer mathematics. It is shown that the number and strength of connections between qubits affect the dimensionality of problems solved by algorithms of quantum computer mathematics. It is proposed to consider two approaches to calculating combinatorial optimization problems on quantum computers: universal, using quantum gates, and specialized, based on a parameterization of physical processes. Examples of constructing a half-adder for two qubits of an IBM quantum processor and an example of solving the problem of finding the maximum independent set for the IBM and D-wave quantum computers are given. Conclusions. Today, quantum computers are available online through cloud services for research and commercial use. At present, quantum processors do not have enough qubits to replace semiconductor computers in universal computing. The search for a solution to a combinatorial optimization problem is performed by achieving the minimum energy of the system of coupled qubits, on which the task is mapped, and the data are the initial conditions. Approaches to solving combinatorial optimization problems on quantum computers are considered and the results of solving the problem of finding the maximum independent set on the IBM and D-wave quantum computers are given. Keywords: quantum computer, quantum computer mathematics, qubit, maximal independent set for a graph.

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A benchmark for quantum optimization: the traveling salesman

Quantum Information and Computation ◽

10.26421/qic21.7-8-2 ◽

2021 ◽

Vol 21 (7&8) ◽

pp. 557-562

Author(s):

Richard H. Warren

Keyword(s):

Quantum Computing ◽

Combinatorial Optimization ◽

Optimization Problems ◽

Traveling Salesman ◽

Traveling Salesman Problems ◽

Combinatorial Optimization Problems ◽

Quantum Optimization

We present compelling reasons for symmetric traveling salesman problems (TSPs) to be the benchmark for quantum computing of combinatorial optimization problems for all types of quantum hardware. There are seven reasons for endorsing these TSPs to be the benchmark and no shortcomings.

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Heuristic Approaches to Stochastic Quadratic Assignment Problem: VaR and CVar Cases

MENDEL ◽

10.13164/mendel.2017.1.073 ◽

2017 ◽

Vol 23 (1) ◽

pp. 73-78 ◽

Cited By ~ 2

Author(s):

Radomil Matousek ◽

Pavel Popela ◽

Jakub Kudela

Keyword(s):

At Risk ◽

Assignment Problem ◽

Value At Risk ◽

Optimization Problems ◽

Quadratic Assignment Problem ◽

Risk Measures ◽

Conditional Value At Risk ◽

Quadratic Assignment ◽

Heuristic Approaches ◽

Combinatorial Optimization Problems

The goal of this paper is to continue our investigation of the heuristic approaches of solving thestochastic quadratic assignment problem (StoQAP) and provide additional insight into the behavior of di erentformulations that arise through the stochastic nature of the problem. The deterministic Quadratic AssignmentProblem (QAP) belongs to a class of well-known hard combinatorial optimization problems. Working with severalreal-world applications we have found that their QAP parameters can (and should) be considered as stochasticones. Thus, we review the StoQAP as a stochastic program and discuss its suitable deterministic reformulations.The two formulations we are going to investigate include two of the most used risk measures - Value at Risk(VaR) and Conditional Value at Risk (CVaR). The focus is on VaR and CVaR formulations and results of testcomputations for various instances of StoQAP solved by a genetic algorithm, which are presented and discussed.

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