Embedding Equality Constraints of Optimization Problems into a Quantum Annealer

Tomas Vyskocil,Hristo Djidjev

doi:10.3390/a12040077

Tomas Vyskocil, Hristo Djidjev

Open Access

https://doi.org/10.3390/a12040077

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Journal: Algorithms	Publication Date: Apr 17, 2019
Citations: 18	License type: CC BY 4.0

Affiliation: Los Alamos National Laboratory

Abstract

Quantum annealers such as D-Wave machines are designed to propose solutions for quadratic unconstrained binary optimization (QUBO) problems by mapping them onto the quantum processing unit, which tries to find a solution by measuring the parameters of a minimum-energy state of the quantum system. While many NP-hard problems can be easily formulated as binary quadratic optimization problems, such formulations almost always contain one or more constraints, which are not allowed in a QUBO. Embedding such constraints as quadratic penalties is the standard approach for addressing this issue, but it has drawbacks such as the introduction of large coefficients and using too many additional qubits. In this paper, we propose an alternative approach for implementing constraints based on a combinatorial design and solving mixed-integer linear programming (MILP) problems in order to find better embeddings of constraints of the type ∑ x i = k for binary variables x i. Our approach is scalable to any number of variables and uses a linear number of ancillary variables for a fixed k.

Highlights

Quantum annealing (QA) computers such as the commercially available D-Wave 2X and D-Wave2000Q are designed to seek solutions to problems that are hard for the conventional computers, such as many NP-hard problems [1]
We propose a method that allows embedding in the Chimera graph linear constraints of any number of variables
We experimented with the case k = 1 since it allows larger problems to be embedded and because that design is less sensitive to faults in the D-Wave hardware

Summary

Introduction

Quantum annealing (QA) computers such as the commercially available D-Wave 2X and D-Wave2000Q are designed to seek solutions to problems that are hard for the conventional computers, such as many NP-hard problems [1]. Quantum annealing (QA) computers such as the commercially available D-Wave 2X and D-Wave. The type of problems such computers can solve directly in their quantum hardware is minimizing a quadratic form of the type. I =1 where variables xi are either in the set {−1, 1}, in which case the problem is called an Ising problem, or in the set {0, 1}, in which case it is called a Quadratic Unconstrained Binary Optimization (QUBO). To minimize Q, QA computers are trying to find a low-energy state of the quantum system whose Hamiltonian corresponds to Q [3,4]. Solving an optimization problem using a quantum annealer involves the following several steps. (i) Represent the optimization problem of interest as a QUBO or Ising problem.

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