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Title: Embedding Equality Constraints of Optimization Problems into a Quantum Annealer

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 Σ 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.

Authors:
ORCiD logo [1]; ORCiD logo [2]
  1. The State University of New Jersey, Piscataway, NJ (United States)
  2. Los Alamos National Laboratory; Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program
OSTI Identifier:
1544690
Report Number(s):
LA-UR-19-20224
Journal ID: ISSN 1999-4893; ALGOCH
Grant/Contract Number:  
89233218CNA000001
Resource Type:
Accepted Manuscript
Journal Name:
Algorithms
Additional Journal Information:
Journal Volume: 12; Journal Issue: 4; Journal ID: ISSN 1999-4893
Publisher:
MDPI
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; quantum annealing; D-Wave; QUBO; constrained optimization; mixed-integer programming

Citation Formats

Vyskocil, Tomas, and Djidjev, Hristo Nikolov. Embedding Equality Constraints of Optimization Problems into a Quantum Annealer. United States: N. p., 2019. Web. doi:10.3390/a12040077.
Vyskocil, Tomas, & Djidjev, Hristo Nikolov. Embedding Equality Constraints of Optimization Problems into a Quantum Annealer. United States. doi:10.3390/a12040077.
Vyskocil, Tomas, and Djidjev, Hristo Nikolov. Wed . "Embedding Equality Constraints of Optimization Problems into a Quantum Annealer". United States. doi:10.3390/a12040077. https://www.osti.gov/servlets/purl/1544690.
@article{osti_1544690,
title = {Embedding Equality Constraints of Optimization Problems into a Quantum Annealer},
author = {Vyskocil, Tomas and Djidjev, Hristo Nikolov},
abstractNote = {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 Σ xi = k for binary variables xi. Our approach is scalable to any number of variables and uses a linear number of ancillary variables for a fixed k.},
doi = {10.3390/a12040077},
journal = {Algorithms},
number = 4,
volume = 12,
place = {United States},
year = {2019},
month = {4}
}

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