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Embedding Equality Constraints of Optimization Problems into a Quantum Annealer

Department of Computer Science, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Author to whom correspondence should be addressed.
The results in this paper have previously appeared in the Proceedings of the IEEE International Conference on Rebooting Computing (ICRC 2018) and in the technical report LA-UR-18-30971, Los Alamos National Laboratory, November 2018.
Algorithms 2019, 12(4), 77;
Received: 10 January 2019 / Revised: 4 April 2019 / Accepted: 13 April 2019 / Published: 17 April 2019
(This article belongs to the Special Issue Quantum Optimization Theory, Algorithms, and Applications)
PDF [2210 KB, uploaded 17 April 2019]


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. View Full-Text
Keywords: quantum annealing; D-Wave; QUBO; constrained optimization; mixed-integer programming quantum annealing; D-Wave; QUBO; constrained optimization; mixed-integer programming

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Vyskocil, T.; Djidjev, H. Embedding Equality Constraints of Optimization Problems into a Quantum Annealer. Algorithms 2019, 12, 77.

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