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Algorithms

Embedding Equality Constraints of Optimization Problems into a Quantum Annealer

Tomáš Vyskočil, Hristo N. Djidjev

Published Apr 17, 2019• 22 citations
Publisher

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.

DOI
10.3390/a12040077

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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.
Quantum annealers such as the commercially available D-Wave machines are designed to natively solve quadratic unconstrained binary optimization (QUBO) problems. While most of the well-known NP-hard optimization problems can easily be formulated as quadratic binary problems, such formulations also contain constraints, which commonly are added to the objective function in the form of penalties to obtain a QUBO version. However, the standard method for defining such penalties leads to QUBOs that are dense and therefore take too much of the resources of the quantum annealer. In this paper, we describe an alternative approach to the constraint embedding problem that uses mixed-integer linear programming (MILP) and is scalable to problems of arbitrary number of variables.
Quantum annealers such as the D-Wave 2X computer are designed to natively solve Quadratic Unconstrained Binary Optimization (QUBO) problems to optimality or near optimality. Most NP-hard problems, which are hard for classical computers, can be naturally described as quadratic binary problems that contain a quadratic binary objective function and one or more constraints. Since a QUBO cannot have constraints, each such constraint has to be added to the objective function as a penalty, in order to solve on D-Wave. For a minimization problem, for instance, such penalty can be a quadratic term that gets a value zero, if the constraint is satisfied, and a large value, if it is not. In many cases, however, the penalty can significantly increase the number of quadratic terms in the resulting QUBO and make it too large to embed into the D-Wave hardware. In this paper, we develop an alternative method for formulating and embedding constraints of the type Σi=1sxi=1, which is much more scalable than the existing ones, and analyze the properties of the resulting embeddings.
Quantum annealing is a model for quantum computing that is aimed at solving hard optimization problems by representing them as quadratic unconstrained binary optimization (QUBO) problems. Although many NP-hard problems can easily be formulated as binary-variable problems with a quadratic objective function, such formulations typically also include constraints, which are not allowed in a QUBO. Hence, such constraints are usually incorporated in the objective function as additive penalty terms. While there is substantial previous work on implementing linear equality constraints, the case of inequality constraints has not much been studied. In this paper, we propose a new approach for formulating and embedding inequality constraints as penalties and describe early implementation results.
Mixed-integer linear programming problems are extensively used in industry for a wide range of optimization tasks. However, as they get larger, they present computational challenges for classical solvers within practical time limits. Quantum annealers can, in principle, accelerate the solution of problems formulated as quadratic unconstrained binary optimization instances, but their limited scale currently prevents achieving practical speedups. Quantum-classical algorithms have been proposed to take advantage of both paradigms and to allow current quantum computers to be used in larger problems. In this work, a hardware-agnostic Benders' decomposition algorithm and a series of enhancements with the goal of taking the most advantage of quantum computing are presented. The decomposition consists of a master problem with integer variables, which is reformulated as a quadratic unconstrained binary optimization problem and solved with a quantum annealer, and a linear subproblem solved by a classical computer. The enhancements consist, among others, of different embedding processes that substantially reduce the pre-processing time of the embedding computation without compromising solution quality, a conservative handling of cut constraints, and a stopping criterion that accounts for the limited size of current quantum computers and their heuristic nature. The proposed algorithm is benchmarked against classical approaches using a D-Wave quantum annealer for a scalable family of transmission network expansion planning problems.
Quantum annealing computers are designed to produce high-quality solutions to optimization problems that can be formulated as quadratic unconstrained binary optimization (QUBO) problems. While most of the well known NP-hard problems can easily be represented as quadratic binary problems, such formulations usually contain constraints that have to be added as penalties to the objective function in order to obtain QUBOs. In this paper, we propose a method based on finite automaton representation of the constraints for generating penalty implementations for them, which uses fewer qubits than the alternatives and is general enough to be applied to a whole class of constraints.
Quantum annealing is a promising approach for solving combinatorial optimization problems. However, its performance is often limited by the overhead of additional qubits required for embedding logical QUBO models onto quantum annealers. This overhead becomes severe when logical QUBO models have dense connectivity. Such dense structures frequently arise when formulating equality and inequality constraints. To address this issue, we propose a method to construct a significantly sparser logical QUBO model for these constraints. By adding auxiliary variables based on specific network structures, our approach decomposes the original constraint into smaller, more manageable constraints. We demonstrate that this method reduces the number of edges (quadratic terms) from O(N2)O(N^2)O(N2) to O(N)O(N)O(N) for the one-hot constraint and to O(Nlog⁡N)O(N\log N)O(NlogN) in the worst case for general equality constraints, where NNN is the number of variables. Experimental results on D-Wave's hardware show that our formulation leads to substantial reductions in the number of qubits required for embedding, shorter average chain lengths, lower chain break rates, and higher feasible solution rates compared to conventional methods. This work provides a practical tool for efficiently solving constrained optimization problems on current and future quantum annealers.