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arXiv:2307.10032

Automatic Conversion of MiniZinc Programs to QUBO

Wolf, Armin, Cristian Grozea

Published Jul 19, 2023• 0 citations
arXivHTMLPDF

Abstract

Obtaining Quadratic Unconstrained Binary Optimisation models for various optimisation problems, in order to solve those on physical quantum computers (such as the the DWave annealers) is nowadays a lengthy and tedious process that requires one to remodel all problem variables as binary variables and squeeze the target function and the constraints into a single quadratic polynomial into these new variables. We report here on the basis of our automatic converter from MiniZinc to QUBO, which is able to process a large set of constraint optimisation and constraint satisfaction problems and turn them into equivalent QUBOs, effectively optimising the whole process.

DOI
10.48550/arxiv.2307.10032

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Constraint satisfaction or optimisation models -- even if they are formulated in high-level modelling languages -- need to be reduced into an equivalent format before they can be solved by the use of Quantum Computing. In this paper we show how Boolean and integer FlatZinc builtins over finite-domain integer variables can be equivalently reformulated as linear equations, linear inequalities or binary products of those variables, i.e. as finite-domain quadratic integer programs. Those quadratic integer programs can be further transformed into equivalent Quadratic Unconstrained Binary Optimisation problem models, i.e. a general format for optimisation problems to be solved on Quantum Computers especially on Quantum Annealers.
Practical discrete optimization problems often contain multidimensional arrays of variables with linear constraints, which complicates their conversion to QUBO (quadratic unconstrained binary optimization) form. The article proposes a systematic approach to transforming such problems, which includes three key steps: the transition from a multidimensional representation of variables to a one-dimensional one using the Kronecker product of matrices, the reduction of mixed variables to binary ones, and the introduction of linear constraints into the objective function through quadratic penalties. Explicit computational formulas are obtained for each stage, simplifying their software implementation. The developed method is illustrated with examples from graph theory and combinatorial optimization, including classical formulations, confirming its universality. The results of the article allow standardizing the process of adapting problems for solving on quantum annealing algorithms (e.g., D-Wave) and classical QUBO solvers.
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The broad applicability of Quadratic Unconstrained Binary Optimization (QUBO) constitutes a general-purpose modeling framework for combinatorial optimization problems and are a required format for gate array and quantum annealing computers. QUBO annealers as well as other solution approaches benefit from starting with a diverse set of solutions with local optimality an additional benefit. This paper presents a new method for generating a set of one-flip local optima leveraging constraint programming. Further, as demonstrated in experimental testing, analysis of the solution set allows the generation of soft constraints to help guide the optimization process.