Conversion of Boolean and Integer FlatZinc Builtins to Quadratic or Linear Integer Problems

TL;DR

The paper tackles transforming FlatZinc-based CSP/COP models into Quadratic Integer Programs over finite domains (QIP(FD)) to enable solving with quantum devices, notably Quantum Annealers. It systematically derives constructive reformulations for Boolean and integer builtins, using one-hot encodings, auxiliary variables, and linear/quadratic constraints to express complex operations (e.g., element access, maxima/minima, division, logical relations, and set membership). The main contributions include explicit encodings for a broad suite of builtins (3.1–3.3) and the demonstration that these can be embedded into QIP(FD) and subsequently into QUBO form, supporting a practical MiniZinc-to-QUBO workflow. This work enables leveraging quantum computing resources to tackle a wide class of CSP/COP instances by providing a concrete pipeline from high-level FlatZinc models to quantum-solvable representations. The practical impact lies in broadening the accessibility of quantum-accelerated constraint solving for optimisation problems expressed in MiniZinc/FlatZinc, with future work focusing on implementing the end-to-end toolchain.

Abstract

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.