Resource-Scalable Fully Quantum Metropolis-Hastings for Integer Linear Programming

TL;DR

This paper presents a fully quantum Metropolis-Hastings algorithm for integer linear programming that operates coherently without classical assistance, demonstrating linear resource scaling and potential for quantum optimization speedups.

Contribution

It introduces a novel quantum algorithm for ILP that uses reversible circuits and linear qubit scaling, advancing quantum constraint programming.

Findings

Supports linear objectives and mixed constraints.

Numerical simulations validate resource scaling.

Progressive thermalization toward feasible solutions.

Abstract

Integer linear programming (ILP) remains computationally challenging due to its NP-complete nature despite its central role in scheduling, logistics, and design optimization. We introduce a fully quantum Metropolis-Hastings algorithm for ILP that implements a coherent random walk over the discrete feasible region using only reversible quantum circuits, without quantum-RAM assumptions or classical pre/post-processing. Each walk step is a unitary update that prepares coherent candidate moves, evaluates the objective and constraints reversibly -- including a constraint-satisfaction counter to enforce feasibility -- and encodes Metropolis acceptance amplitudes via a low-overhead linearized rule. At the logical level, the construction uses $\mathcal{O}(n\log_2 N)$ qubits to represent $n$ integer variables over the interval $[-N,\,N-1]$, and the Toffoli-equivalent cost per Metropolis step grows linearly with the total logical qubit count. Using explicit ripple-carry adder constructions, we support linear objectives and mixed equality/inequality constraints. Numerical circuit-level simulations on a broad ensemble of randomly generated instances validate the predicted linear resource scaling and exhibit progressive thermalization toward low-cost feasible solutions under the annealing schedule. Overall, the method provides a coherent, resource-characterized baseline for fully quantum constraint programming and a foundation for incorporating additional quantum speedups in combinatorial optimization.