Alleviating the quantum Big-$M$ problem

TL;DR

This paper addresses the quantum Big-$M$ problem in QUBO formulations, revealing its NP-hardness, establishing spectral gap bounds, and proposing an SDP-based translation algorithm that improves quantum solver performance.

Contribution

It introduces a systematic analysis of the quantum Big-$M$ problem, proves its NP-hardness, and presents a novel SDP relaxation method that enhances spectral gaps and solution quality.

Findings

The proposed algorithm yields spectral gaps orders of magnitude larger.

Numerical benchmarks show improved performance over previous methods.

Experimental results on IonQ devices demonstrate faster solutions and better quality.

Abstract

A major obstacle for quantum optimizers is the reformulation of constraints as a quadratic unconstrained binary optimization (QUBO). Current QUBO translators exaggerate the weight $M$ of the penalty terms. Classically known as the "Big-$M$" problem, the issue becomes even more daunting for quantum solvers, since it affects the physical energy scale. We take a systematic, encompassing look at the quantum big-$M$ problem, revealing NP-hardness in finding the optimal $M$ and establishing bounds on the Hamiltonian spectral gap $\Delta$, inversely related to the expected run-time of quantum solvers. We propose a practical translation algorithm, based on SDP relaxation, that outperforms previous methods in numerical benchmarks. Our algorithm gives values of $\Delta$ orders of magnitude greater, e.g. for portfolio optimization instances. Solving such instances with an adiabatic algorithm on 6-qubits of an IonQ device, we observe significant advantages in time to solution and average solution quality. Our findings are relevant to quantum and quantum-inspired solvers alike.