Optimization of Quantum Systems Emulation via a Variant of the Bandwidth Minimization Problem

TL;DR

This paper presents a novel optimization approach called weighted-BMP for quantum system emulation, improving site ordering to reduce costs and outperforming heuristic methods through exact MILP solutions and algorithmic enhancements.

Contribution

It introduces weighted-BMP, an exact MILP-based method for site ordering in quantum emulation, with symmetry-breaking and lower bounds, demonstrating significant performance improvements.

Findings

25.61% average CPU time reduction with enhancements

24.48% average memory reduction over RCM heuristic

First application of exact optimization considering interactions in quantum emulation

Abstract

This paper introduces weighted-BMP, a variant of the Bandwidth Minimization Problem (BMP), with a significant application in optimizing quantum emulation. Weighted-BMP optimizes particles ordering to reduce the emulation costs, by designing a particle interaction matrix where strong interactions are placed as close as possible to the diagonal. We formulate the problem using a Mixed Integer Linear Program (MILP) and solve it to optimality with a state of the art solver. To strengthen our MILP model, we introduce symmetry-breaking inequalities and establish a lower bound. Through extensive numerical analysis, we examine the impacts of these enhancements on the solver's performance. The introduced reinforcements result in an average CPU time reduction of 25.61 percent. Additionally, we conduct quantum emulations of realistic instances. Our numerical tests show that the weighted-BMP approach outperforms the Reverse Cuthill-McKee (RCM) algorithm, an efficient heuristic used for site ordering tasks in quantum emulation, achieving an average memory storage reduction of 24.48 percent. From an application standpoint, this study is the first to apply an exact optimization method, weighted-BMP, that considers interactions for site ordering in quantum emulation pre-processing, and shows its crucial role in cost reduction. From an algorithmic perspective, it contributes by introducing important reinforcements and lays the groundwork for future research on further enhancements, particularly on strengthening the weak linear relaxation of the MILP.