Realistic Runtime Analysis for Quantum Simplex Computation

TL;DR

This paper introduces a quantum runtime analysis framework for optimization problems like Linear Programming, using hybrid benchmarking to estimate practical performance and assess the feasibility of quantum advantage given current hardware limitations.

Contribution

It develops a quantum analog for classical runtime analysis of optimization algorithms and proposes hybrid benchmarking to evaluate quantum methods on classical systems.

Findings

Quantum algorithms for Linear Programming show potential asymptotic speedup.

Practical quantum advantage requires gate operation times much faster than current hardware.

Hybrid benchmarking effectively estimates quantum performance without large-scale quantum hardware.

Abstract

In recent years, strong expectations have been raised for the possible power of quantum computing for solving difficult optimization problems, based on theoretical, asymptotic worst-case bounds. Can we expect this to have consequences for Linear and Integer Programming when solving instances of practically relevant size, a fundamental goal of Mathematical Programming, Operations Research and Algorithm Engineering? Answering this question faces a crucial impediment: The lack of sufficiently large quantum platforms prevents performing real-world tests for comparison with classical methods. In this paper, we present a quantum analog for classical runtime analysis when solving real-world instances of important optimization problems. To this end, we measure the expected practical performance of quantum computers by analyzing the expected gate complexity of a quantum algorithm. The lack of practical quantum platforms for experimental comparison is addressed by hybrid benchmarking, in which the algorithm is performed on a classical system, logging the expected cost of the various subroutines that are employed by the quantum versions. In particular, we provide an analysis of quantum methods for Linear Programming, for which recent work has provided asymptotic speedup through quantum subroutines for the Simplex method. We show that a practical quantum advantage for realistic problem sizes would require quantum gate operation times that are considerably below current physical limitations.