Efficient Use of Quantum Linear System Algorithms in Interior Point Methods for Linear Optimization

TL;DR

This paper explores the effective integration of quantum linear system algorithms into interior point methods for linear optimization, addressing challenges like ill-conditioning and errors, and proposing an inexact quantum interior point method with iterative refinement.

Contribution

It introduces an Inexact Infeasible Quantum Interior Point Method for linear optimization and discusses how to achieve exact solutions efficiently using iterative refinement.

Findings

Quantum interior point methods can be enhanced with inexact solvers.

Iterative refinement improves solution accuracy without excessive quantum computation.

Computational experiments demonstrate the method's potential with quantum simulators.

Abstract

Quantum computing has attracted significant interest in the optimization community because it potentially can solve classes of optimization problems faster than conventional supercomputers. Several researchers proposed quantum computing methods, especially Quantum Interior Point Methods (QIPMs), to solve convex optimization problems, such as Linear Optimization, Semidefinite Optimization, and Second-order Cone Optimization problems. Most of them have applied a Quantum Linear System Algorithm at each iteration to compute a Newton step. However, using quantum linear solvers in QIPMs comes with many challenges, such as having ill-conditioned systems and the considerable error of quantum solvers. This paper investigates how one can efficiently use quantum linear solvers in QIPMs. Accordingly, an Inexact Infeasible Quantum Interior Point Method is developed to solve linear optimization problems. We also discuss how can we get an exact solution by Iterative Refinement without excessive time of quantum solvers. Finally, computational results with QISKIT implementation of our QIPM using quantum simulators are analyzed.