Quantum Interior Point Methods: A Review of Developments and An Optimally Scaling Framework

TL;DR

This paper reviews recent developments in quantum interior point methods (QIPMs), highlighting an almost-exact hybrid quantum-classical framework that achieves optimal scalability for large-scale optimization problems.

Contribution

It introduces an almost-exact QIPM framework that fully leverages quantum hardware for matrix-vector operations, surpassing existing methods in scalability.

Findings

Achieves optimal worst-case scalability with respect to problem dimension.

Constructs and solves Newton systems entirely on quantum hardware.

Surpasses the scalability of existing classical and quantum IPMs.

Abstract

The growing demand for solving large-scale, data-intensive linear and conic optimization problems, particularly in applications such as artificial intelligence and machine learning, has highlighted the limitations of classical interior point methods (IPMs). Despite their favorable polynomial-time convergence, conventional IPMs often suffer from high per-iteration computational costs, especially for dense problem instances. Recent advances in quantum computing, particularly quantum linear system solvers, offer promising avenues to accelerate the most computationally intensive steps of IPMs. However, practical challenges such as quantum error, hardware noise, and sensitivity to poorly conditioned systems remain significant obstacles. In response, a series of Quantum IPMs (QIPMs) has been developed to address these challenges, incorporating techniques such as feasibility maintenance, iterative refinement, and preconditioning. In this work, we review this line of research with a focus on our recent contributions, including an almost-exact QIPM framework. This hybrid quantum-classical approach constructs and solves the Newton system entirely on a quantum computer, while performing solution updates classically. Crucially, all matrix-vector operations are executed on quantum hardware, enabling the method to achieve an optimal worst-case scalability w.r.t dimension, surpassing the scalability of existing classical and quantum IPMs.