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

mohammadhossein mohammadisiahroudi, Zeguan Wu, Pouya Sampourmahani, Adrian Harkness, Tamas Terlaky

Keywords: Quantum Interior Point Method, Quantum Linear System Algorithm, Iterative Refinement, Preconditioning, Linear Optimization, Conic Optimization

TL;DR: Paper reviews recent advancements in QIPMs and presnets the state-of-the-art

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) have 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 a novel 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.

Submission Number: 20