Variational Quantum Algorithms for Semidefinite Programming

TL;DR

This paper introduces variational quantum algorithms designed to approximately solve semidefinite programs, with theoretical convergence analysis and numerical simulations demonstrating effectiveness even in noisy environments.

Contribution

It proposes new variational quantum algorithms for SDPs, including convergence proofs for weakly constrained cases and algorithms for more general SDPs.

Findings

Algorithms converge to approximate solutions under certain conditions

Numerical simulations show effectiveness in MaxCut problems

Convergence persists in noisy quantum settings

Abstract

A semidefinite program (SDP) is a particular kind of convex optimization problem with applications in operations research, combinatorial optimization, quantum information science, and beyond. In this work, we propose variational quantum algorithms for approximately solving SDPs. For one class of SDPs, we provide a rigorous analysis of their convergence to approximate locally optimal solutions, under the assumption that they are weakly constrained (i.e., $N\gg M$, where $N$ is the dimension of the input matrices and $M$ is the number of constraints). We also provide algorithms for a more general class of SDPs that requires fewer assumptions. Finally, we numerically simulate our quantum algorithms for applications such as MaxCut, and the results of these simulations provide evidence that convergence still occurs in noisy settings.