A quantum dual logarithmic barrier method for linear optimization

TL;DR

This paper introduces a quantum-enhanced dual logarithmic barrier method for linear optimization, achieving quadratic convergence and improved iteration complexity using quantum linear system algorithms and iterative refinement.

Contribution

It develops a quantum version of the dual logarithmic barrier method with inexact directions, providing the best-known iteration complexity and sublinear quantum complexity for certain problem sizes.

Findings

Quadratic convergence toward the central path.

Best-known iteration complexity of O(√n log(nμ₀/ζ)).

Sublinear quantum complexity for problems with more constraints than variables.

Abstract

Quantum computing has the potential to speed up some optimization methods. One can use quantum computers to solve linear systems via Quantum Linear System Algorithms (QLSAs). QLSAs can be used as a subroutine for algorithms that require solving linear systems, such as the dual logarithmic barrier method (DLBM) for solving linear optimization (LO) problems. In this paper, we use a QLSA to solve the linear systems arising in each iteration of the DLBM. To use the QLSA in a hybrid setting, we read out quantum states via a tomography procedure which introduces considerable error and noise. Thus, this paper first proposes an inexact-feasible variant of DLBM for LO problems and then extends it to a quantum version. Our quantum approach has quadratic convergence toward the central path with inexact directions and we show that this method has the best-known $\mathcal{O}(\sqrt{n} \log (n \mu_0 /\zeta))$ iteration complexity, where $n$ is the number of variables, $\mu_0$ is the initial duality gap, and $\zeta$ is the desired accuracy. We further use iterative refinement to improve the time complexity dependence on accuracy. For LO problems with quadratically more constraints than variables, the quantum complexity of our method has a sublinear dependence on dimension.