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arXiv:2412.15977 · INFORMS Journal on Optimization

A Quantum Dual Logarithmic Barrier Method for Linear Optimization

Zeguan Wu, Pouya Sampourmahani, Mohammadhossein Mohammadisiahroudi, Tamás Terlaky B

Published Dec 5, 2025• 1 citation
arXivHTMLPDF

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 that 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 [Formula: see text] iteration complexity, where n is the number of variables, [Formula: see text] is the initial duality gap, and [Formula: see text] is the accuracy. We further use iterative refinement to improve the complexity dependence on accuracy and condition number. For LO problems with quadratically more constraints than variables, the quantum complexity of our method has a sublinear dependence on dimension. Funding: This research was supported by Defense Advanced Research Projects Agency [Grant W911NF2010022] and the National Science Foundation [Grant 2128527].

DOI
10.1287/ijoo.2024.0062

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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 we can get an exact solution by Iterative Refinement without excessive time of quantum solvers. Finally, computational results with a QISKIT implementation of our QIPM using quantum simulators are analyzed.
Solving linear systems is at the foundation of many algorithms. Recently, quantum linear system algorithms (QLSAs) have attracted great attention since they converge to a solution exponentially faster than classical algorithms in terms of the problem dimension. However, low-complexity circuit implementations of the oracles assumed in these QLSAs constitute the major bottleneck for practical quantum speed-up in solving linear systems. In this work, we focus on the application of QLSAs for linear systems that are expressed as a low rank tensor sums, which arise in solving discretized PDEs. Previous works uses modified Krylov subspace methods to solve such linear systems with a per-iteration complexity being polylogarithmic of the dimension but with no guarantees on the total convergence cost. We propose a quantum algorithm based on the recent advances on adiabatic-inspired QLSA and perform a detailed analysis of the circuit depth of its implementation. We rigorously show that the total complexity of our implementation is polylogarithmic in the dimension, which is comparable to the per-iteration complexity of the classical heuristic methods.
We propose a novel quantum algorithm for solving linear optimization problems by quantum-mechanical simulation of the central path. While interior point methods follow the central path with an iterative algorithm that works with successive linearizations of the perturbed KKT conditions, we perform a simulation working directly with the nonlinear complementarity equations. Using a two-stage quantum eigenpath traversal schedule, this approach yields an algorithm for solving linear optimization problems involving mmm constraints and nnn variables to ε\varepsilonε-optimality using O~ ⁣((m+n)3/2 ε−1+(m+n)1/2 ε−3/2)\widetilde{O}\!\left((m+n)^{3/2}\,\varepsilon^{-1}+(m+n)^{1/2}\,\varepsilon^{-3/2}\right)O((m+n)3/2ε−1+(m+n)1/2ε−3/2) queries to an oracle that evaluates a potential function. In the standard gate model (i.e., without access to quantum RAM), our algorithm can obtain ε\varepsilonε-precise solutions to LO problems using at most O~ ⁣([(m+n)3/2ε+(m+n)1/2ε3/2]⋅nnz(A))\widetilde{O}\!\left(\left[\frac{(m+n)^{3/2}}{\varepsilon} + \frac{(m+n)^{1/2}}{\varepsilon^{3/2}}\right] \cdot \mathrm{nnz}(A)\right)O([ε(m+n)3/2​+ε3/2(m+n)1/2​]⋅nnz(A)) elementary gates, where nnz(A)\mathrm{nnz}(A)nnz(A) is the total number of non-zero elements found in the constraint matrix.
We present and analyze an iterative refinement (IR) framework for improving the precision dependence of algorithms that combine a quantum linear system algorithm (QLSA) with quantum state tomography. Existing QLSAs achieve polylogarithmic dependence on the inverse error tolerance to prepare a quantum state encoding the solution, but extracting a classical description of such a state via tomography typically introduces a polynomial dependence on the target precision. To retain polylogarithmic dependence in the inverse error tolerance throughout the entire process, we develop an IR scheme that solves a sequence of related linear systems to progressively refine the solution while requiring only fixed-precision quantum subroutines, finally obtaining a classical description of a high-precision solution. We analyze the proposed framework under three input models: quantum-read/classical-write RAM (QRAM), linear combinations of unitaries (LCU), and sparse-access oracles. We evaluate the proposed scheme through numerical experiments on both quantum simulators and real quantum hardware. The results demonstrate that iterative refinement efficiently improves the precision of the solution and exhibits robustness to hardware noise.
Quantum linear system algorithms (QLSAs) have the potential to speed up algorithms that rely on solving linear systems. Interior point methods (IPMs) yield a fundamental family of polynomial-time algorithms for solving optimization problems. IPMs solve a Newton linear system at each iteration to compute the search direction; thus, QLSAs can potentially speed up IPMs. Due to the noise in contemporary quantum computers, quantum-assisted IPMs (QIPMs) only admit an inexact solution to the Newton linear system. Typically, an inexact search direction leads to an infeasible solution, so, to overcome this, we propose an inexact-feasible QIPM (IF-QIPM) for solving linearly constrained quadratic optimization problems. We also apply the algorithm to ℓ1-norm soft margin support vector machine (SVM) problems, and demonstrate that our algorithm enjoys a speedup in the dimension over existing approaches. This complexity bound is better than any existing classical or quantum algorithm that produces a classical solution.