Efficient Use of Quantum Linear System Algorithms in Interior Point Methods for Linear Optimization

Abstract

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.

An Inexact Feasible Quantum Interior Point Method for Linearly Constrained Quadratic Optimization

“…This is a vital consideration, as the feasible neighborhood of the central path as outlined in ( 5) is a subset of the primal-dual feasible set; if primal and dual feasibility are not satisfied exactly at any point in the algorithm, the iterates leave this neighborhood and the method fails. Our work fills this gap by using a method inspired by the QIPMs of [13,14].…”

“…A similar system was proposed and called "Orthogonal Subspaces System" (OSS) in [13,14], and we use the same name in this work. The matrix in the OSS system ( 6) is of size n × n, and it is nonsingular.…”

“…As discussed in [14], to solve the OSS system (6) using quantum algorithms, we can first rewrite it as the normalized Hermitian OSS system…”

“…In particular, quantum IPMs (QIPMs) that utilize QLSAs to solve the Newton linear system arising in each iteration have been proposed for LO problems [11,12] and semidefinite optimization problems [13]. To maintain the feasibility of the iterates using quantum subroutines, the authors of [13,14] introduce the so-called orthogonal subspace system (OSS) for SDO and LO problems, and, in particular, demonstrate that a feasible solution to the original Newton system can be recovered from an inexact solution to the OSS. However, linearly constrained quadratic optimization problems, which are fundamental to both optimization and machine learning, have yet to be formally studied in the quantum literature.…”

“…In this work, we generalize the OSS for LO problems in [14] to LCQO problems and provide an efficient method for constructing the OSS using a quantum computer. Using the OSS, we can obtain an inexact feasible IPM, solving for the search directions inexactly but maintaining the feasibility of the iterates throughout the process of our IPM.…”