Near Term Algorithms for Linear Systems of Equations

TL;DR

This paper explores near-term quantum algorithms for solving linear systems, introducing new variational methods and demonstrating their implementation on real quantum hardware to address the limitations of existing algorithms.

Contribution

It introduces several novel variational quantum linear solver methods, including the Evolutionary Ansatz, Logical Ansatz, and Adiabatic Ansatz, with experimental demonstrations.

Findings

First application of Evolutionary Ansatz to VQLS

First implementation of Logical Ansatz VQLS on hardware

Demonstration of CQS method on real quantum devices

Abstract

Finding solutions to systems of linear equations is a common prob\-lem in many areas of science and engineering, with much potential for a speedup on quantum devices. While the Harrow-Hassidim-Lloyd (HHL) quantum algorithm yields up to an exponential speed-up over classical algorithms in some cases, it requires a fault-tolerant quantum computer, which is unlikely to be available in the near term. Thus, attention has turned to the investigation of quantum algorithms for noisy intermediate-scale quantum (NISQ) devices where several near-term approaches to solving systems of linear equations have been proposed. This paper focuses on the Variational Quantum Linear Solvers (VQLS), and other closely related methods. This paper makes several contributions that include: the first application of the Evolutionary Ansatz to the VQLS (EAVQLS), the first implementation of the Logical Ansatz VQLS (LAVQLS), based on the Classical Combination of Quantum States (CQS) method, the first proof of principle demonstration of the CQS method on real quantum hardware and a method for the implementation of the Adiabatic Ansatz (AAVQLS). These approaches are implemented and contrasted.