Matrix Inversion by Quantum Walk

TL;DR

This paper introduces a simplified quantum algorithm for matrix inversion by replacing complex phase estimation steps with a continuous time quantum walk, leveraging weak couplings and perturbation theory.

Contribution

It presents a novel approach that simplifies the HHL algorithm using quantum walks, reducing complexity in quantum matrix inversion.

Findings

Simplified the HHL algorithm with quantum walks

Achieved efficient matrix inversion using weak couplings

Potentially broad applications in quantum computing

Abstract

The HHL algorithm for matrix inversion is a landmark algorithm in quantum computation. Its ability to produce a state $|x\rangle$ that is the solution of $Ax=b$, given the input state $|b\rangle$, is envisaged to have diverse applications. In this paper, we substantially simplify the algorithm, originally formed of a complex sequence of phase estimations, amplitude amplifications and Hamiltonian simulations, by replacing the phase estimations with a continuous time quantum walk. The key technique is the use of weak couplings to access the matrix inversion embedded in perturbation theory.