An HHL-Based Algorithm for Computing Hitting Probabilities of Quantum Random Walks

TL;DR

This paper introduces a quantum algorithm based on HHL for efficiently computing hitting probabilities in one-dimensional quantum random walks with absorbing boundaries, outperforming classical methods.

Contribution

It applies the HHL quantum algorithm to a new problem in quantum walks, providing a faster solution for calculating hitting probabilities.

Findings

Quantum algorithm reduces computation time compared to classical methods.

HHL-based approach successfully computes hitting probabilities for general quantum walks.

Numerical experiments confirm the efficiency of the proposed quantum algorithm.

Abstract

We present a novel application of the HHL (Harrow-Hassidim-Lloyd) algorithm -- a quantum algorithm solving systems of linear equations -- in solving an open problem about quantum random walks, namely computing hitting (or absorption) probabilities of a general (not only Hadamard) one-dimensional quantum random walks with two absorbing boundaries. This is achieved by a simple observation that the problem of computing hitting probabilities of quantum random walks can be reduced to inverting a matrix. Then a quantum algorithm with the HHL algorithm as a subroutine is developed for solving the problem, which is faster than the known classical algorithms by numerical experiments.