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.
@article{arxiv.2009.03618,
title = {An HHL-Based Algorithm for Computing Hitting Probabilities of Quantum Random Walks},
author = {Ji Guan and Qisheng Wang and Mingsheng Ying},
journal= {arXiv preprint arXiv:2009.03618},
year = {2022}
}