Finding a marked node on any graph by continuous-time quantum walk

TL;DR

This paper demonstrates that continuous-time quantum walks can be used to find marked nodes on any graph in quadratically faster time than classical methods, bridging the gap with discrete-time quantum walk algorithms.

Contribution

It introduces a new continuous-time quantum walk algorithm that finds marked nodes on any ergodic, reversible Markov chain in square root of the classical hitting time.

Findings

New continuous-time quantum walk algorithm for marked node search

Achieves quadratic speedup in hitting time for any ergodic, reversible Markov chain

Establishes connection between discrete-time and continuous-time quantum walks

Abstract

Spatial search by discrete-time quantum walk can find a marked node on any ergodic, reversible Markov chain $P$ quadratically faster than its classical counterpart, i.e.\ in a time that is in the square root of the hitting time of $P$. However, in the framework of continuous-time quantum walks, it was previously unknown whether such general speed-up is possible. In fact, in this framework, the widely used quantum algorithm by Childs and Goldstone fails to achieve such a speedup. Furthermore, it is not clear how to apply this algorithm for searching any Markov chain $P$. In this article, we aim to reconcile the apparent differences between the running times of spatial search algorithms in these two frameworks. We first present a modified version of the Childs and Goldstone algorithm which can search for a marked element for any ergodic, reversible $P$ by performing a quantum walk on its edges. Although this approach improves the algorithmic running time for several instances, it cannot provide a generic quadratic speedup for any $P$. Secondly, using the framework of interpolated Markov chains, we provide a new spatial search algorithm by continuous-time quantum walk which can find a marked node on any $P$ in the square root of the classical hitting time. In the scenario where multiple nodes are marked, the algorithmic running time scales as the square root of a quantity known as the extended hitting time. Our results establish a novel connection between discrete-time and continuous-time quantum walks and can be used to develop a number of Markov chain-based quantum algorithms.