Quantum walks can find a marked element on any graph

TL;DR

This paper introduces a new quantum walk method that efficiently finds marked vertices in any graph, improving upon classical methods and applicable even when certain parameters are only approximately known.

Contribution

The authors develop a simpler, more general quantum walk approach that detects and finds marked vertices without requiring the walk to be state-transitive.

Findings

Quantum walks find marked vertices quadratically faster than classical hitting times.

The method works for multiple marked elements using extended hitting time.

Algorithms are provided for cases with approximate parameter knowledge.

Abstract

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set $M$ consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time $HT(P,M)$ of any reversible random walk $P$ on the graph. In the case of multiple marked elements, the number of steps is given in terms of a related quantity $HT^+(\mathit{P,M})$ which we call extended hitting time. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the random walk $P$ and the absorbing walk $P'$, whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of this interpolation. Contrary to previous approaches, our results remain valid when the random walk $P$ is not state-transitive. We also provide algorithms in the cases when only approximations or bounds on parameters $p_M$ (the probability of picking a marked vertex from the stationary distribution) and $HT^+(\mathit{P,M})$ are known.