Restart expedites quantum walk hitting times

TL;DR

This paper investigates how restart strategies influence quantum walk hitting times, revealing that restart can overcome detection issues like dark states and induce optimal, staircase-like restart timings.

Contribution

It introduces a monitored quantum walk framework to analyze restart effects, highlighting the elimination of dark states and the emergence of staircase optimal restart times.

Findings

Restart removes dark state issues in quantum walks.

Optimal restart times exhibit staircase behavior with sudden drops.

Restart makes detection insensitive to sampling period.

Abstract

Classical first-passage times under restart are used in a wide variety of models, yet the quantum version of the problem still misses key concepts. We study the quantum hitting time with restart using a monitored quantum walk. The restart strategy eliminates the problem of dark states, i.e. cases where the particle evades detection, while maintaining the ballistic propagation which is important for fast search. We find profound effects of quantum oscillations on the restart problem, namely a type of instability of the mean detection time, and optimal restart times that form staircases, with sudden drops as the rate of sampling is modified. In the absence of restart and in the Zeno limit, the detection of the walker is not possible and we examine how restart overcomes this well-known problem, showing that the optimal restart time becomes insensitive to the sampling period.