Instability in the quantum restart problem

TL;DR

This paper investigates the quantum restart problem, revealing an instability in the optimal mean hitting time due to quantum oscillations, which affects the optimal restart and sampling times in quantum walks.

Contribution

It uncovers a novel instability in the quantum restart problem, demonstrating how quantum oscillations cause large variations in optimal restart times and identifying specific staircase patterns.

Findings

Instability causes large changes in optimal restart time.

Two types of staircase structures depend on parity of distance.

Global minimum of hitting time depends on both restart and sampling times.

Abstract

Repeatedly-monitored quantum walks with a rate $1/\tau$ yield discrete-time trajectories which are inherently random. With these paths the first-hitting time with sharp restart is studied. We find an instability in the optimal mean hitting time, which is not found in the corresponding classical random walk process. This instability implies that a small change in parameters can lead to a rather large change of the optimal restart time. We show that the optimal restart time versus $\tau$, as a control parameter, exhibits sets of staircases and plunges. The plunges, are due to the mentioned instability, which in turn is related to the quantum oscillations of the first-hitting time probability, in the absence of restarts. Furthermore, we prove that there are only two patterns of staircase structures, dependent on the parity of the distance between the target and the source in units of lattice constant. The global minimum of the hitting time, is controlled not only by the restart time, as in classical problems, but also by the sampling time $\tau$. We provide numerical evidence that this global minimum occurs for the $\tau$ minimizing the mean hitting time, given restarts taking place after each measurement. Last but not least, we numerically show that the instability found in this work is relatively robust against stochastic perturbations in the sampling time $\tau$.