Hitting statistics from quantum jumps

TL;DR

This paper introduces a method to analyze hitting times in continuous-time open quantum walks using quantum jumps, deriving distributions and moments, and ensuring consistency with previous discrete-time models.

Contribution

It develops a quantum jump-based framework for continuous-time hitting times, including explicit formulas and a remedy for non-incoherent states.

Findings

Derived the distribution of hitting times for quantum walks.

Established consistency with previous discrete-time models.

Proposed a remedy for non-incoherent final states.

Abstract

We define the hitting time for a model of continuous-time open quantum walks in terms of quantum jumps. Our starting point is a master equation in Lindblad form, which can be taken as the quantum analogue of the rate equation for a classical continuous-time Markov chain. The quantum jump method is well known in the quantum optics community and has also been applied to simulate open quantum walks in discrete time. This method however, is well-suited to continuous-time problems. It is shown here that a continuous-time hitting problem is amenable to analysis via quantum jumps: The hitting time can be defined as the time of the first jump. Using this fact, we derive the distribution of hitting times and explicit exressions for its statistical moments. Simple examples are considered to illustrate the final results. We then show that the hitting statistics obtained via quantum jumps is consistent with a previous definition for a measured walk in discrete time [Phys.~Rev.~A~{\bf 73}, 032341 (2006)] (when generalised to allow for non-unitary evolution and in the limit of small time steps). A caveat of the quantum-jump approach is that it relies on the final state (the state which we want to hit) to share only incoherent edges with other vertices in the graph. We propose a simple remedy to restore the applicability of quantum jumps when this is not the case and show that the hitting-time statistics will again converge to that obtained from the measured discrete walk in appropriate limits.