Survival probability of the Grover walk on the ladder graph

TL;DR

This paper analyzes the survival probability of the Grover quantum walk on a ladder graph with an absorbing sink, revealing how graph modifications influence quantum transport behavior.

Contribution

It provides a closed-form formula for survival probability and shows how attaching a loop alters its dependence on the ladder length.

Findings

Survival probability can increase or decrease with ladder length depending on graph modifications.

A closed formula for survival probability is derived using an orthonormal basis in the dark subspace.

Attaching a loop to the ladder changes the asymptotic behavior of survival probability.

Abstract

We provide a detailed analysis of the survival probability of the Grover walk on the ladder graph with an absorbing sink. This model was discussed in Mare\v s et al., Phys. Rev. A 101, 032113 (2020), as an example of counter-intuitive behaviour in quantum transport where it was found that the survival probability decreases with the length of the ladder $L$, despite the fact that the number of dark states increases. An orthonormal basis in the dark subspace is constructed, which allows us to derive a closed formula for the survival probability. It is shown that the course of the survival probability as a function of $L$ can change from increasing and converging exponentially quickly to decreasing and converging like $L^{-1}$ simply by attaching a loop to one of the corners of the ladder. The interplay between the initial state and the graph configuration is investigated.