Conditional Probability Distributions of Finite Absorbing Quantum Walks

TL;DR

This paper analyzes the global behavior of finite absorbing one-dimensional quantum walks, revealing distinct wave phenomena and quantum revivals over different timescales through spectral analysis.

Contribution

It introduces a spectral approach to study finite absorbing quantum walks and demonstrates the existence of quantum revivals in regular quantum walk systems.

Findings

Distinct wave reflections at timescale t=O(n)

Quantum revivals observed at t=O(n^2)

Stability of distributions at t=O(n^3)

Abstract

Quantum walks are known to have nontrivial interactions with absorbing boundaries. In particular it has been shown that an absorbing boundary in the one dimensional quantum walk partially reflects information, as observed by absorption probability computations. In this paper, we shift our sights from the local phenomena of absorption probabilities to the global behavior of finite absorbing quantum walks in one dimension. We conduct our analysis by approximating the eigenbasis of the associated absorbing quantum walk operator matrix Q_n where n is the lattice size. The conditional probability distributions of these finite absorbing quantum walks exhibit distinct behavior at various timescales, namely wavelike reflections for t = O(n), fractional quantum revivals for t = O(n^2), and stability for t = O(n^3). At the end of this paper, we demonstrate the existence of these quantum revivals in other sufficiently regular quantum walk systems.