Monitored Recurrence of a One-parameter Family of Three-state Quantum Walks

TL;DR

This paper investigates the recurrence properties of a one-parameter family of three-state quantum walks on a line, revealing how the Polya number depends on coin parameters and initial states, with insights into quantum state recurrence.

Contribution

It introduces a simplified basis for analysis and explores how coin parameters influence recurrence probabilities in three-state quantum walks.

Findings

Polya number depends on coin parameter and initial coin state

Walks can return to origin with certainty under certain conditions

Analysis of quantum state recurrence provides deeper understanding

Abstract

Monitored recurrence of a one-parameter set of three-state quantum walks on a line is investigated. The calculations are considerably simplified by choosing a suitable basis of the coin space. We show that the Polya number (i.e. the site recurrence probability) depends on the coin parameter and the probability that the walker is initially in a particular coin state for which the walk returns to the origin with certainty. Finally, we present a brief investigation of the exact quantum state recurrence.