Nonreversal and nonrepeating quantum walks

TL;DR

This paper introduces two novel variations of discrete-time quantum walks, the nonreversal and nonrepeating walks, which modify the walk dynamics using new coin operators, leading to unique properties and initial condition independence.

Contribution

The paper presents the design and analysis of two new quantum walk models with specific constraints, and proves their moment properties are independent of initial conditions.

Findings

Even-order joint moments are initial-condition independent for the nonrepeating walk.

Numerical evidence suggests similar independence for the nonreversal walk.

Contrasts with traditional coins like the Grover operator, where initial conditions influence the walk.

Abstract

We introduce a variation of the discrete time quantum walk, the nonreversal quantum walk, which does not step back onto a position which it has just occupied. This allows us to simulate a dimer and we achieve it by introducing a new type of coin operator. The nonrepeating walk, which never moves in the same direction in consecutive time steps, arises by a permutation of this coin operator. We describe the basic properties of both walks and prove that the even-order joint moments of the nonrepeating walker are independent of the initial condition, being determined by five parameters derived from the coin instead. Numerical evidence suggests that the same is the case for the nonreversal walk. This contrasts strongly with previously studied coins, such as the Grover operator, where the initial condition can be used to control the standard deviation of the walker.