Invariance in quantum walks with time-dependent coin operators

TL;DR

This paper investigates how certain phase parameters in a time-dependent quantum walk influence its behavior, revealing an invariance in the walk's probabilistic properties due to underlying symmetries.

Contribution

It demonstrates that specific phase adjustments in the coin operator do not affect the quantum walk's probability distribution, highlighting a symmetry that preserves its core features.

Findings

Probabilistic properties remain unchanged under phase adjustments

Identifies symmetry responsible for invariance

Provides insight into control of quantum walk dynamics

Abstract

In this paper we unveil some features of a discrete-time quantum walk on the line whose coin depends on the temporal variable. After considering the most general form of the unitary coin operator, we focus on the role played by the two phase factors that one can incorporate there, and show how both terms influence the evolution of the system. A closer analysis reveals that the probabilistic properties of the motion of the walker remain unaltered when the update rule of these phases is chosen adequately. This invariance is based on a symmetry with consequences not yet fully explored.