One Dimensional Quantum Walks with Memory

TL;DR

This paper explores one-dimensional quantum walks with a two-step memory, deriving their amplitudes and probabilities, and highlighting differences from classical and memoryless quantum walks.

Contribution

It introduces a quantum walk model with memory of two previous steps and derives its fundamental properties, extending the understanding of quantum walk dynamics.

Findings

Quantum walks with memory differ significantly from classical and memoryless quantum walks.

Explicit formulas for amplitudes and probabilities are derived.

Memory influences the walk's behavior and potential applications.

Abstract

We investigate the quantum versions of a one-dimensional random walk, whose corresponding Markov Chain is of order 2. This corresponds to the walk having a memory of up to two previous steps. We derive the amplitudes and probabilities for these walks, and point out how they differ from both classical random walks, and quantum walks without memory.