The Elephant Quantum Walk

TL;DR

This paper introduces the Elephant Quantum Walk, a quantum analogue of the classical Elephant Random Walk, demonstrating robust superballistic diffusion driven primarily by memory effects rather than microscopic parameters.

Contribution

It presents a quantum walk model with long-range memory, showing superballistic behavior independent of the coin operator, highlighting the dominant role of memory extension.

Findings

The quantum walk exhibits superballistic diffusion with $\sigma \,\propto \, t^{3/2}$.

Superdiffusivity is mainly governed by memory extension, not microscopic parameters.

The diffusion coefficient depends on parameters reflecting the process's memory extension.

Abstract

We explore the impact of long-range memory on the properties of a family of quantum walks in a one-dimensional lattice and discrete time, which can be understood as the quantum version of the classical "Elephant Random Walk" non-Markovian process. This Elephant Quantum Walk is robustly superballistic with the standard deviation showing a constant exponent, $\sigma \propto t^{3/ 2}$ , whatever the quantum coin operator, on which the diffusion coefficient is dependent. On the one hand, this result indicates that contrarily to the classical case, the degree of superdiffusivity in quantum non- Markovian processes of this kind is mainly ruled by the extension of memory rather than other microscopic parameters that explicitly define the process. On the other hand, these parameters reflect on the diffusion coefficient.