Quantum walk hydrodynamics

TL;DR

This paper introduces a hydrodynamic interpretation of discrete-time quantum walks via a relativistic Madelung transform, demonstrating shock formation and analyzing asymptotic structures with numerical and analytical methods.

Contribution

It presents a novel relativistic generalization of the Madelung transform for quantum walks, linking quantum dynamics to hydrodynamics and analyzing shock phenomena.

Findings

Quantum walks can produce hydrodynamical shocks under suitable initial conditions

An analytical form of the quantum shock structure is derived

Numerical simulations confirm shock formation and structure

Abstract

A simple Discrete-Time Quantum Walk on the line is revisited and given an hydrodynamic interpretation through a novel relativistic generalization of the Madelung transform. Numerical results are presented which show that suitable initial conditions indeed produce hydrodynamical shocks. An analytical computation of the asymptotic quantum shock structure is presented. The non-relativistic limit is explored in the Supplementary Material (SM).