Large Qudit Limit of One-dimensional Quantum Walks

TL;DR

This paper investigates the long-time behavior of one-dimensional quantum walks with increasing internal states, revealing a transition from multiple peaks to a universal convex distribution as the number of components grows large.

Contribution

It introduces a family of quantum walk models with variable internal states and analyzes their limit distributions, uncovering a universal convex structure in the large-component limit.

Findings

Limit distributions show multiple peaks for finite components.

As the number of components increases, peaks vanish and a convex distribution emerges.

The results suggest a connection between quantum walks and classical diffusion in the large-component limit.

Abstract

We study a series of one-dimensional discrete-time quantum-walk models labeled by half integers $j=1/2, 1, 3/2, ...$, introduced by Miyazaki {\it et al.}, each of which the walker's wave function has $2j+1$ components and hopping range at each time step is $2j$. In long-time limit the density functions of pseudovelocity-distributions are generally given by superposition of appropriately scaled Konno's density function. Since Konno's density function has a finite open support and it diverges at the boundaries of support, limit distribution of pseudovelocities in the $(2j+1)$-component model can have $2j+1$ pikes, when $2j+1$ is even. When $j$ becomes very large, however, we found that these pikes vanish and a universal and monotone convex structure appears around the origin in limit distributions. We discuss a possible route from quantum walks to classical diffusion associated with the $j \to \infty$ limit.