Universal Behavior of Quantum Walks with Long-Range Steps

TL;DR

This paper demonstrates that quantum walks with long-range steps exhibit universal behavior for all decay exponents b3  2, with similar probability distributions and displacements, contrasting classical walks which have different universality classes.

Contribution

The study reveals a universal behavior in quantum walks with long-range steps for all b3  2, extending previous understanding and interpolating results for arbitrary b3  2.

Findings

Quantum walks with b3  2 show similar probability distributions.

Mean square displacements are of the same form for b3=2, 4, and nearest neighbor steps.

Universal behavior extends across all b3  2, including interpolations for arbitrary b3  2.

Abstract

Quantum walks with long-range steps $R^{-\gamma}$ ($R$ being the distance between sites) on a discrete line behave in similar ways for all $\gamma\geq2$. This is in contrast to classical random walks, which for $\gamma >3$ belong to a different universality class than for $\gamma \leq 3$. We show that the average probabilities to be at the initial site after time $t$ as well as the mean square displacements are of the same functional form for quantum walks with $\gamma=2$, 4, and with nearest neighbor steps. We interpolate this result to arbitrary $\gamma\geq2$.