Classification of coined quantum walks on the line and comparison to correlated classical random walks

TL;DR

This paper classifies one-dimensional coined quantum walks, corrects previous amplitude formulas, and compares their spreading and limiting distributions with correlated classical random walks, highlighting quantum advantages.

Contribution

It provides a complete classification of symmetric and general quantum walks, corrected amplitude formulas, and a detailed comparison with classical correlated random walks.

Findings

Quantum walks exhibit quadratic variance growth, unlike classical walks.

Corrected formulas for walk amplitudes improve accuracy of previous models.

Quantum limiting distributions differ significantly from classical counterparts.

Abstract

We present a comprehensive classification of one-dimensional coined quantum walks on the infinite line, focusing on the spatial probability distributions they induce. Building on prior results, we identify all initial coin states that lead to symmetric quantum walks for arbitrary coins, and provide a bijective parametrisation of all symmetric quantum walks modulo distributional equivalence. Extending beyond the symmetric case, we also give a surjective parametrisation of all coined quantum walks under the same equivalence relation and a bijective parametrisation modulo equivalence of the walks' limiting distributions. Furthermore, we derive corrected closed-form expressions for the walk amplitudes, resolving inaccuracies in previous literature, and generalise the approach to the correlated classical random walk. This unified framework enables a direct comparison between quantum and classical dynamics. Additionally, we discuss the asymptotic scaling of variances for both models, identifying quadratic spreading as a hallmark of non-trivial quantum walks and contrasting it with the linear behaviour of classical walks, except at the extremal points of maximal correlation. Finally, we compare the limiting distributions arising from quantum walks with the ones in the classical case.