On the Optimal Rate of Convergence for Translation-Invariant 1D Quantum Walks

TL;DR

This paper investigates the convergence rate of translation-invariant 1D quantum walks, establishing an optimal rate of n^{-1/3} for the distribution of the scaled position after n steps, with implications for quantum dynamics analysis.

Contribution

The paper proves the convergence rate of quantum walk distributions in 1D and confirms its optimality for specific cases, advancing understanding of quantum walk dynamics.

Findings

Convergence rate of n^{-1/3} in the Lévy metric for 1D quantum walks.

Optimality of the convergence rate for two-dimensional coin quantum walks.

Established the rate for the supremum distance in special quantum walk cases.

Abstract

We study the convergence rate of translation-invariant discrete-time quantum dynamics on a one-dimensional lattice. We prove that the cumulative distributions function of the ballistically scaled position $X(n)/{n}$ after $n$ steps converges at a rate of $n^{-1/3}$ in the L\'evy metric as $n\to\infty$. In the special case of step-coin quantum walks with two-dimensional coin space, we recover the same convergence rate for the supremum distance and prove optimality.