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

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 $X(n)\mathrm{/}n$ after $n$ steps converges at a rate of $n^{-1\mathrm{/}3}$ in the Lévy metric as $n\to \mathrm{\infty }$. 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.