Asymptotic velocity of a position-dependent quantum walk

TL;DR

This paper analyzes the asymptotic behavior of a position-dependent quantum walk on the integers, establishing convergence of the scaled position operator to an asymptotic velocity operator and describing the limiting distribution of the walker's position.

Contribution

It proves the convergence of the position operator to an asymptotic velocity operator under position-dependent coin conditions and characterizes the limiting distribution of the walker's scaled position.

Findings

Convergence of the position operator to the asymptotic velocity operator.

Description of the limiting distribution of the scaled position.

Conditions under which the spectral measure influences the walker's asymptotic behavior.

Abstract

We consider a position-dependent coined quantum walk on $\mathbb{Z}$ and assume that the coin operator $C(x)$ satisfies \[ \|C(x) - C_0 \| \leq c_1|x|^{-1-\epsilon}, \quad x \in \mathbb{Z} \] with positive $c_1$ and $\epsilon$ and $C_0 \in U(2)$. We show that the Heisenberg operator $\hat x(t)$ of the position operator converges to the asymptotic velocity operator $\hat v_+$ so that \[ \mbox{s-}\lim_{t \to \infty} {\rm exp}\left( i \xi \frac{\hat x(t)}{t} \right) = \Pi_{\rm p}(U) + {\rm exp}(i \xi \hat v_+) \Pi_{\rm ac}(U) \] provided that $U$ has no singular continuous spectrum. Here $\Pi_{\rm p}(U)$ (resp. $\Pi_{\rm ac}(U)$) is the orthogonal projection onto the direct sum of all eigenspaces (resp. the subspace of absolute continuity) of $U$. We also prove that for the random variable $X_t$ denoting the position of a quantum walker at time $t \in \mathbb{N}$, $X_t/t$ converges in law to a random variable $V$ with the probability distribution \[ \mu_V = \|\Pi_{\rm p}(U)\Psi_0\|^2\delta_0 + \|E_{\hat v_+}(\cdot) \Pi_{\rm ac}(U)\Psi_0\|^2, \] where $\Psi_0$ is the initial state, $\delta_0$ the Dirac measure at zero, and $E_{\hat v_+}$ the spectral measure of $\hat v_+$.