Convergence of quantum random walks with decoherence

TL;DR

This paper analyzes how quantum random walks with decoherence converge to normal distributions under certain spectral conditions, providing explicit formulas, examples, and universality classes for their scaling limits and moments.

Contribution

It establishes spectral conditions for convergence of decoherent quantum walks to normal distributions and offers a complete description of their behavior beyond these conditions.

Findings

Convergence to normal distributions under specific eigenvalue conditions.

Explicit formulas for O(2) quantum walks and their moments.

Identification of universality classes for critical exponents.

Abstract

In this paper, we study the discrete-time quantum random walks on a line subject to decoherence. The convergence of the rescaled position probability distribution $p(x,t)$ depends mainly on the spectrum of the superoperator $\mathcal{L}_{kk}$. We show that if 1 is an eigenvalue of the superoperator with multiplicity one and there is no other eigenvalue whose modulus equals to 1, then $\hat {P}(\frac{\nu} {\sqrt t},t)$ converges to a convex combination of normal distributions. In terms of position space, the rescaled probability mass function $p_t (x, t) \equiv p(\sqrt t x, t)$, $ x \in Z/\sqrt t$, converges in distribution to a continuous convex combination of normal distributions. We give an necessary and sufficient condition for a U(2) decoherent quantum walk that satisfies the eigenvalue conditions. We also give a complete description of the behavior of quantum walks whose eigenvalues do not satisfy these assumptions. Specific examples such as the Hadamard walk, walks under real and complex rotations are illustrated. For the O(2) quantum random walks, an explicit formula is provided for the scaling limit of $p(x,t)$ and their moments. We also obtain exact critical exponents for their moments at the critical point and show universality classes with respect to these critical exponents.