Limit distributions of two-dimensional quantum walks

TL;DR

This paper analytically investigates the long-term behavior of a family of two-dimensional quantum walks, deriving their limit distributions and analyzing how initial conditions and parameters influence their symmetry and localization.

Contribution

It provides the first comprehensive analytical derivation of the limit distributions for a broad class of two-dimensional quantum walks, including the Grover walk, and explores parameter and initial state effects.

Findings

Limit distributions are explicitly derived for the quantum walk family.

The dependence of the distribution on coin parameters and initial states is characterized.

Symmetry and localization properties of the limit distribution are fully described.

Abstract

One-parameter family of discrete-time quantum-walk models on the square lattice, which includes the Grover-walk model as a special case, is analytically studied. Convergence in the long-time limit $t \to \infty$ of all joint moments of two components of walker's pseudovelocity, $X_t/t$ and $Y_t/t$, is proved and the probability density of limit distribution is derived. Dependence of the two-dimensional limit density function on the parameter of quantum coin and initial four-component qudit of quantum walker is determined. Symmetry of limit distribution on a plane and localization around the origin are completely controlled. Comparison with numerical results of direct computer-simulations is also shown.