Quantum random walks in one dimension

TL;DR

This paper analyzes one-dimensional quantum random walks using matrix methods, deriving moments, limit theorems, and symmetry conditions, highlighting fundamental differences from classical random walks.

Contribution

It provides a combinatorial expression for moments, explores dependence on initial states and unitary matrices, and establishes new limit theorems and symmetry conditions.

Findings

Derived combinatorial expression for the mth moment.

Established new limit theorems for quantum walks.

Identified conditions for symmetry of the walk's distribution.

Abstract

This letter treats the quantum random walk on the line determined by a 2 times 2 unitary matrix U. A combinatorial expression for the mth moment of the quantum random walk is presented by using 4 matrices, P, Q, R and S given by U. The dependence of the mth moment on U and initial qubit state phi is clarified. A new type of limit theorems for the quantum walk is given. Furthermore necessary and sufficient conditions for symmetry of distribution for the quantum walk is presented. Our results show that the behavior of quantum random walk is striking different from that of the classical ramdom walk.