Quantum walks and elliptic integrals

TL;DR

This paper derives a novel expression for the return probability generating function of a one-dimensional quantum walk using elliptic integrals, paralleling Polya's classical result for random walks.

Contribution

It introduces a new elliptic integral representation for the generating function of a 1D quantum walk's return probability, extending classical random walk theory.

Findings

Derived an elliptic integral expression for the quantum walk's return probability

Established a connection between quantum walks and elliptic integrals

Extended Polya's classical results to quantum walk scenarios

Abstract

Polya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional quantum walk.