Recurrence of biased quantum walks on a line

TL;DR

This paper investigates the recurrence properties of biased quantum walks on a line, demonstrating that unlike classical walks, quantum walks can remain recurrent under bias, with specific parameter ranges identified.

Contribution

It introduces a new analysis of biased quantum walks' recurrence behavior, showing their stability and identifying conditions for recurrence.

Findings

Biased quantum walks can be recurrent under certain parameters.

Recurrence in quantum walks is more stable against bias than in classical walks.

Existence of genuinely biased quantum walks that are recurrent.

Abstract

The Polya number of a classical random walk on a regular lattice is known to depend solely on the dimension of the lattice. For one and two dimensions it equals one, meaning unit probability to return to the origin. This result is extremely sensitive to the directional symmetry, any deviation from the equal probability to travel in each direction results in a change of the character of the walk from recurrent to transient. Applying our definition of the Polya number to quantum walks on a line we show that the recurrence character of quantum walks is more stable against bias. We determine the range of parameters for which biased quantum walks remain recurrent. We find that there exist genuine biased quantum walks which are recurrent.