One dimensional lazy quantum walks and occupancy rate

TL;DR

This paper analyzes lazy quantum walks, showing they have similar moments to normal walks and higher occupancy rates, indicating more uniform probability distribution across positions.

Contribution

It introduces occupancy number and rate for lazy quantum walks, demonstrating their higher occupancy compared to other quantum and classical walks.

Findings

Lazy quantum walks have the same order of moments as normal quantum walks.

Lazy quantum walks with DFT coin have similar distribution to Hadamard walks.

Lazy quantum walks exhibit higher occupancy rates than other walk types.

Abstract

Lazy quantum walks were presented by Andrew M. Childs to prove that the continuous-time quantum walk is a limit of the discrete-time quantum walk [Commun.Math.Phys.294,581-603(2010)]. In this paper, we discuss properties of lazy quantum walks. Our analysis shows that lazy quantum walks have $O(t^n)$ order of the n-th moment of the corresponding probability distribution, which is the same as that for normal quantum walks. Also, the lazy quantum walk with DFT (Discrete Fourier Transform) coin operator has a similar probability distribution concentrated interval to that of the normal Hadamard quantum walk. Most importantly, we introduce the concepts of occupancy number and occupancy rate to measure the extent to which the walk has a (relatively) high probability at every position in its range. We conclude that lazy quantum walks have a higher occupancy rate than other walks such as normal quantum walks, classical walks and lazy classical walks.