Moments of Coinless Quantum Walks on Lattices

TL;DR

This paper derives analytical expressions for the moments of coinless quantum walks on lattices, showing their ballistic spreading behavior and comparing their efficiency with coined quantum walks.

Contribution

It provides the first detailed analytical analysis of moments in coinless quantum walks on lattices, including explicit calculations and parameter optimization.

Findings

Mean square displacement grows ballistically over time.

Optimal parameters maximize the spread of the walk.

Approximations remain accurate even for small time steps.

Abstract

The properties of the coinless quantum walk model have not been as thoroughly analyzed as those of the coined model. Both evolve in discrete time steps but the former uses a smaller Hilbert space, which is spanned merely by the site basis. Besides, the evolution operator can be obtained using a process of lattice tessellation, which is very appealing. The moments of the probability distribution play an important role in the context of quantum walks. The ballistic behavior of the mean square displacement indicates that quantum-walk-based algorithms are faster than random-walk-based ones. In this paper, we obtain analytical expressions for the moments of the coinless model on $d$-dimensional lattices. The mean square displacement for large times is explicitly calculated for the one- and two-dimensional lattices and, using optimization methods, the parameter values that give the largest spread are calculated and compared with the equivalent ones of the coined model. Although we have employed asymptotic methods, our approximations are accurate even for small numbers of time steps.