Alternate two-dimensional quantum walk with a single-qubit coin

TL;DR

This paper introduces an alternate two-dimensional quantum walk using a single-qubit coin, demonstrating its equivalence to the Grover walk for certain states, and showing it generates more spatial entanglement.

Contribution

It provides a detailed proof of the equivalence between the alternate walk and the Grover walk and extends the analysis to broader initial states and classes of quantum walks.

Findings

The alternate walk reproduces the Grover walk's probability distribution for specific initial states.

It outperforms the Grover walk in generating x-y spatial entanglement.

The equivalence is generalized to wider classes of quantum walks with a limit theorem.

Abstract

We have recently proposed a two-dimensional quantum walk where the requirement of a higher dimensionality of the coin space is substituted with the alternance of the directions in which the walker can move [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. {\bf 106}, 080502 (2011)]. For a particular initial state of the coin, this walk is able to perfectly reproduce the spatial probability distribution of the non-localized case of the Grover walk. Here, we present a more detailed proof of this equivalence. We also extend the analysis to other initial states, in order to provide a more complete picture of our walk. We show that this scheme outperforms the Grover walk in the generation of $x$-$y$ spatial entanglement for any initial condition, with the maximum entanglement obtained in the case of the particular aforementioned state. Finally, the equivalence is generalized to wider classes of quantum walks and a limit theorem for the alternate walk in this context is presented.