Mimicking the probability distribution of a two-dimensional Grover walk with a single-qubit coin

TL;DR

This paper demonstrates that a two-dimensional Grover walk's probability distribution can be mimicked using a simpler quantum walk with a single-qubit coin, reducing complexity while maintaining key properties.

Contribution

The authors introduce a novel scheme that reproduces the non-localized spatial density of a 2D Grover walk using only a single-qubit coin, simplifying experimental implementation.

Findings

The scheme accurately reproduces the spatial probability distribution of the Grover walk.

It achieves more efficient entanglement between orthogonal directions.

The approach simplifies experimental realization of multi-dimensional quantum walks.

Abstract

Multi-dimensional quantum walks usually require large coin spaces. Here we show that the non-localized case of the spatial density probability of the two-dimensional Grover walk can be obtained using only a two-dimensional coin space and a quantum walk in alternate directions. We present a formal proof of this correspondence and analyze the behavior of the coin-position entanglement as well as the x-y spatial entanglement in our scheme with respect to the Grover one. We show that our experimentally simpler scheme allows to entangle the two orthogonal directions of the walk more efficiently.