Spatial search using the discrete time quantum walk

TL;DR

This paper investigates the efficiency of the discrete time quantum walk search algorithm across different spatial dimensions, revealing how its performance depends on graph structure and connectivity, especially in 2D.

Contribution

The study provides detailed numerical analysis of the quantum walk search algorithm's dimension-dependent efficiency and how graph properties influence its performance.

Findings

Quantum walk search in 2D scales as O(√N log N).

Prefactors depend on graph degree and symmetry.

Higher connectivity reduces search time complexity.

Abstract

We study the quantum walk search algorithm of Shenvi, Kempe and Whaley [PRA 67 052307 (2003)] on data structures of one to two spatial dimensions, on which the algorithm is thought to be less efficient than in three or more spatial dimensions. Our aim is to understand why the quantum algorithm is dimension dependent whereas the best classical algorithm is not, and to show in more detail how the efficiency of the quantum algorithm varies with spatial dimension or accessibility of the data. Our numerical results agree with the expected scaling in 2D of $O(\sqrt{N \log N})$, and show how the prefactors display significant dependence on both the degree and symmetry of the graph. Specifically, we see, as expected, the prefactor of the time complexity dropping as the degree (connectivity) of the structure is increased.