Quantum searches on highly symmetric graphs

TL;DR

This paper investigates quantum walks on highly symmetric graphs for search problems, demonstrating quadratic speedups and providing analytical solutions due to the graphs' automorphism properties.

Contribution

It introduces a quantum walk framework on symmetric graphs with a quantum oracle for search, analyzing specific graph types and showing quadratic speedups.

Findings

Quadratic quantum speedup in search on symmetric graphs

Analytical solutions due to low-dimensional Hilbert spaces

Effective quantum circuit implementation for these walks

Abstract

We study scattering quantum walks on highly symmetric graphs and use the walks to solve search problems on these graphs. The particle making the walk resides on the edges of the graph, and at each time step scatters at the vertices. All of the vertices have the same scattering properties except for a subset of special vertices. The object of the search is to find a special vertex. A quantum circuit implementation of these walks is presented in which the set of special vertices is specified by a quantum oracle. We consider the complete graph, a complete bipartite graph, and an $M$-partite graph. In all cases, the dimension of the Hilbert space in which the time evolution of the walk takes place is small (between three and six), so the walks can be completely analyzed analytically. Such dimensional reduction is due to the fact that these graphs have large automorphism groups. We find the usual quadratic quantum speedups in all cases considered.