Optimizing Quantum Walk Search on a Reduced Uniform Complete Multi-Partite Graph

TL;DR

This paper applies the invariant subspace method to optimize quantum walk search algorithms on reduced multi-partite graphs, enhancing understanding of their spectral properties and optimal parameters.

Contribution

It introduces a framework for analyzing and optimizing quantum walk search on complex graphs using eigenbasis formulation and coupling adjustments.

Findings

Eigenbasis formulation facilitates transport between lowest energy states.

Optimal coupling factor preserves search efficiency.

Method improves understanding of quantum walk dynamics on multi-partite graphs.

Abstract

In a recent work by Novo et al. (Sci. Rep. 5, 13304, 2015), the invariant subspace method was applied to the study of continuous-time quantum walk (CTQW). The method helps to reduce a graph into a simpler version that allows more transparent analyses of the quantum walk model. In this work, we adopt the aforementioned method to investigate the optimality of a quantum walk search of a marked element on a complete multi-partite graph. We formulate the eigenbasis that would facilitate the transport between the two lowest energy eigenstates and demonstrate how to set the appropriate coupling factor to preserve the optimality.