Diagrammatic Approach to Quantum Search

TL;DR

This paper presents a diagrammatic method for analyzing quantum search algorithms on graphs, enabling estimation of search dynamics in complex subspaces using perturbation theory.

Contribution

It introduces a novel diagrammatic approach combined with degenerate perturbation theory to analyze quantum search on large, complex graphs.

Findings

First example of degenerate perturbation theory applied to graph-based quantum search

Method allows estimation of search dynamics in high-dimensional subspaces

Provides a new tool for analyzing quantum walks on complex graphs

Abstract

We introduce a simple diagrammatic approach for estimating how a randomly walking quantum particle searches on a graph in continuous-time, which involves sketching small weighted graphs with self-loops and considering degenerate perturbation theory's effects on them. Using this method, we give the first example of degenerate perturbation theory solving search on a graph whose evolution occurs in a subspace whose dimension grows with $N$.