Quantum search in structured database using local adiabatic evolution and spectral methods

TL;DR

This paper introduces a quantum search algorithm for structured databases modeled as graphs, utilizing local adiabatic evolution, spectral methods, and Krylov subspace techniques to efficiently find marked elements.

Contribution

It presents a novel systematic approach combining spectral and perturbation methods for quantum search in structured graph-based databases.

Findings

Provides a systematic way to estimate search time for regular graphs

Uses spectral distribution and Krylov methods for algorithm design

Applicable to any undirected regular connected graph

Abstract

Since Grover's seminal work which provides a way to speed up combinatorial search, quantum search has been studied in great detail. We propose a new method for designing quantum search algorithms for finding a marked element in the state space of a graph. The algorithm is based on a local diabatic evolution of the Hamiltonian associated with the graph. The main new idea is to apply some techniques such as Krylov bspace projection methods, Lanczos algorithm and spectral distribution methods. Indeed, using these techniques together with the second-order perturbation theory, we give a systematic method for calculating the approximate search time at which the marked state can be reached. That is, for any undirected regular connected graph which is considered as the state space of the database, the introduced algorithm provides a systematic and programmable way for evaluation of the search time, in terms of the corresponding graph polynomials.