Quantum Search with a Generalized Laplacian

TL;DR

This paper introduces a generalized Laplacian for quantum walks in spin networks, demonstrating enhanced search success probabilities on weighted graphs and proposing a two-stage algorithm for near-perfect results.

Contribution

It extends quantum walk models to a generalized Laplacian, enabling improved search algorithms on weighted graphs with tunable parameters.

Findings

Success probability boosted to over 0.84 with specific weights.

Two-stage algorithm achieves success probability of 0.996.

Standard and signless Laplacians have limited weight configurations.

Abstract

A single excitation in a quantum spin network described by the Heisenberg model can effect a variety of continuous-time quantum walks on unweighted graphs, including those governed by the discrete Laplacian, adjacency matrix, and signless Laplacian. In this paper, we show that the Heisenberg model can effect these three quantum walks on signed weighted graphs, as well as a generalized Laplacian equal to the discrete Laplacian plus a real-valued multiple of the degree matrix, for which the standard Laplacian, adjacency matrix, and signless Laplacian are special cases. We explore the algorithmic consequence of this generalized Laplacian quantum walk when searching a weighted barbell graph consisting of two equal-sized, unweighted cliques connected by a single signed weighted edge or bridge, with the search oracle constituting an external magnetic field in the spin network. We prove that there are two weights for the bridge (which could both be positive, both negative, or one of each, depending on the multiple of the degree matrix) that allow amplitude to cross from one clique to the other -- except for the standard and signless Laplacians that respectively only have one negative or positive weight bridge -- boosting the success probability from 0.5 to 0.820 or 0.843 for each weight. Moreover, one of the weights leads to a two-stage algorithm that further boosts the success probability to 0.996.