Gaussian Amplitude Amplification for Quantum Pathfinding

TL;DR

This paper introduces a quantum amplitude amplification method tailored for pathfinding on weighted graphs, leveraging Gaussian distributions to enhance solution probabilities and applying it to problems like the Traveling Salesman.

Contribution

It develops a novel oracle and circuit design that utilizes Gaussian distributions for improved quantum pathfinding and explores its effectiveness on randomized and classical optimization problems.

Findings

Effective amplification of minimum/maximum solutions on weighted graphs.

Potential application to the Traveling Salesman problem.

Demonstrated capability on randomized weight cases.

Abstract

We study an oracle operation, along with its circuit design, which combined with the Grover diffusion operator boosts the probability of finding minimum or maximum solutions on a weighted directed graph. We focus on a geometry of sequentially connected bipartite graphs, which naturally gives rise to solution spaces describable by gaussian distributions. We then demonstrate how an oracle which encodes these distributions can be used to solve for the optimal path via amplitude amplification. And finally, we explore the degree to which this algorithm is capable of solving cases which are generated using randomized weights, as well as a theoretical application for solving the Traveling Salesman problem.