Quantum Walks, Feynman Propagators and Graph Topology on an IBM Quantum Computer

TL;DR

This paper demonstrates how quantum walk algorithms can be used on IBM quantum computers to analyze graph topologies, revealing features faster than classical methods by leveraging quantum superposition.

Contribution

It introduces a method for applying quantum walks to topological data analysis and relates quantum walk amplitudes to Feynman propagators, with experimental validation on IBM quantum hardware.

Findings

Quantum walks can identify graph features efficiently.

Results on IBM quantum hardware agree with classical simulations.

Quantum approach offers potential speedups in topological data analysis.

Abstract

Topological data analysis is a rapidly developing area of data science where one tries to discover topological patterns in data sets to generate insight and knowledge discovery. In this project we use quantum walk algorithms to discover features of a data graph on which the walk takes place. This can be done faster on quantum computers where all paths can be explored using superposition. We begin with simple walks on a polygon and move up to graphs described by higher dimensional meshes. We use insight from the physics description of quantum walks defined in terms of probability amplitudes to go from one site on a graph to another distant site and show how this relates to the Feynman propagator or Kernel in the physics terminology. Our results from quantum computation using IBM's Qiskit quantum computing software were in good agreement with those obtained using classical computing methods.