Classical approach to the graph isomorphism problem using quantum walks

TL;DR

This paper explores quantum walks as a basis for graph isomorphism algorithms, demonstrating their effectiveness and potential advantages over classical methods in identifying graph isomorphism.

Contribution

It develops both classical and quantum algorithms based on quantum walks for graph isomorphism, highlighting their efficiency and applicability to large graph databases.

Findings

Quantum walks differ significantly from classical walks in properties.

Quantum walk-based algorithms effectively identify isomorphism classes of large graphs.

Classical simulation of quantum walks is computationally efficient for practical purposes.

Abstract

Given the extensive application of classical random walks to classical algorithms in a variety of fields, their quantum analogue in quantum walks is expected to provide a fruitful source of quantum algorithms. So far, however, such algorithms have been scarce. In this work, we enumerate some important differences between quantum and classical walks, leading to their markedly different properties. We show that for many practical purposes, the implementation of quantum walks can be efficiently achieved using a classical computer. We then develop both classical and quantum graph isomorphism algorithms based on discrete-time quantum walks. We show that they are effective in identifying isomorphism classes of large databases of graphs, in particular groups of strongly regular graphs. We consider this approach to represent a promising candidate for an efficient solution to the graph isomorphism problem, and believe that similar methods employing quantum walks, or derivatives of these walks, may prove beneficial in constructing other algorithms for a variety of purposes.