Quantum invariants and the graph isomorphism problem

P W Mills,V M Dwyer,R P Rundle,Todd Tilma,Simon J Devitt,Mark J Everitt,J H Samson

doi:10.1103/physreva.100.052317

Abstract

Three new graph invariants are introduced which may be measured from a quantum graph state and form examples of a framework under which other graph invariants can be constructed. Each invariant is based on distinguishing a different number of qubits. This is done by applying alternate measurements to the qubits to be distinguished. The performance of these invariants is evaluated and compared to classical invariants. We verify that the invariants can distinguish all non-isomorphic graphs with 9 or fewer nodes. The invariants have also been applied to `classically hard' strongly regular graphs, successfully distinguishing all strongly regular graphs of up to 29 nodes, and preliminarily to weighted graphs. We have found that although it is possible to prepare states with a polynomial number of operations, the average number of preparations required to distinguish non-isomorphic graph states scales exponentially with the number of nodes. We have so far been unable to find operators which reliably compare graphs and reduce the required number of preparations to feasible levels.

Highlights

A graph is a set of nodes connected by edges, and two graphs are termed isomorphic if one may be obtained from the other by permuting the labels of their nodes [1]
III B measurements from the equal-angle slice of the Wigner function are used which have the form of Eq (4) and treat no qubits differently. This results in a natural graph invariant which identifies more than 99.8% of the graphs we have tested, outperforming all of the classical invariants we consider
We have shown in calculations that α = 1 retains the ability to distinguish graph states, yet provides a much improved signal-to-noise ratio (SNR)

Summary

Introduction

A graph is a set of nodes connected by edges, and two graphs are termed isomorphic if one may be obtained from the other by permuting the labels of their nodes [1]. In the field of image recognition, including registration problems in computer vision [4] and medical imaging (for example, automated histology analysis [5]), graphs are used as effective structural descriptors due to their ability to represent relational information in which nodes are associated to image components and edges are associated to the relationships between them.

Methods

Findings

Discussion

Conclusion