On quantum invariants and the graph isomorphism problem

TL;DR

This paper introduces three quantum graph invariants derived from quantum states, demonstrating their ability to distinguish various classes of graphs, but also revealing scalability challenges in their practical application.

Contribution

It presents a novel framework for quantum graph invariants based on measurements, expanding the tools for graph isomorphism testing using quantum states.

Findings

Successfully distinguished all non-isomorphic graphs with up to 9 nodes

Distinguished all strongly regular graphs up to 29 nodes

Preparation complexity scales exponentially with graph size

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.