Quantum invariants and the graph isomorphism problem

Three 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 different 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 nonisomorphic graphs with nine 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 nonisomorphic 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.