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Computational advantages gained by quantum algorithms rely largely on thecoherence of quantum devices and are generally compromised by decoherence. Asan exception, we present a quantum algorithm for graph isomorphism testingwhose performance is optimal when operating in the partially coherent regime,as opposed to the extremes of fully coherent or classical regimes. Thealgorithm builds on continuous-time quantum stochastic walks (QSWs) on graphsand the algorithmic performance is quantified by the distinguishing power (DIP)between non-isomorphic graphs. The QSW explores the entire graph and acquiresinformation about the underlying structure, which is extracted by monitoringstochastic jumps across an auxiliary edge. The resulting counting statistics ofstochastic jumps is used to identify the spectrum of the dynamical generator ofthe QSW, serving as a novel graph invariant, based on which non-isomorphicgraphs are distinguished. We provide specific examples of non-isomorphic graphsthat are only distinguishable by QSWs in the presence of decoherence.

Decoherence enhances performance of quantum walks applied to graph isomorphism testing