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