  Distributed consensus algorithm over networks of quantum systems has been the
focus of recent studies in the context of quantum computing and distributed
control. Most of the progress in this category have been on the convergence
conditions and optimizing the convergence rate of the algorithm, for quantum
networks with undirected underlying topology. This paper aims to address the
extension of this problem over quantum networks with directed underlying
graphs. In doing so, the convergence to two different stable states namely,
consensus and synchronous states have been studied. Based on the intertwining
relation between the eigenvalues, it is shown that for determining the
convergence rate to the consensus state, all induced graphs should be
considered while for the synchronous state only the underlying graph suffices.
Furthermore, it is illustrated that for the range of weights that the Aldous'
conjecture holds true, the convergence rate to both states are equal. Using the
Pareto region for convergence rates of the algorithm, the global and Pareto
optimal points for several topologies have been provided.
