Continuous Time Quantum Consensus & Quantum Synchronisation
For researchers in quantum distributed control, this provides a theoretical extension to directed graphs, but the results are incremental as they rely on known eigenvalue methods and Aldous' conjecture.
This paper extends quantum consensus and synchronization algorithms to directed quantum networks, showing that convergence to consensus depends on all induced graphs while synchronization depends only on the underlying graph, and that under Aldous' conjecture both rates are equal. Pareto optimal convergence rates are provided for several topologies.
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.