Continuous Time Quantum Consensus & Quantum Synchronisation

TL;DR

This paper extends quantum consensus algorithms to directed networks, analyzing convergence to consensus and synchronization states, and identifying optimal conditions and rates for various topologies.

Contribution

It introduces convergence analysis for quantum networks with directed graphs and compares conditions for consensus and synchronization states.

Findings

Convergence to consensus depends on all induced graphs.

Synchronization convergence depends only on the underlying graph.

Optimal convergence rates are identified for different topologies.

Abstract

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.