Reaching Quantum Consensus with Directed Links: Missing Symmetry and Switching Interactions

Abstract: In this paper, we study consensus seeking of quantum networks under directed interactions defined by a set of permutation operators among a network of qubits. The state evolution of the quantum network is described by a continuous-time master equation, for which we establish an unconditional convergence result indicating that the network state always converges with the limit determined by the generating subgroup of the permutations making use of the Perron-Frobenius theory. We also give a tight graphical criterion regarding when such limit admits a reduced-state consensus. Further, we provide a clear description to the missing symmetry in the reduced-state consensus from a graphical point of view, where the information-flow hierarchy in quantum permutation operators is characterized by different layers of information-induced graphs. Finally, we investigate quantum synchronization in the presence of network Hamiltonian, study quantum consensus conditions under switching interactions, and present a few numerical examples illustrating the obtained results.