  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.
