Quantum network reduced-state synchronization part II-the missing symmetry and switching interactions

Abstract

We consider reduced-state synchronization of qubit networks with the aim of driving the qubits' reduced states to a common trajectory. The evolution of the quantum network's state is described by a master equation, where the network Hamiltonian is either a direct sum or a tensor product of identical qubit Hamiltonians, and the coupling terms are given by a set of permutation operators over the network. The permutations introduce naturally quantum directed interactions. Part I of the paper establishes synchronization conditions for fixed quantum interactions. In this part of the paper, we further investigate the missing symmetry in the reducedstate synchronization from a graphical point of view. The information-flow hierarchy in quantum permutation operators is characterized by different layers of information-induced graphs, based on which a clear bridge between quantum and classical consensus dynamics is built. We show that the quantum synchronization equation is by nature equivalent to a cut-balanced consensus process. Then a necessary and sufficient condition is obtained for reaching quantum reducedstate synchronization in light of recent work by Hendrickx and Tsitsiklis [19].

Optimizing the Convergence Rate of the Continuous-Time Quantum Consensus

“…Furthermore, they derive the general conditions for convergence and show that convergence is guaranteed provided that some mild assumptions are held. Using the results on convergence, they prove that it ensures asymptotic convergence as well.Authors in [27,28,29] propose a new approach based on the induced graphs of the quantum interaction graph for relating the quantum consensus over the N -qubit network to the classical consensus dynamics. They have shown how to carry out convergence speed optimization of the equivalent classical consensus via convex programming.…”

“…Authors in [27,28,29] propose a new approach based on the induced graphs of the quantum interaction graph for relating the quantum consensus over the N -qubit network to the classical consensus dynamics. They have shown how to carry out convergence speed optimization of the equivalent classical consensus via convex programming.…”

Continuous time quantum consensus & quantum synchronisation

“…Consensus state is the symmetric state which is invariant to all permutations [7]. Synchronous state is the state where the reduced states of the quantum network are driven to a common trajectory [9], [10]. Employing the intertwining relation [11] between the eigenvalues of the Laplacian matrices of the induced graphs, we have shown that the convergence rate to the consensus state is obtained from the spectrum of all induced graphs combined.…”

“…The extension of this class of algorithms to the quantum domain has been addressed in [7] where four different generalizations of classical consensus states have been proposed. In [8], [9], [10] the necessary and sufficient conditions for asymptotic convergence of the quantum consensus algorithm is studied. Optimizing the convergence rate of the algorithm to the consensus state has been addressed in [11], [12].…”

Quantum network reduced-state synchronization part I-convergence under directed interactions

“…In Part II of this work [27], we will further explore the missing symmetry in the reduced-state synchronization from a graphical point of view, and also investigate the convergence to reduced-state synchronization under switching permutations.…”