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arXiv:1410.5924

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

Guodong Shi, Shuangshuang Fu, Ian R. Petersen

Published Oct 22, 2014• 3 citations
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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.

DOI
10.48550/arxiv.1410.5924

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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 reduced-state 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 reduced-state synchronization in light of recent work by Hendrickx and Tsitsiklis [19].
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In this paper, we investigate the convergence of the state of a quantum network to a consensus (symmetric) state. The state evolution of the quantum network with continuous-time swapping operators can be described by a Lindblad master equation, which also introduces an underlying interaction graph for the network. For a fixed quantum interaction graph, we prove that the state of a quantum network with continuous-time Markovian dynamics converges to a consensus state, with convergence rate given by the smallest nonzero eigenvalue of a matrix serving as the Laplacian of the quantum interaction graph. We show that this convergence rate can be optimized via standard convex programming given a fixed amount of edge weights. For switching quantum interaction graphs, we establish necessary and sufficient conditions for exponential quantum consensus and asymptotic quantum consensus, respectively. The convergence analysis is based on a bridge built between the proposed quantum consensus scheme and classical consensus dynamics, in that quantum consensus of n qubits naturally defines a consensus process on an induced classical graph with 22nnodes. Existing consensus results on classical networks can thus be adopted to establish the quantum consensus convergence.