We establish a thorough treatment of reduced-state synchronization for 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. We show that reduced-state synchronization is achieved if and only if the quantum interaction graphs corresponding to the permutation operators form a strongly connected union graph. The proof is based on an algebraic analysis making use of the Perron-Frobenius theorem for non-negative matrices. The convergence rate and the limiting orbit are explicitly characterized. Numerical examples are provided illustrating the obtained results. Further, we 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. We show that the quantum synchronization equation is by nature equivalent to several parallel cut-balanced consensus processes, and a necessary and sufficient condition is obtained for quantum reduced-state synchronization under switching interactions applying recent work of Hendrickx and Tsitsiklis.
Developments in continuous-time quantum consensus seeking were made in @cite_12 @cite_13 @cite_4 for Markovian quantum dynamics governed by master equations @cite_11 . In @cite_12 , using a group-theoretic analysis, the authors proposed a consensus master equation involving permutation operators and showed that a symmetric state consensus can be achieved under such evolutions. In @cite_4 , a graphical method was systematically introduced for building the connection between the quantum consensus dynamics and its classical analogue by splitting the evolution of the entries in the network density operator. The idea of breaking down large density operators of multiple qubits can in fact be traced back to @cite_19 using Stokes tensors. A type of quantum synchronization was also shown in the sense that the network trajectory tends to an orbit determined by the network Hamiltonian and the symmetrization of the initial state, which implies that all quantum nodes asymptotically reach the same orbit @cite_4 . For research on quantum network control and information processing we refer to @cite_18 @cite_26 @cite_17 ; for a survey for quantum control theory we refer to @cite_10 .
@cite_18 @cite_26 @cite_4 @cite_10 @cite_17 @cite_19 @cite_13 @cite_12 @cite_11
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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 22n nodes. Existing consensus results on classical networks can thus be adopted to establish the quantum consensus convergence. AbstractIn this paper we propose a continuous-time, dissipative Markov dynamics that asymp-totically drives a network of n-dimensional quantum systems to the set of states that areinvariant under the action of the subsystem permutation group. The Lindblad-type gen-erator of the dynamics is built with two-body subsystem swap operators, thus satisfyinglocality constraints, and preserve symmetric observables. The potential use of the pro-posed generator in combination with local control and measurement actions is illustratedwith two applications: the generation of a global pure state and the estimation of thenetwork size. 1 INTRODUCTION Classical consensus algorithms and the related distributed control problems have recentlygenerated an impressive body of literature, motivated by applications in distributed com-putation and multi-agent coordination, see e.g. [1, 2, 3, 4, 5, 6, 7, 8, 9]. We have recentlyrecast the problem, and a class of algorithms for its solution, as a dynamical symmetriza-tion problem in a group-theoretic framework [10]. This allows to extend the use of simpleand robust algorithms, e.g. of gossip type [11], to new settings and applications. Amongthese, we have studied a quantum version of consensus problems and its applications [12],which can be seen as symmetrization with respect to the subsystem-permutation group, aswell as control methods like quantum dynamical decoupling [13], in which we generally donot have a multipartite structure and symmetrization is attained with respect to other ﬁnitegroups. The emerging dynamics are intrinsically in discrete time, and suitable for sequentialimplementation in dissipative quantum simulators [14]. In this paper, we propose and study a master-equation based approach to drive a quantum network with @math qubits to a consensus (symmetric) state introduced by The state evolution of the quantum network is described by a Lindblad master equation with the Lindblad terms generated by continuous-time swapping operators, which also introduce an underlying interaction graph. We establish a graphical method that bridges the proposed quantum consensus scheme and classical consensus dynamics by studying an induced graph (with @math nodes) of the quantum interaction graph (with @math qubits). A fundamental connection is then shown that quantum consensus over the quantum graph is equivalent to componentwise classical consensus over the induced graph, which allows various existing works on classical consensus to be applicable to the quantum setting. Some basic scaling and structural properties of the quantum induced graph are established via combinatorial analysis. Necessary and sufficient conditions for exponential and asymptotic quantum consensus are obtained, respectively, for switching quantum interaction graphs. As a quantum analogue of classical synchronization of coupled oscillators, quantum synchronization conditions are also presented, in which the reduced states of all qubits tend to a common trajectory. This paper extends the consensus framework, widely studied in the literature on distributed computing and control algorithms, to networks of quantum systems. We define consensus situations on the basis of invariance and symmetry properties, finding four different generalizations of classical consensus states. This new viewpoint can be directly used to study consensus for probability distributions, as these can be seen as a particular case of quantum statistical states: in this light, our analysis is also relevant for classical problems. We then extend the gossip consensus algorithm to the quantum setting and prove it converges to symmetric states while preserving the expectation of permutation-invariant global observables. Applications of the framework and the algorithms to estimation and control problems on quantum networks are discussed.