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.
Consensus and synchronization of node states in a network of coupled dynamics have been extensively studied in the past decades @cite_24 @cite_36 @cite_31 . The basic idea is that @cite_23 nodes in a network with suitable connectivity can reach a common state or trajectory where all nodes' initial values are encoded with distributed node interactions, and the strong involvement of graph-theoretic approaches has reshaped the study of networked control systems @cite_16 . Excellent results have been derived towards the understandings of how bidirectional or directed, fixed or switching, deterministic or random, node interactions influence the consensus value and the speed of consensus, e.g., @cite_28 @cite_30 @cite_0 @cite_32 .
@cite_30 @cite_36 @cite_28 @cite_32 @cite_24 @cite_0 @cite_23 @cite_31 @cite_16
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We consider the following distributed consensus problem: Each node in a complete communication network of size n initially holds an opinion, which is chosen arbitrarily from a finite set Σ. The system must converge toward a consensus state in which all, or almost all nodes, hold the same opinion. Moreover, this opinion should be valid, i.e., it should be one among those initially present in the system. This condition should be met even in the presence of a malicious adversary who can modify the opinions of a bounded subset of nodes, adaptively chosen in every round. We consider the 3-majority dynamics: At every round, every node pulls the opinion from three random neighbors and sets his new opinion to the majority one (ties are broken arbitrarily). Let k be the number of valid opinions. We show that, if k ≤ nα, where α is a suitable positive constant, the 3-majority dynamics converges in time polynomial in k and log n with high probability even in the presence of an adversary who can affect up to o([EQUATION]) nodes at each round. Previously, the convergence of the 3-majority protocol was known for |Σ| = 2 only, with an argument that is robust to adversarial errors. On the other hand, no anonymous, uniform-gossip protocol that is robust to adversarial errors was known for |Σ| > 2. We study several variants of coordinated consensus in dynamic networks. We assume a synchronous model, where the communication graph for each round is chosen by a worst-case adversary. The network topology is always connected, but can change completely from one round to the next. The model captures mobile and wireless networks, where communication can be unpredictable. In this setting we study the fundamental problems of eventual, simultaneous, and Δ-coordinated consensus, as well as their relationship to other distributed problems, such as determining the size of the network. We show that in the absence of a good initial upper bound on the size of the network, eventual consensus is as hard as computing deterministic functions of the input, e.g., the minimum or maximum of inputs to the nodes. We also give an algorithm for computing such functions that is optimal in every execution. Next, we show that simultaneous consensus can never be achieved in less than n - 1 rounds in any execution, where n is the size of the network; consequently, simultaneous consensus is as hard as computing an upper bound on the number of nodes in the network. For Δ-coordinated consensus, we show that if the ratio between nodes with input 0 and input 1 is bounded away from 1, it is possible to decide in time n-Θ(√ nΔ), where Δ bounds the time from the first decision until all nodes decide. If the dynamic graph has diameter D, the time to decide is min O(nD Δ),n-Ω(nΔ D) , even if D is not known in advance. Finally, we show that (a) there is a dynamic graph such that for every input, no node can decide before time n-O(Δ0.28n0.72); and (b) for any diameter D = O(Δ), there is an execution with diameter D where no node can decide before time Ω(nD Δ). To our knowledge, our work constitutes the first study of Δ-coordinated consensus in general graphs. The paper investigates the synchronization of a network of identical linear state-space models under a possibly time-varying and directed interconnection structure. The main result is the construction of a dynamic output feedback coupling that achieves synchronization if the decoupled systems have no exponentially unstable mode and if the communication graph is uniformly connected. The result can be interpreted as a generalization of classical consensus algorithms. Stronger conditions are shown to be sufficient-but to some extent, also necessary-to ensure synchronization with the diffusive static output coupling often considered in the literature. We consider the binary consensus problem where each node in the network initially observes one of two states and the goal for each node is to eventually decide which one of the two states was initially held by the majority of the nodes. Each node contacts other nodes and updates its current state based on the state communicated by the last contacted node. We assume that both signaling (the information exchanged at node contacts) and memory (computation state at each node) are limited and restrict our attention to systems where each node can contact any other node (i.e., complete graphs). It is well known that for systems with binary signaling and memory, the probability of reaching incorrect consensus is equal to the fraction of nodes that initially held the minority state. We show that extending both the signaling and memory by just one state dramatically improves the reliability and speed of reaching the correct consensus. Specifically, we show that the probability of error decays exponentially with the number of nodes N and the convergence time is logarithmic in N for large N. We also examine the case when the state is ternary and signaling is binary. The convergence of this system to consensus is again shown to be logarithmic in N for large N, and is therefore faster than purely binary systems. The type of distributed consensus problems that we study arises in the context of decentralized peer-to-peer networks, e.g. sensor networks and opinion formation in social networks - our results suggest that robust and efficient protocols can be built with rather limited signaling and memory.