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arXiv:1605.06976 · Australian Control Conference

Continuous time quantum consensus & quantum synchronisation

Saber Jafarizadeh

Published Nov 1, 2016• 3 citations
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Abstract

Distributed consensus algorithm over networks of quantum systems has been the focus of recent studies in the context of quantum computing and distributed control. Most of the progress in this category have been on the convergence conditions and optimizing the convergence rate of the algorithm, for quantum networks with undirected underlying topology. This paper aims to address the extension of this problem over quantum networks with directed underlying graphs. In doing so, the convergence to two different stable states namely, consensus and synchronous states have been studied. Based on the intertwining relation between the eigenvalues, it is shown that for determining the convergence rate to the consensus state, all induced graphs should be considered while for the synchronous state only the underlying graph suffices. Furthermore, it is illustrated that for the range of weights that the Aldouss conjecture holds true, the convergence rate to both states are equal. Using the Pareto region for convergence rates of the algorithm, the global and Pareto optimal points for several topologies have been provided.

DOI
10.1109/aucc.2016.7868219

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Distributed consensus algorithm over networks of quantum systems has been the focus of recent studies in the context of quantum computing and distributed control. Most of the progress in this category have been on the convergence conditions and optimizing the convergence rate of the algorithm, for quantum networks with undirected underlying topology. This paper aims to address the extension of this problem over quantum networks with directed underlying graphs. In doing so, the convergence to two different stable states namely, consensus and synchronous states have been studied. Based on the intertwining relation between the eigenvalues, it is shown that for determining the convergence rate to the consensus state, all induced graphs should be considered while for the synchronous state only the underlying graph suffices. Furthermore, it is illustrated that for the range of weights that the Aldouss conjecture holds true, the convergence rate to both states are equal. Using the Pareto region for convergence rates of the algorithm, the global and Pareto optimal points for several topologies have been provided.
Inspired by the recent developments in the field of distributed quantum computing, distributed quantum systems are analyzed as networks of quantum systems. This gives rise to the distributed quantum consensus algorithms. Focus of this paper is on optimizing the convergence rate of the continuous-time quantum consensus algorithm over a quantum network with N qudits. It is shown that the optimal convergence rate is independent of the value of d in qudits. By classifying the induced graphs as the Schreier graphs, they are categorized in terms of the partitions of integer N. The intertwining relation is established between one-level dominant partitions in the Hasse Diagram of integer N. Based on this result, the proof of the Aldous' conjecture is extended to all possible induced graphs, and the original optimization problem is reduced to optimizing algebraic connectivity of the smallest induced graph. Utilizing the generalization of Aldous' conjecture, it is shown that the convergence rates of the algorithm to both the consensus state and the reduced quantum state consensus are the same. By providing the analytical solution to semidefinite programming formulation of the obtained problem, closed-form expressions for the optimal results are provided for a range of topologies.
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.
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 introduce 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 2 nodes. Existing consensus results on classical networks can thus be adopted to establish the quantum consensus convergence.
This paper presents a control-theoretic survey of four representative quantum consensus algorithms, with a focus on their convergence properties and implementation feasibility. The selected algorithms—ranging from gossip-based method to measurement-based feedback method and geometric control methods—exhibit fundamentally different convergence targets, including symmetric state consensus and directional alignment of Bloch vectors. Each approach is analyzed in terms of its dynamical structure, physical requirements, and control complexity. To show their performance, numerical simulations are conducted on a 3-qubit network under a fully connected topology. The results imply the trade-offs between convergence strength, coherence preservation, and practical implementability, offering insights for the design of distributed control protocols in quantum networks.
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Quantum hybrid networks where classical and quantum components coexist play a fundamental role in quantum communications. We consider a basic quantum hybrid network model consisting of a number of nodes each holding a qubit, for which the aim is to drive the network to a consensus in the sense that all qubits reach a common state. Projective measurements are applied serving as control means, and the measurement results are exchanged among the nodes via classical communication channels. We show how to carry out centralized optimal path planning for the network with all-to-all classical communications, in which case the problem becomes a stochastic optimal control problem with a continuous action space. To overcome the computation and communication obstacles facing the centralized solutions, we also develop a distributed Pairwise Qubit Projection (PQP) algorithm, where pairs of nodes meet at a given time and respectively perform measurements at their geometric average. We show that the qubit states are driven to a consensus almost surely along the proposed PQP algorithm, and that the expected qubit density operators converge to the average of the network's initial values.