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arXiv:1509.05823 · IEEE Transactions on Automatic Control

Optimizing the Convergence Rate of the Continuous-Time Quantum Consensus

Saber Jafarizadeh

Published Apr 24, 2017• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/tac.2017.2696707

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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.
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.
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.
Distributed gossip algorithm has been studied in literature for practical implementation of the distributed consensus algorithm as a fundamental algorithm for the purpose of in-network collaborative processing. This paper focuses on optimizing the convergence rate of the gossip algorithm for both classical and quantum networks. A novel model of the gossip algorithm with non-uniform clock distribution is proposed which can reach the optimal convergence rate of the continuous-time consensus algorithm. It is described that how the non-uniform clock distribution is achievable by modifying the rate of the Poisson process modeling the clock of the gossip algorithm. The minimization problem for optimizing the asymptotic convergence rate of the proposed gossip algorithm and its corresponding semidefinite programming formulation is addressed analytically. It is shown that the optimal results obtained for uniform clock distribution are suboptimal compared to those of the non-uniform one and for non-uniform distribution the optimal answer is not unique i.e. there is more than one set of probabilities that can achieve the optimal convergence rate. Based on the optimal continuous-time consensus algorithm and the detailed balance property, an effective method of obtaining one of these optimal answers is proposed. Regarding quantum gossip algorithm, by expanding the density matrix in terms of the generalized Gell-Mann matrices, the evolution equation of the quantum gossip algorithm is transformed to the state update equation of the classical gossip algorithm. By defining the quantum gossip operator, the original optimization problem is formulated as a convex optimization problem, which can be addressed by semidefinite programming.
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 proposes a quantum framework for the design of communication topologies in consensus-based multi-agent systems. The communication graph is selected online by solving a mixed-integer quadratic program (MIQP) that minimizes a cost combining communication and distance penalties with degree-regularization terms, while enforcing exact connectivity through a flow-based formulation. To cope with the combinatorial complexity of this NP-hard problem, we develop a three-block ADMM scheme that decomposes the MIQP into a convex quadratic program in relaxed edge and flow variables, a pure binary unconstrained subproblem, and a closed-form auxiliary update. The binary subproblem is mapped to a quadratic unconstrained binary optimization (QUBO) Hamiltonian and approximately solved via quantum imaginary time evolution (QITE). The resulting time-varying, optimizer-generated Laplacians are applied to linear first- and second-order consensus dynamics. Numerical simulations on networks demonstrate that the proposed method produces connected topologies that satisfy degree constraints, achieve consensus, and incur costs comparable to those of classical mixed-integer solvers, thereby illustrating how quantum algorithms can be embedded as topology optimizers within closed-loop distributed control architectures.
In this article, we consider the partial quantum consensus problem of a qubit network in a distributed view. The local quantum operation is designed based on the Hamiltonian by using the local information of each quantum system in a network of qubits. We construct the unitary transformation for each quantum system to achieve the partial quantum consensus, that is, the directions of the quantum states in the Bloch ball will reach an agreement. A simple case of two-qubit quantum systems is considered first, and a minimum completing time of reaching partial consensus is obtained based on the geometric configuration of each qubit. Furthermore, we extend the approaches to deal with the more general N -qubit networks. Two partial quantum consensus protocols, based on the Lyapunov method for chain graphs and the geometry method for connected graphs, are proposed. The geometry method can be utilized to deal with more general connected graphs, while for the Lyapunov method, the global consensus can be obtained. The numerical simulation over a qubit network is demonstrated to verify the validity and the effectiveness of the theoretical results.