Consensus for Quantum Networks: Symmetry From Gossip Interactions

et al. 2017

Sci Rep

We investigate the evolution of the network entropy for consensus dynamics in classical and quantum networks. We show that in the classical case, the network differential entropy is monotonically nonincreasing if the node initial values are continuous random variables. While for quantum consensus dynamics, the network's von Neumann entropy is in contrast non-decreasing. In light of this inconsistency, we compare several distributed algorithms with random or deterministic coefficients for classical or quantum networks, and show that quantum algorithms with deterministic coefficients are physically related to classical algorithms with random coefficients.How agreement emerging among a group of agents interacting each other is an intriguing subject in various research disciplines [1][2][3][4][5] . The fundamental idea lies in that, cooperative decisions can lead to consensus in node states throughout a network even each node can only interact with a few neighboring nodes. Inspired by this, consensus control over networks has been systematically studied in the past decade 6,7 , becoming one of the foundational blocks for engineering solutions to distributed coordination problems in multi-agent systems 8 . Recent work 9, 10 , further developed consensus dynamics for quantum networks, where each node corresponds to a qubit 11 . The concepts regarding the network density matrix as statistical ensembles of pure quantum states, reaching a quantum consensus were systematically developed 9 , and it has been shown that a quantum consensus can be reached with the help of quantum swapping operators for both continuous-time and discrete-time dynamics 9,12 . In fact, the two categories of dynamics over classical and quantum networks can be put together into a group-theoretic framework 10 , and quantum consensus dynamics can in fact be equivalently mapped into certain parallel classical dynamics over disjoint subsets of the entries of the network density matrix 12 . This line of research on consensus dynamics is related to the work on quantum walks over complex networks 13,14 , where associated with the network Laplacian there is a Hermitian operator defining the evolution of the network state in a quantum space.Despite of their algorithmic consistency from a high level between classical and quantum consensus dynamics 10,12 , it is worth investigating the two categories of physical processes from an information perspective. Classical consensus dynamics is realized by nodes observing their neighbors' states (or, their relative states) and then taking real-time feedback. On the other hand, such precise state observation among different components in a quantum network is proven to be impossible 11 , and quantum consensus dynamics is realized via nodes interacting not directly, but with the help of local environments which by themselves are quantum subsystems. Tracing out these local environments, the state evolution of the quantum network follows a master equation with the same attraction nature 12 .To this end, we investigate the ...

Section: Resultsmentioning

confidence: 83%

Entropy Evolution in Consensus Networks

et al. 2017

Sci Rep

“…which corresponds to the quantum symmetric consensus state introduced in [9]. As has been shown in [9], for any ρ, its P-average ρ P is symmetric in the sense that it is invariant under any permutation operation, which immediately implies that the reduced states at each qubit are identical in ρ P .…”

Section: Resultsmentioning

confidence: 83%

“…The system (3) with H = 0 was studied in [14]. The aim of [23], [14] is to drive the quantum system to a symmetric state as introduced in [9]. As a result, P * must be a generating subset of the entire permutation group P in [23], [14].…”

Section: B State Evolutionmentioning

confidence: 99%

“…Sepulchre et al [8] generalized consensus algorithms to noncommutative spaces and presented convergence results for quantum stochastic maps to a fully mixed state. Mazzarella et al [9] made a systematic study of consensus-seeking in quantum networks, where four classes of consensus quantum states based on invariance and symmetry properties were introduced and a quantum gossip algorithm [10] was proposed for reaching a symmetric consensus state over a quantum network. The class of quantum gossip algorithms can be further extended to the so-called symmetrization problems in a group-theoretic framework and be applied to consensus on probability distributions and quantum dynamical decoupling G. Shi, S. Fu are with the College of Engineering and Computer Science, The Australian National University, Canberra ACT 0200, Australia.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quantum network reduced-state synchronization part I-convergence under directed interactions

2015 American Control Conference (ACC)

2015

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. This part of the paper focuses on convergence conditions. We show that reduced-state synchronization is achieved if and only if the quantum permutations 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.

“…As an analogue of classical distributed consensus control [2], [3], [7], [1], recently consensus and synchronization seeking in quantum networks has also drawn attention [8], [9], [11], [14], [21], [22], partially due to the fact that composite quantum systems naturally inherit a network structure [4], [6], [5]. Under quantum mechanics, one must obey the quantum state-space and quantum evolution postulates [4], while the central idea remains the same for collaborative consensus seeking [8], [9].…”

Section: Introductionmentioning

confidence: 99%

Quantum network reduced-state synchronization part II-the missing symmetry and switching interactions

2015 American Control Conference (ACC)

2015

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 reducedstate 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 reducedstate synchronization in light of recent work by Hendrickx and Tsitsiklis [19].