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Consensus for Quantum Networks: Symmetry From Gossip Interactions

Luca Mazzarella, Alain Sarlette, Francesco Ticozzi

Published Dec 3, 2015• 58 citations
OpenAlex

Abstract

Abstract—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 consen-sus 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. Index Terms—Consensus, distributed algorithms, Markov pro-cesses, quantum information and control. I.

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Consensus for Quantum Networks: Symmetry From Gossip Interactions

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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.

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This paper extends the gossip algorithm, widely studied in the literature on distributed computing and control algorithms, to networks of quantum systems. In doing so, we reinterpret the classical algorithm and the average consensus task as a symmetrization problem with respect to the action of the permutation group. This allows us to extend in a natural way the gossip consensus algorithm to the quantum setting and prove its convergence properties to symmetric states while preserving the expectation of permutation-invariant global observables.

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Reaching Quantum Consensus with Directed Links: Missing Symmetry and Switching Interactions

Guodong Shi, Shuangshuang Fu, Ian R. Petersen

In this paper, we study consensus seeking of quantum networks under directed interactions defined by a set of permutation operators among a network of qubits. The state evolution of the quantum network is described by a continuous-time master equation, for which we establish an unconditional convergence result indicating that the network state always converges with the limit determined by the generating subgroup of the permutations making use of the Perron-Frobenius theory. We also give a tight graphical criterion regarding when such limit admits a reduced-state consensus. Further, we provide a clear description to the missing symmetry in the reduced-state consensus from a graphical point of view, where the information-flow hierarchy in quantum permutation operators is characterized by different layers of information-induced graphs. Finally, we investigate quantum synchronization in the presence of network Hamiltonian, study quantum consensus conditions under switching interactions, and present a few numerical examples illustrating the obtained results.

arXiv:1410.5924•Oct 22, 2014•3 citations
Abstract—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 consen-sus 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. Index Terms—Consensus, distributed algorithms, Markov pro-cesses, quantum information and control. I.
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.
This paper extends the gossip algorithm, widely studied in the literature on distributed computing and control algorithms, to networks of quantum systems. In doing so, we reinterpret the classical algorithm and the average consensus task as a symmetrization problem with respect to the action of the permutation group. This allows us to extend in a natural way the gossip consensus algorithm to the quantum setting and prove its convergence properties to symmetric states while preserving the expectation of permutation-invariant global observables.
Consensus algorithms are mechanisms driving nodes of a network to agreement. In quantum networks they also redistribute information encoded in the global quantum state, allowing global properties to be inferred from local measurements without reconstructing the global state. Motivated by this, we investigate decentralized quantum consensus in qudit networks governed by discrete random unitary dynamics. We characterize the analytic asymptotic dynamics, identify global information preserved by the evolution, and derive conditions for which different notions of quantum consensus emerge. A general theorem is derived that establishes a unified framework for extracting globally encoded information from local measurements beyond consensus states. First, we analyze CNOT networks and determine which global information can be recovered by individual nodes and which requires joint measurements on cooperating nodes. The CNOT model is well understood and presents a case with entanglement creation - a characteristic quantum feature of networks worth analyzing in detail. Second, we identify the complete class of two-qubit interactions that asymptotically achieve consensus on the average number of excitations, thereby establishing a quantum analogue of classical average consensus. We further show that joint measurements on k cooperating nodes recover exactly the first k moments of the excitation-number distribution.
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.
In this study, we deal with quantum networked systems consisting ofNNNquantum subsystems havingDDDstates. In previous studies, it was proved that a symmetric-state consensus (SSC) can be attained by an algorithm composed of local measurements and a local feedback control in cases ofD=2D=2D=2and 3. However, a proof for generalDDDremained an open problem. In this study, we rigorously prove that an SSC can be attained by the same algorithm in the case of generalDDD.
In this paper, we study consensus seeking of quantum networks under directed interactions defined by a set of permutation operators among a network of qubits. The state evolution of the quantum network is described by a continuous-time master equation, for which we establish an unconditional convergence result indicating that the network state always converges with the limit determined by the generating subgroup of the permutations making use of the Perron-Frobenius theory. We also give a tight graphical criterion regarding when such limit admits a reduced-state consensus. Further, we provide a clear description to the missing symmetry in the reduced-state consensus from a graphical point of view, where the information-flow hierarchy in quantum permutation operators is characterized by different layers of information-induced graphs. Finally, we investigate quantum synchronization in the presence of network Hamiltonian, study quantum consensus conditions under switching interactions, and present a few numerical examples illustrating the obtained results.