Symmetrizing quantum dynamics beyond gossip-type algorithms

Recently, consensus-type problems have been formulated in the quantum domain. Obtaining average quantum consensus consists in the dynamical symmetrization of a multipartite quantum system while preserving the expectation of a given global observable. In this paper, two improved ways of obtaining consensus via dissipative engineering are introduced, which employ on quasi local preparation of mixtures of symmetric pure states, and show better performance in terms of purity dynamics with respect to existing algorithms. In addition, the first method can be used in combination with simple control resources in order to engineer pure Dicke states, while the second method guarantees a stronger type of consensus, namely single-measurement consensus. This implies that outcomes of local measurements on different subsystems are perfectly correlated when consensus is achieved. Both dynamics can be randomized and are suitable for feedback implementation.Comment: 11 pages, 3 figure

Symmetrization for Quantum Networks: a continuous-time approach

In this paper we propose a continuous-time, dissipative Markov dynamics that asymptotically drives a network of n-dimensional quantum systems to the set of states that are invariant under the action of the subsystem permutation group. The Lindblad-type generator of the dynamics is built with two-body subsystem swap operators, thus satisfying locality constraints, and preserve symmetric observables. The potential use of the proposed generator in combination with local control and measurement actions is illustrated with two applications: the generation of a global pure state and the estimation of the network size.Comment: submitted to MTNS 201

Symmetrizing dynamics: from classical to quantum applications

Among the issues regarding networked systems, the “consensus problem” and the related algorithms have received a significant share of attention during the last ten years. In this problem the network agents asymptotically have to attain agreement on the value of some objective variable under local communication constraints. A number of algorithms have been developed to address this problem, among which the celebrated gossip algorithm. The latter relays on switching dynamics and, under rather weak assumptions, exhibits robust convergence under variations in the interaction constraints, i.e. the network topology. In this dissertation we reinterpret the goal of the consensus problem as a symmetrisation problem, and we address it by a switching-type dynamics based on convex combinations of actions of a finite group. In order to study the convergence of our class of algorithms we lift the dynamics to an abstract, group-theoretic level that allow us to derive general conditions for convergence. Such conditions, in fact, are independent of the particular group action, and focus only on the group itself and the way the iterations are selected. Convergence is guaranteed provided that some mild assumptions on the selection rule for the iterations are fulfilled. Furthermore, this class of algorithms retains the robustness features and unsupervised character of the consensus algorithm. Our reformulation allow to devise algorithms for application as diverse as randomized discrete Fourier transform and random state generation. We pose a special emphasis on the extension of the consensus problem to the quantum domain. In this setting we highlight how, due to the richer mathematical structure over which the internal state is encoded, the definition of the consensus goal admits various extensions, each of them exhibiting different features. We also propose a suitable dissipative dynamics enacting the symmetrising gossip interactions and then use our general result on convergence to prove it ensures asymptotic convergence. Beside the technical results, one of the main contributions of our work is a new, generalized view point on consensus, which allows us to extend the robustness of consensus-inspired algorithms to new problems in apparently unrelated fields. This reinforces the role of consensus algorithms as fundamental tools for distributed computing, both in the classical and the quantum setting

Iterated projection methods with classical and quantum applications

This work studies the methods of alternating projection, especially they are applied to particular quantum maps. After a first analysis of these methods in the classical case, MAPs are extended to quantum CPTP maps. As application, they are used in order to study the preparation of a quantum state under Quasi-locality constrictions. finally, Bregman theory, as an extension of MAPs, is analyze