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arXiv:1702.00041 · Journal of Physics A Mathematical and Theoretical

A model of synchronization over quantum networks

Paolo Antonelli, Pierangelo Marcati

Published Jun 15, 2017• 21 citations
arXivHTMLPDF

Abstract

Abstract We investigate a non-Abelian generalization of the Kuramoto model proposed by Lohe and given by N quantum oscillators (‘nodes’) connected by a quantum network where the wavefunction at each node is distributed over quantum channels to all other connected nodes. It leads to a system of Schrödinger equations coupled by nonlinear self-interacting potentials given by their correlations. We give a complete picture of synchronization results, given on the relative size of the natural frequency and the coupling constant, for two non-identical oscillators and show complete phase synchronization for arbitrary N > 2 identical oscillators. Our results are mainly based on the analysis of the ODE system satisfied by the correlations and on the introduction of a quantum order parameter, which is analogous to the one defined by Kuramoto in the classical model. As a consequence of the previous results, we obtain the synchronization of the probability and the current densities defined via the Madelung transformations.

DOI
10.1088/1751-8121/aa79c9

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Abstract We investigate a non-Abelian generalization of the Kuramoto model proposed by Lohe and given by N quantum oscillators (‘nodes’) connected by a quantum network where the wavefunction at each node is distributed over quantum channels to all other connected nodes. It leads to a system of Schrödinger equations coupled by nonlinear self-interacting potentials given by their correlations. We give a complete picture of synchronization results, given on the relative size of the natural frequency and the coupling constant, for two non-identical oscillators and show complete phase synchronization for arbitrary N > 2 identical oscillators. Our results are mainly based on the analysis of the ODE system satisfied by the correlations and on the introduction of a quantum order parameter, which is analogous to the one defined by Kuramoto in the classical model. As a consequence of the previous results, we obtain the synchronization of the probability and the current densities defined via the Madelung transformations.
We consider a network of quantum oscillators in which quantum states are distributed among connected nodes by means of unitary transformations. The distributed states interact with each local state according to a time-dependent interaction Hamiltonian, which is modeled by a Hermitian operator constructed in terms of the states themselves, thereby introducing nonlinear network interactions. For qubit nodes of differing natural frequencies, we show numerically that for a sufficiently large coupling constant, synchronization of quantum nodes occurs in which the spins of all qubits are mutually aligned with a common frequency of oscillation, following initial transient configurations. We discuss the significance of quantum synchronization as a means to create copies of unknown quantum states.
We present a quantum synchronization estimate of the Schrödinger–Lohe (S–L) model introduced by Lohe (2010 J. Phys. A: Math. Theor. 43 465301 ). The S–L model describes the dynamics of quantum oscillators on the nodes of a quantum network. When the coupling strength is positive and the maximal L 2 distances between normalized initial wave functions are smaller than , we show that the L 2 distances between wave functions converge to zero exponentially fast. Our result generalizes earlier work by Chi et al (2014 J. Math. Phys. 55 052703 ) for the Lohe model.
Synchronization is a collective phenomenon in which interacting nonlinear oscillators develop a persistent relation between their phases and frequencies. In quantum systems, its description requires connecting the classical dynamics of limit-cycle oscillators with quantum fluctuations, dissipation, and the notion of phase. In this tutorial review, we offer a systematic framework for analyzing synchronization in classical and quantum dynamical systems. Starting from amplitude-phase dynamics, we introduce successively topics of relevance, such as phase locking, the Adler equation, Arnold tongues, and the Kuramoto model, and use these concepts to analyze coupled oscillators and their stability. We then formulate the corresponding description for quantum limit-cycle oscillators using Lindblad master equations and phase-space methods, with explicit examples of two and many coupled oscillators. We further discuss noise, synchronization measures, and the extreme quantum regime, including synchronization in spin systems. The emphasis throughout is on analytical derivations and a practical methodology for identifying synchronized states, determining their stability, and characterizing synchronization in quantum systems. The review is designed as a self-contained guide from which we hope that students can learn the essential concepts and techniques, while researchers can readily find the analytical tools and methodological insights needed to approach new synchronization problems.
Higher-order networks with multiway interactions can exhibit collective dynamical phenomena that are absent in traditional pairwise network models. However, analyzing such dynamics becomes computationally prohibitive as their state space grows combinatorially in the multiway interaction order. Here we develop quantum algorithms for two central tasks -- synchronization estimation and certification of the no-phase-locking regime -- in the simplicial Kuramoto model. This model is a higher-order generalization of the celebrated Kuramoto model for coupled oscillators on graph-based networks. Under explicit assumptions on data access and types, and simplicial structure, we derive end-to-end quantum gate complexities and identify regimes with polynomial quantum advantage for synchronization estimation and super-polynomial quantum advantage for no-phase-locking certification over classical methods. More broadly, these results extend quantum algorithms for higher-order networks from structural analysis to nonlinear dynamical diagnostics, easing a major computational bottleneck and opening a route to quantum methods for probing higher-order phenomena beyond the reach of direct classical approaches.
The main investigation in this thesis is the research on a future quantum internet. The focus is laid on the network structure of this quantum network. The network is rst examined by the investigation of the Kuramoto model for classical oscillators. The main property of the network is the ability to synchronise all nodes such that all oscillate with the same frequency. Several parameters of the oscillators are varied to verify why and when synchronisation takes place. After modelling the calculated equations of motion, an interesting conclusion arises. For a symmetric ring network, a stable conguration is not always found but by introducing asymmetric oscillators in the network, the synchronous state can be found. This conclusion leads to the idea that a quantum network requires certain dierences in its structure in order to guarantee transmission to take place. Furthermore, the existence of synchronisation heavily depends on the parameters used for the oscillators. The analysis is then continued in the quantum domain where optomechanical systems are introduced. At rst, two systems are connected to each other by a gaseous interaction and an electric interaction via a Dung circuit. The main investigation is again into the synchronisation, which implies that the operators belonging to both systems behave the same as a function of time. Then several synchronisation measures are introduced to measure the ability of the systems to synchronise. As predicted, both systems synchronise in terms of the operators for each system. Then a quantum network is introduced, where a complex yet ecient network is created. This quantum network is a small world network, where a transmitter node is able to connect to the receiver node by only a few links. Furthermore, multiple transmissions can take place at the same time and links which are not connected do not synchronise with the transmitter node. After modelling this network, the results are in compliance with the theory available. Challenges for a practical quantum network include the experimental basis of being able to utilise optomechanical systems outside laboratory circumstances. Also, the network should be tested for practical use and expanded in a much larger size both in theoretical analysis as well as in experiments.