Consensus for Quantum Networks: Symmetry From Gossip Interactions

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.

[1]  Matthew R. James,et al.  An Introduction to Quantum Filtering , 2006, SIAM Journal of Control and Optimization.

[2]  Matthew R. James,et al.  Quantum Dissipative Systems and Feedback Control Design by Interconnection , 2007, IEEE Transactions on Automatic Control.

[3]  Julia Kempe,et al.  Quantum random walks: An introductory overview , 2003, quant-ph/0303081.

[4]  Lorenza Viola,et al.  Stabilizing entangled states with quasi-local quantum dynamical semigroups , 2011, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[5]  J Eisert,et al.  Precisely timing dissipative quantum information processing. , 2012, Physical review letters.

[6]  Matthew R. James,et al.  The Series Product and Its Application to Quantum Feedforward and Feedback Networks , 2007, IEEE Transactions on Automatic Control.

[7]  Hans Maassen.  Quantum probability applied to the damped harmonic oscillator , 2004 .

[8]  Francesco Ticozzi,et al.  Discrete-time controllability for feedback quantum dynamics , 2010, Autom..

[9]  Francesco Bullo,et al.  Distributed Control of Robotic Networks , 2009 .

[10]  P. Zoller,et al.  Preparation of entangled states by quantum Markov processes , 2008, 0803.1463.

[11]  Stephen P. Boyd,et al.  Randomized gossip algorithms , 2006, IEEE Transactions on Information Theory.

[12]  Alessandro Chiuso,et al.  Gossip Algorithms for Simultaneous Distributed Estimation and Classification in Sensor Networks , 2011, IEEE Journal of Selected Topics in Signal Processing.

[13]  Lorenza Viola,et al.  Quantum Information Encoding, Protection, and Correction from Trace-Norm Isometries , 2009, 0912.0963.

[14]  Pierre Rouchon,et al.  Consensus in non-commutative spaces , 2010, 49th IEEE Conference on Decision and Control (CDC).

[15]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[16]  K. Lendi,et al.  Quantum Dynamical Semigroups and Applications , 1987 .

[17]  G. J. Milburn,et al.  Quantum error correction for continuously detected errors , 2003 .

[18]  Claudio Altafini,et al.  Modeling and Control of Quantum Systems: An Introduction , 2012, IEEE Transactions on Automatic Control.

[19]  Luc Moreau,et al.  Stability of multiagent systems with time-dependent communication links , 2005, IEEE Transactions on Automatic Control.

[20]  John N. Tsitsiklis,et al.  Problems in decentralized decision making and computation , 1984 .

[21]  Stephen P. Boyd,et al.  Design of Low-Bandwidth Spatially Distributed Feedback , 2008, IEEE Transactions on Automatic Control.

[22]  Mikhail N. Vyalyi,et al.  Classical and Quantum Computation , 2002, Graduate studies in mathematics.

[23]  J. Hespanha,et al.  Estimation on Graphs from Relative Measurements Distributed algorithms and fundamental limits , 2022 .

[24]  Amílcar Sernadas,et al.  Quantum Computation and Information , 2006 .

[25]  Geir E. Dullerud,et al.  Distributed control design for spatially interconnected systems , 2003, IEEE Trans. Autom. Control..

[26]  Raymond Laflamme,et al.  A Theory of Quantum Error-Correcting Codes , 1996 .

[27]  Fernando Paganini,et al.  Distributed control of spatially invariant systems , 2002, IEEE Trans. Autom. Control..

[28]  Jie Lin,et al.  Coordination of groups of mobile autonomous agents using nearest neighbor rules , 2003, IEEE Trans. Autom. Control..

[29]  Naomi Ehrich Leonard,et al.  Collective Motion, Sensor Networks, and Ocean Sampling , 2007, Proceedings of the IEEE.

[30]  G. Milburn,et al.  Quantum Measurement and Control , 2009 .

[31]  Lorenza Viola,et al.  Quantum Markovian Subsystems: Invariance, Attractivity, and Control , 2007, IEEE Transactions on Automatic Control.

[32]  Mark M. Wilde,et al.  From Classical to Quantum Shannon Theory , 2011, ArXiv.

[33]  Lorenza Viola,et al.  Random decoupling schemes for quantum dynamical control and error suppression. , 2005, Physical review letters.

[34]  K. Kraus,et al.  States, effects, and operations : fundamental notions of quantum theory : lectures in mathematical physics at the University of Texas at Austin , 1983 .

[35]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

[36]  J. J. Sakurai,et al.  Modern Quantum Mechanics , 1986 .

[37]  D. Poulin,et al.  Information-preserving structures: A general framework for quantum zero-error information , 2010, 1006.1358.

[38]  Howard E. Brandt,et al.  Quantum computation and information : AMS Special Session Quantum Computation and Information, January 19-21, 2000, Washington, D.C. , 2002 .

[39]  Michael M. Wolf,et al.  Unital Quantum Channels – Convex Structure and Revivals of Birkhoff’s Theorem , 2008, 0806.2820.

[40]  Saverio Bolognani,et al.  Engineering Stable Discrete-Time Quantum Dynamics via a Canonical QR Decomposition , 2009, IEEE Transactions on Automatic Control.

[41]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[42]  T. Monz,et al.  An open-system quantum simulator with trapped ions , 2011, Nature.

[43]  Lorenza Viola,et al.  Single-bit feedback and quantum-dynamical decoupling , 2006 .

[44]  V. Borkar,et al.  Asymptotic agreement in distributed estimation , 1982 .

[45]  R. Werner,et al.  Separability properties of tripartite states with U ⊗ U ⊗ U symmetry , 2000, quant-ph/0003008.

[46]  Lorenza Viola,et al.  Engineering quantum dynamics , 2001 .

[47]  F. Verstraete,et al.  Quantum computation and quantum-state engineering driven by dissipation , 2009 .

[48]  M. Szegedy,et al.  Quantum Walk Based Search Algorithms , 2008, TAMC.

[49]  J. Hespanha,et al.  Estimation on graphs from relative measurements , 2007, IEEE Control Systems.

[50]  D. D’Alessandro.  Introduction to Quantum Control and Dynamics , 2007 .

[51]  P. Olver.  Nonlinear Systems , 2013 .

[52]  Hans Peter Büchler,et al.  Preparation of Entangled States by Dissipative Quantum Markov Processes , 2008 .

[53]  Alain Sarlette,et al.  Symmetrization for Quantum Networks: a continuous-time approach , 2014 .