Probabilistic bisimilarities between quantum processes

Modeling and reasoning about concurrent quantum systems is very important both for distributed quantum computing and for quantum protocol verification. As a consequence, a general framework describing formally the communication and concurrency in complex quantum systems is necessary. For this purpose, we propose a model qCCS which is a natural quantum extension of classical value-passing CCS with the input and output of quantum states, and unitary transformations and measurements on quantum systems. The operational semantics of qCCS is given based on probabilistic labeled transition system. This semantics has many different features compared with the proposals in literature in order to describe input and output of quantum systems which are possibly correlated with other components. Based on this operational semantics, we introduce the notions of strong probabilistic bisimilarity and weak probabilistic bisimilarity between quantum processes and discuss some properties of them, such as congruence under various combinators.

[1]  Lov K. Grover.  Quantum Mechanics Helps in Searching for a Needle in a Haystack , 1997, quant-ph/9706033.

[2]  W. Wootters,et al.  A single quantum cannot be cloned , 1982, Nature.

[3]  Nancy A. Lynch,et al.  Probabilistic Simulations for Probabilistic Processes , 1994, Nord. J. Comput..

[4]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[5]  Jeff W. Sanders,et al.  Quantum Programming , 2000, MPC.

[6]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and EPR channels , 1993 .

[7]  Yuan Feng,et al.  Catalyst-assisted probabilistic entanglement transformation , 2005, IEEE Transactions on Information Theory.

[8]  Robin Milner,et al.  A Calculus of Mobile Processes, II , 1992, Inf. Comput..

[9]  Rajagopal Nagarajan,et al.  Communicating quantum processes , 2004, POPL '05.

[10]  Quantum mechanics and faster-than-light communication: Methodological considerations , 1983 .

[11]  Mingsheng Ying.  Bisimulation indexes and their applications , 2002, Theor. Comput. Sci..

[12]  E. Knill,et al.  Conventions for quantum pseudocode , 1996, 2211.02559.

[13]  C. Petri.  Kommunikation mit Automaten , 1962 .

[14]  Communication and non-communication in Einstein-Rosen experiments , 1987 .

[15]  J. Neumann.  Mathematical Foundations of Quantum Mechanics , 1955 .

[16]  Bent Thomsen,et al.  Calculi for higher order communicating systems , 1990 .

[17]  Mingsheng Ying,et al.  Reasoning about probabilistic sequential programs in a probabilistic logic , 2003, Acta Informatica.

[18]  C. A. R. Hoare,et al.  Communicating sequential processes , 1978, CACM.

[19]  Peter Selinger,et al.  A Brief Survey of Quantum Programming Languages , 2004, FLOPS.

[20]  Mingsheng Ying,et al.  Additive models of probabilistic processes , 2002, Theor. Comput. Sci..

[21]  J GaySimon,et al.  Quantum programming languages: survey and bibliography , 2006 .

[22]  M. Nielsen.  Conditions for a Class of Entanglement Transformations , 1998, quant-ph/9811053.

[23]  André van Tonder,et al.  A Lambda Calculus for Quantum Computation , 2003, SIAM J. Comput..

[24]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[25]  R. Feynman.  Simulating physics with computers , 1999 .

[26]  Mingsheng Ying.  Topology in process calculus - approximate correctness and infinite evolution of concurrent programs , 2001 .

[27]  Mingsheng Ying,et al.  π-calculus with noisy channels , 2005, Acta Informatica.

[28]  Robin Milner,et al.  Communication and concurrency , 1989, PHI Series in computer science.

[29]  Luciano Serafini,et al.  Toward an architecture for quantum programming , 2001, ArXiv.

[30]  Davide Sangiorgi,et al.  Bisimulation for Higher-Order Process Calculi , 1994, Inf. Comput..

[31]  Bent Thomsen,et al.  Plain CHOCS A second generation calculus for higher order processes , 2005, Acta Informatica.

[32]  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 .

[33]  Marie Lalire,et al.  Relations among quantum processes: bisimilarity and congruence , 2006, Mathematical Structures in Computer Science.

[34]  Robin Milner,et al.  A Calculus of Mobile Processes, II , 1992, Inf. Comput..

[35]  Anna Ingólfsdóttir,et al.  A Theory of Communicating Processes with Value-Passing , 1990, ICALP.

[36]  Ekert,et al.  Quantum cryptography based on Bell's theorem. , 1991, Physical review letters.

[37]  Matthew Hennessy,et al.  A proof system for communicating processes with value-passing , 1991, Formal Aspects of Computing.

[38]  Marie Lalire.  A probabilistic branching bisimulation for quantum processes , 2005, ArXiv.

[39]  Martin Wirsing,et al.  Recursive equations in higher-order process calculi , 2001, Theor. Comput. Sci..

[40]  Robin Milner,et al.  A Calculus of Communicating Systems , 1980, Lecture Notes in Computer Science.

[41]  Matthew Hennessy,et al.  Proof systems for message-passing process algebras , 2005, Formal Aspects of Computing.

[42]  Yuan Feng,et al.  When catalysis is useful for probabilistic entanglement transformation (5 pages) , 2004 .

[43]  Charles H. Bennett,et al.  Quantum cryptography using any two nonorthogonal states. , 1992, Physical review letters.

[44]  Martin Wirsing,et al.  Approximate Bisimilarity , 2000, AMAST.

[45]  Kim G. Larsen,et al.  Bisimulation through Probabilistic Testing , 1991, Inf. Comput..

[46]  Peter Selinger,et al.  Towards a quantum programming language , 2004, Mathematical Structures in Computer Science.

[47]  Philippe Jorrand,et al.  Toward a quantum process algebra , 2004, CF '04.

[48]  Anna Ingólfsdóttir,et al.  A Theory of Communicating Processes with Value Passing , 1993, Inf. Comput..

[49]  Bernhard Steffen,et al.  Reactive, Generative and Stratified Models of Probabilistic Processes , 1995, Inf. Comput..

[50]  Jan A. Bergstra,et al.  Algebra of Communicating Processes with Abstraction , 1985, Theor. Comput. Sci..

[51]  Bernhard Ömer,et al.  A Procedural Formalism for Quantum Computing , 2002 .

[52]  Charles H. Bennett,et al.  Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. , 1992, Physical review letters.

[53]  Paolo Zuliani,et al.  Quantum Programming With Mixed States , 2007, QPL.

[54]  M. Peev,et al.  Practical quantum key distribution with polarization entangled photons , 2005, EQEC '05. European Quantum Electronics Conference, 2005..

[55]  Peter W. Shor,et al.  Algorithms for Quantum Computation: Discrete Log and Factoring (Extended Abstract) , 1994, FOCS 1994.