Model checking quantum Markov chains

Although the security of quantum cryptography is provable based on the principles of quantum mechanics, it can be compromised by the flaws in the design of quantum protocols and the noise in their physical implementations. So, it is indispensable to develop techniques of verifying and debugging quantum cryptographic systems. Model-checking has proved to be effective in the verification of classical cryptographic protocols, but an essential difficulty arises when it is applied to quantum systems: the state space of a quantum system is always a continuum even when its dimension is finite. To overcome this difficulty, we introduce a novel notion of quantum Markov chain, specially suited to model quantum cryptographic protocols, in which quantum effects are entirely encoded into super-operators labelling transitions, leaving the location information (nodes) being classical. Then we define a quantum extension of probabilistic computation tree logic (PCTL) and develop a model-checking algorithm for quantum Markov chains.

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

[2]  Yuan Feng,et al.  Proof rules for the correctness of quantum programs , 2007, Theor. Comput. Sci..

[3]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[4]  Stephan Merz,et al.  Model Checking , 2000 .

[5]  Rohit Chadha,et al.  QUANTUM COMPUTATION TREE LOGIC — MODEL CHECKING AND COMPLETE CALCULUS , 2008 .

[6]  R. Xu,et al.  Theory of open quantum systems , 2002 .

[7]  Rajagopal Nagarajan,et al.  Probabilistic Model-Checking of Quantum Protocols , 2005, ArXiv.

[8]  Stanley Gudder,et al.  Quantum Markov chains , 2008 .

[9]  Gilles Brassard,et al.  Quantum cryptography: Public key distribution and coin tossing , 2014, Theor. Comput. Sci..

[10]  Yuan Feng,et al.  Verification of Quantum Programs , 2011, Sci. Comput. Program..

[11]  Guowu Yang,et al.  Quantum logic synthesis by symbolic reachability analysis , 2004, Proceedings. 41st Design Automation Conference, 2004..

[12]  Ulrich Faigle,et al.  A Markovian Model for Joint Observations, Bell's Inequality and Hidden States , 2010 .

[13]  Nikolaos Papanikolaou,et al.  Model – Checking Quantum Protocols , 2006 .

[14]  Yuan Feng,et al.  Quantum loop programs , 2010, Acta Informatica.

[15]  J. Diestel,et al.  On vector measures , 1974 .

[16]  Bengt Jonsson,et al.  A logic for reasoning about time and reliability , 1990, Formal Aspects of Computing.

[17]  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 .

[18]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[19]  Rajagopal Nagarajan,et al.  QMC: A Model Checker for Quantum Systems , 2007, CAV.

[20]  Keiji Matsumoto,et al.  Exact Quantum Algorithms for the Leader Election Problem , 2005, TOCT.

[21]  Gavin Lowe,et al.  Breaking and Fixing the Needham-Schroeder Public-Key Protocol Using FDR , 1996, Softw. Concepts Tools.

[22]  E. Allen Emerson,et al.  Temporal and Modal Logic , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[23]  Guowu Yang,et al.  Optimal synthesis of multiple output Boolean functions using a set of quantum gates by symbolic reachability analysis , 2006, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

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

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

[26]  Moshe Y. Vardi.  Automatic verification of probabilistic concurrent finite state programs , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[27]  Feihu Xu,et al.  Experimental demonstration of phase-remapping attack in a practical quantum key distribution system , 2010, 1005.2376.

[28]  J. Norris.  Appendix: probability and measure , 1997 .

[29]  Luigi Accardi,et al.  Nonrelativistic Quantum Mechanics as a Noncommutative Markof Process , 1976 .

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

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

[32]  Christel Baier,et al.  Principles of model checking , 2008 .

[33]  Alexander Schönhuth,et al.  Discrete Quantum Markov Chains , 2010, ArXiv.

[34]  Marta Z. Kwiatkowska,et al.  Stochastic Model Checking , 2007, SFM.