Modèles formels du calcul quantique : ressources, machines abstraites et calcul par mesure. (Formal models of quantum computing: resources, abstract machines and measurement-based quantum computing)

L'etude des structures fondamentales du traitement de l'information quantique est un defi majeur, dont l'un des objectifs est de mieux cerner les capacites et les limites de l'ordinateur quantique, tout en contribuant a sa realisation physique notamment en s'interessant aux ressources du calcul quantique. Les ressources d'un calcul quantique incluent le temps et l'espace mais egalement la taille des operations utilisees et la quantite d'intrication. Cette these contribue de plusieurs manieres a la recherche de ressources minimales dans le cadre de modeles de calcul quantique ouvrant de prometteuses perspectives de realisations physiques. Ces modeles sont le calcul par consommation d'intrication et le calcul par mesures projectives. Cette these a egalement permis de reduire les ressources en temps et en espace necessaires a la preparation de certains etats quantiques, les etats graphes. Etudier la reduction des ressources necessite l'abstraction et la formalisation des modeles de calcul quantique mettant en evidence les structures meme du traitement de l'information quantique. Le q-calcul et les machines de Turing controlees classiquement, introduits dans cette these, ont cet objectif. Des modeles plus specifiques au calcul par consommation d'intrication, ou au calcul par mesures projectives sont egalement consideres.

[1]  Simon Perdrix.  STATE TRANSFER INSTEAD OF TELEPORTATION IN MEASUREMENT-BASED QUANTUM COMPUTATION , 2005 .

[2]  Masanori Ohya,et al.  Generalized Quantum Turing Machine and its Application to the SAT Chaos Algorithm , 2004 .

[3]  P. Grangier,et al.  Experimental Tests of Realistic Local Theories via Bell's Theorem , 1981 .

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

[5]  John Watrous,et al.  Succinct quantum proofs for properties of finite groups , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[6]  Simon Perdrix,et al.  Classically controlled quantum computation , 2004, Mathematical Structures in Computer Science.

[7]  Stephen A. Fenner,et al.  Universal quantum computation with two- and three-qubit projective measurements , 2001 .

[8]  Simon Perdrix.  Towards minimal resources of measurement-based quantum computation , 2007 .

[9]  Simon Perdrix,et al.  Unifying quantum computation with projective measurements only and one-way quantum computation , 2004, Other Conferences.

[10]  Simon Perdrix,et al.  Measurement-Based Quantum Turing Machines and Questions of Universalities , 2004 .

[11]  A. Gleason.  Measures on the Closed Subspaces of a Hilbert Space , 1957 .

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

[13]  Samson Abramsky,et al.  High-level methods for quantum computation and information , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[14]  Yaoyun Shi.  Both Toffoli and controlled-NOT need little help to do universal quantum computing , 2003, Quantum Inf. Comput..

[15]  H. Dishkant,et al.  Logic of Quantum Mechanics , 1976 .

[16]  P. Benioff.  The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines , 1980 .

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

[18]  Samson Abramsky,et al.  Domain theory , 1995, LICS 1995.

[19]  Andrew Chi-Chih Yao,et al.  Quantum Circuit Complexity , 1993, FOCS.

[20]  Yuan Feng,et al.  Probabilistic bisimilarities between quantum processes , 2006, ArXiv.

[21]  Simon Perdrix,et al.  Quantum Patterns and Types for Entanglement and Separability , 2007, QPL.

[22]  Simon J. Gay,et al.  Quantum Programming Languages Survey and Bibliography , 2006 .

[23]  Benoît Valiron,et al.  A Lambda Calculus for Quantum Computation with Classical Control , 2005, TLCA.

[24]  Samson Abramsky,et al.  A categorical quantum logic , 2006, Math. Struct. Comput. Sci..

[25]  G. Vidal,et al.  Classical simulation versus universality in measurement-based quantum computation , 2006, quant-ph/0608060.

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

[27]  D. Bohm.  A SUGGESTED INTERPRETATION OF THE QUANTUM THEORY IN TERMS OF "HIDDEN" VARIABLES. II , 1952 .

[28]  Jonathan Grattage.  A functional quantum programming language , 2005, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05).

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

[30]  J. Eisert,et al.  Entanglement in Graph States and its Applications , 2006, quant-ph/0602096.

[31]  Bart De Moor,et al.  Graphical description of the action of local Clifford transformations on graph states , 2003, quant-ph/0308151.

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

[33]  Yong Zhang.  Algebraic Structures Underlying Quantum Information Protocols , 2006 .

[34]  Umesh V. Vazirani,et al.  Quantum complexity theory , 1993, STOC.

[35]  Jim Geelen,et al.  Matchings, Matroids and Unimodular Matrices , 1995 .

[36]  Hans J. Briegel,et al.  The one-way quantum computer--a non-network model of quantum computation , 2001, quant-ph/0108118.

[37]  A. Turing.  On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[38]  Sang-il Oum,et al.  Approximating rank-width and clique-width quickly , 2005, TALG.

[39]  Samson Abramsky,et al.  A categorical semantics of quantum protocols , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[40]  P. Jorrand,et al.  Measurement-Based Quantum Turing Machines and their Universality , 2004, quant-ph/0404146.

[41]  Debbie W. Leung,et al.  Quantum computation by measurements , 2003 .

[42]  Peter Selinger,et al.  Dagger Compact Closed Categories and Completely Positive Maps: (Extended Abstract) , 2007, QPL.

[43]  Hans-J. Briegel,et al.  Computational model underlying the one-way quantum computer , 2002, Quantum Inf. Comput..

[44]  R. Raussendorf,et al.  Computational Model for the One‐Way Quantum Computer: Concepts and Summary , 2005 .

[45]  Harumichi Nishimura,et al.  Computational complexity of uniform quantum circuit families and quantum Turing machines , 2002, Theor. Comput. Sci..

[46]  Elham Kashefi.  Quantum Domain Theory - Definitions and Applications , 2003, ArXiv.

[47]  S. Braunstein,et al.  Quantum computation , 1996 .

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

[49]  J. Eisert,et al.  Multiparty entanglement in graph states , 2003, quant-ph/0307130.

[50]  Mehdi Mhalla,et al.  Quantum Query Complexity of Some Graph Problems , 2004, SIAM J. Comput..

[51]  C. Jones,et al.  A probabilistic powerdomain of evaluations , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[52]  R. Jozsa.  On the simulation of quantum circuits , 2006, quant-ph/0603163.

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

[54]  Patrick Cousot,et al.  Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints , 1977, POPL.

[55]  D. Deutsch.  Quantum theory, the Church–Turing principle and the universal quantum computer , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[56]  Michael A. Nielsen,et al.  Universal quantum computation using only projective measurement, quantum memory, and preparation of the 0 state , 2001 .

[57]  Samson Abramsky,et al.  Physical Traces: Quantum vs. Classical Information Processing , 2002, CTCS.

[58]  Gilles Dowek,et al.  Operational semantics for formal tensorial calculus , 2004 .

[59]  Simon Perdrix.  Resources for measurement−based quantum computation: A unifying view , 2005 .

[60]  Elham Kashefi,et al.  The measurement calculus , 2004, JACM.

[61]  D. Leung,et al.  Experimental realization of a quantum algorithm , 1998, Nature.

[62]  Michael A. Nielsen,et al.  Quantum computation by measurement and quantum memory , 2003 .

[63]  A. Bouchet.  Connectivity of Isotropic Systems , 1989 .

[64]  Simon Perdrix.  Qubit vs Observable Resource Trade‐Offs in Measurement‐Based Quantum Computation , 2004 .

[65]  I. Chuang,et al.  Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance , 2001, Nature.

[66]  Jr.,et al.  Multivalued logic gates for quantum computation , 2000, quant-ph/0002033.

[67]  S. Lloyd.  Quantum-Mechanical Computers , 1995 .

[68]  Tomoyuki Yamakami.  A Foundation of Programming a Multi-tape Quantum Turing Machine , 1999, MFCS.

[69]  Peter W. Shor,et al.  Algorithms for quantum computation: discrete logarithms and factoring , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

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

[71]  Wolfgang Dür,et al.  Universal resources for measurement-based quantum computation. , 2006, Physical review letters.

[72]  Patrick Cousot,et al.  Types as abstract interpretations , 1997, POPL '97.

[73]  Debbie W. Leung,et al.  Unified derivations of measurement-based schemes for quantum computation , 2005 .

[74]  P. Selinger.  Towards a semantics for higher-order quantum computation , 2004 .

[75]  P. Benioff.  Quantum mechanical hamiltonian models of turing machines , 1982 .

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

[77]  Maarten Van den Nest,et al.  Local equivalence of stabilizer states and codes , 2005 .

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

[79]  Bob Coecke,et al.  Quantum information-flow, concretely, abstractly , 2004 .

[80]  A. Bouchet.  Digraph decompositions and Eulerian systems , 1987 .

[81]  H. Weinfurter,et al.  Experimental quantum teleportation , 1997, Nature.

[82]  Simon Perdrix,et al.  Towards a quantum calculus , 2007 .

[83]  Mehdi Mhalla,et al.  Resources Required for Preparing Graph States , 2006, ISAAC.

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

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

[86]  Mehdi Mhalla,et al.  Complexity of Graph State Preparation , 2004 .