Quantum Complexity: restrictions on algorithms and architectures

A dissertation submitted to the University of Bristol in accordance with the requirements of the degree of Doctor of Philosophy (PhD) in the Faculty of Engineering, Department of Computer Science, July 2009.

[1]  Christoph Meinel,et al.  Structure and importance of logspace-MOD class , 1992, Mathematical systems theory.

[2]  Michael A. Nielsen,et al.  Quantum computing and polynomial equations over the finite field Z 2 , 2004 .

[3]  John Watrous,et al.  On one-dimensional quantum cellular automata , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[4]  Robert Spalek,et al.  Quantum Circuits with Unbounded Fan-out , 2002, STACS.

[5]  Robert Raussendorf Quantum computation via translation-invariant operations on a chain of qubits , 2005 .

[6]  H. Briegel,et al.  Fast simulation of stabilizer circuits using a graph-state representation , 2005, quant-ph/0504117.

[7]  Simon C Benjamin,et al.  Processor core model for quantum computing. , 2006, Physical review letters.

[8]  Max Albert,et al.  The propensity theory: a decision-theoretic restatement , 2007, Synthese.

[9]  Wolfgang Porod,et al.  Quantum cellular automata , 1994 .

[10]  Mikhail N. Vyalyi Hardness of approximating the weight enumerator of a binary linear code , 2003, ArXiv.

[11]  Thomas G. Draper,et al.  A logarithmic-depth quantum carry-lookahead adder , 2006, Quantum Inf. Comput..

[12]  Gilles Brassard,et al.  Strengths and Weaknesses of Quantum Computing , 1997, SIAM J. Comput..

[13]  N. Yoran,et al.  Classical simulation of limited-width cluster-state quantum computation. , 2006, Physical review letters.

[14]  D. Aharonov A Simple Proof that Toffoli and Hadamard are Quantum Universal , 2003, quant-ph/0301040.

[15]  Eric Bach,et al.  Space-bounded quantum computation , 1998 .

[16]  Leonard M. Adleman,et al.  Quantum Computability , 1997, SIAM J. Comput..

[17]  A. Kitaev,et al.  Universal quantum computation with ideal Clifford gates and noisy ancillas (14 pages) , 2004, quant-ph/0403025.

[18]  Christoph Meinel,et al.  Structure and Importance of Logspace-MOD-Classes , 1991, STACS.

[19]  Alexander Russell,et al.  Analyzing Algebraic Quantum Circuits Using Exponential Sums , 2008 .

[20]  Scott Aaronson,et al.  Improved Simulation of Stabilizer Circuits , 2004, ArXiv.

[21]  Yaoyun Shi Quantum and classical tradeoffs , 2005, Theor. Comput. Sci..

[22]  Alexander Yu. Vlasov,et al.  On Quantum Cellular Automata , 2004, ArXiv.

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

[24]  Raymond Laflamme,et al.  Quantum Computation and Quadratically Signed Weight Enumerators , 1999, ArXiv.

[25]  Sandy Irani,et al.  The Power of Quantum Systems on a Line , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[26]  Jason Twamley,et al.  Globally controlled quantum wires for perfect qubit transport, mirroring, and computing. , 2006, Physical review letters.

[27]  A. Kitaev Quantum computations: algorithms and error correction , 1997 .

[28]  James G. Oxley,et al.  Matroid theory , 1992 .

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

[30]  David P. DiVincenzo,et al.  Adaptive quantum computation, constant depth quantum circuits and arthur-merlin games , 2002, Quantum Inf. Comput..

[31]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[32]  Joachim Stolze,et al.  Spin chains as perfect quantum state mirrors , 2005 .

[33]  Andris Ambainis,et al.  Computing with highly mixed states , 2006, JACM.

[34]  John Watrous,et al.  On the complexity of simulating space-bounded quantum computations , 2004, computational complexity.

[35]  S. C. Benjamin,et al.  Quantum Computing in Arrays Coupled by 'Always On' Interactions , 2004, quant-ph/0401071.

[36]  W. Hoeffding Probability Inequalities for sums of Bounded Random Variables , 1963 .

[37]  Michael M. Wolf,et al.  On entropy growth and the hardness of simulating time evolution , 2008, 0801.2078.

[38]  A. Ekert,et al.  Universality in quantum computation , 1995, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[39]  Matthias Christandl,et al.  Perfect state transfer in quantum spin networks. , 2004, Physical review letters.

[40]  Joseph Fitzsimons,et al.  Quantum information processing with delocalized qubits under global control. , 2007, Physical review letters.

[41]  Salil P. Vadhan,et al.  Computational Complexity , 2005, Encyclopedia of Cryptography and Security.

[42]  Alexei Y. Kitaev,et al.  Quantum measurements and the Abelian Stabilizer Problem , 1995, Electron. Colloquium Comput. Complex..

[43]  Carsten Damm,et al.  Problems Complete for +L , 1990, IMYCS.

[44]  Michael A. Nielsen,et al.  The Solovay-Kitaev algorithm , 2006, Quantum Inf. Comput..

[45]  Robert Raussendorf Quantum cellular automaton for universal quantum computation , 2005 .

[46]  Richard Jozsa Quantum computation in algebraic number theory: Hallgren’s efficient quantum algorithm for solving Pell’s equation , 2003 .

[47]  B. Blank The Road to Reality : A Complete Guide to the Laws of the Universe , 2006 .

[48]  Lee Spector The Power of Quantum Computing , 2007 .

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

[50]  Umesh V. Vazirani,et al.  Molecular scale heat engines and scalable quantum computation , 1999, STOC '99.

[51]  Scott Aaronson,et al.  Quantum computing, postselection, and probabilistic polynomial-time , 2004, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[52]  H. Everett "Relative State" Formulation of Quantum Mechanics , 1957 .

[53]  D J Shepherd,et al.  Universally programmable quantum cellular automaton. , 2006, Physical review letters.

[54]  R. Werner,et al.  Reversible quantum cellular automata , 2004, quant-ph/0405174.

[55]  David A. Mix Barrington,et al.  Bounded-width polynomial-size branching programs recognize exactly those languages in NC1 , 1986, STOC '86.

[56]  Dan J. Shepherd On the Role of Hadamard Gates in Quantum Circuits , 2006, Quantum Inf. Process..

[57]  Martin Nilsson,et al.  Parallel Quantum Computation and Quantum Codes , 2001, SIAM J. Comput..

[58]  Maarten Van den Nest,et al.  Classical simulation of quantum computation, the gottesman-Knill theorem, and slightly beyond , 2008, Quantum Inf. Comput..

[59]  E. Knill,et al.  Power of One Bit of Quantum Information , 1998, quant-ph/9802037.

[60]  Yong Zhang,et al.  Bounds on the Power of Constant-Depth Quantum Circuits , 2005, FCT.

[61]  Simone Severini Universal quantum computation with unlabelled qubits , 2006 .

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

[63]  R Raussendorf,et al.  A one-way quantum computer. , 2001, Physical review letters.

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

[65]  Lisa R. Hales The quantum fourier transform and extensions of the abelian hidden subgroup problem , 2002 .

[66]  C. Fuchs Distinguishability and Accessible Information in Quantum Theory , 1996, quant-ph/9601020.

[67]  Alexander Yu. Vlasov,et al.  Programmable Quantum Networks with Pure States , 2005, ArXiv.

[68]  Robert Raußendorf Messungsbasiertes Quantenrechnen mit Clusterzuständen , 2003 .

[69]  N. J. A. Sloane,et al.  The Invariants of the Cli ord , 1999 .

[70]  Robert Spalek,et al.  Quantum Fan-out is Powerful , 2005, Theory Comput..

[71]  R. Jozsa Quantum algorithms and the Fourier transform , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[72]  Tanja Lange,et al.  MATHEMATICAL BACKGROUND OF PUBLIC KEY CRYPTOGRAPHY , 2005 .

[73]  Peter W. Shor,et al.  Estimating Jones polynomials is a complete problem for one clean qubit , 2007, Quantum Inf. Comput..

[74]  M. Bremner,et al.  Temporally unstructured quantum computation , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.