Quantum computation

The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and 2D-magnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like e−αl, where l is a length scale, and α is some positive constant. In contrast, the “presumptive” qubit-model of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about 10−4) before computation can be stabilized. Quantum computation is a catch-all for several models of computation based on a theoretical ability to manufacture, manipulate and measure quantum states. In this context, there are three areas where remarkable algorithms have been found: searching a data base (15 ), abelian groups (factoring and discrete logarithm) (19 , 27 ), and simulating physical systems (5 , 21 ). To this list we may add a fourth class of algorithms which yield approximate,

[1]  H. S. Allen.  The Quantum Theory , 1928, Nature.

[2]  E. Wright,et al.  An Introduction to the Theory of Numbers , 1939 .

[3]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[4]  David E. Muller,et al.  Complexity in Electronic Switching Circuits , 1956, IRE Trans. Electron. Comput..

[5]  Charles H. Bennett,et al.  Logical reversibility of computation , 1973 .

[6]  J. Myrheim,et al.  On the theory of identical particles , 1977 .

[7]  Adi Shamir,et al.  A method for obtaining digital signatures and public-key cryptosystems , 1978, CACM.

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

[9]  Tommaso Toffoli,et al.  Reversible Computing , 1980, ICALP.

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

[11]  T. Toffoli,et al.  Conservative logic , 2002, Collision-Based Computing.

[12]  Frank Wilczek,et al.  Quantum Mechanics of Fractional-Spin Particles , 1982 .

[13]  Vaughan F. R. Jones.  Index for subfactors , 1983 .

[14]  B. Halperin.  Statistics of quasiparticles and the hierarchy of fractional quantized Hall states , 1984 .

[15]  Frank Wilczek,et al.  Fractional Statistics and the Quantum Hall Effect , 1984 .

[16]  Yong-Shi Wu.  General Theory for Quantum Statistics in Two-Dimensions , 1984 .

[17]  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.

[18]  David Deutsch,et al.  Quantum theory as a universal physical theory , 1985 .

[19]  V. Jones.  A polynomial invariant for knots via von Neumann algebras , 1985 .

[20]  R. Feynman.  Quantum mechanical computers , 1986 .

[21]  Louis H. Kauffman,et al.  State Models and the Jones Polynomial , 1987 .

[22]  William H. Press,et al.  Numerical Recipes: The Art of Scientific Computing , 1987 .

[23]  Vaughan F. R. Jones,et al.  Hecke algebra representations of braid groups and link polynomials , 1987 .

[24]  François Jaeger,et al.  Tutte polynomials and link polynomials , 1988 .

[25]  J. Fröhlich,et al.  Quantum field theory of anyons , 1988 .

[26]  Charles H. Bennett.  Time/Space Trade-Offs for Reversible Computation , 1989, SIAM J. Comput..

[27]  D. Deutsch.  Quantum computational networks , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[28]  Edward Witten,et al.  Quantum field theory and the Jones polynomial , 1989 .

[29]  Frank Wilczek,et al.  Fractional statistics and anyon superconductivity , 1990 .

[30]  Einarsson,et al.  Fractional statistics on a torus. , 1990, Physical review letters.

[31]  F. Gabbiani,et al.  Braid statistics in local quantum theory , 1990 .

[32]  M. Atiyah.  On framings of 3-manifolds , 1990 .

[33]  Paul Melvin,et al.  The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2, C) , 1991 .

[34]  Vladimir Turaev,et al.  Invariants of 3-manifolds via link polynomials and quantum groups , 1991 .

[35]  Wen,et al.  Mean-field theory of spin-liquid states with finite energy gap and topological orders. , 1991, Physical review. B, Condensed matter.

[36]  R. Jozsa.  Characterizing classes of functions computable by quantum parallelism , 1991, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[37]  Wilczek,et al.  Paired Hall state at half filling. , 1991, Physical review letters.

[38]  Wen.  Non-Abelian statistics in the fractional quantum Hall states. , 1991, Physical review letters.

[39]  Gregory W. Moore,et al.  Nonabelions in the fractional quantum Hall effect , 1991 .

[40]  Vladimir Turaev,et al.  State sum invariants of 3 manifolds and quantum 6j symbols , 1992 .

[41]  D. Deutsch,et al.  Rapid solution of problems by quantum computation , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[42]  X. Wen,et al.  Many-body systems with non-abelian statistics☆ , 1992 .

[43]  X. Wen.  THEORY OF THE EDGE STATES IN FRACTIONAL QUANTUM HALL EFFECTS , 1992 .

[44]  Nicolai Reshetikhin,et al.  Quantum Groups , 1993 .

[45]  Frederick M. Goodman,et al.  The Temperley-Lieb algebra at roots of unity. , 1993 .

[46]  S Lloyd,et al.  A Potentially Realizable Quantum Computer , 1993, Science.

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

[48]  R. Jozsa.  Fidelity for Mixed Quantum States , 1994 .

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

[50]  J. Barrett,et al.  Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds , 1994 .

[51]  Arjen K. Lenstra,et al.  The Magic Words are Squeamish Ossifrage , 1994, ASIACRYPT.

[52]  V. Turaev.  Quantum Invariants of Knots and 3-Manifolds , 1994, hep-th/9409028.

[53]  D. Coppersmith.  An approximate Fourier transform useful in quantum factoring , 2002, quant-ph/0201067.

[54]  King,et al.  Demonstration of a fundamental quantum logic gate. , 1995, Physical review letters.

[55]  Shor,et al.  Scheme for reducing decoherence in quantum computer memory. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[56]  Barenco,et al.  Conditional Quantum Dynamics and Logic Gates. , 1995, Physical review letters.

[57]  Gardiner,et al.  Decoherence, continuous observation, and quantum computing: A cavity QED model. , 1995, Physical review letters.

[58]  Hood,et al.  Measurement of conditional phase shifts for quantum logic. , 1995, Physical review letters.

[59]  N. Habegger,et al.  Topological Auantum Field Theories derived from the Kauffman bracket , 1995 .

[60]  Barenco,et al.  Elementary gates for quantum computation. , 1995, Physical review. A, Atomic, molecular, and optical physics.

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

[62]  Lloyd,et al.  Almost any quantum logic gate is universal. , 1995, Physical review letters.

[63]  J. Cirac,et al.  Quantum Computations with Cold Trapped Ions. , 1995, Physical review letters.

[64]  Unruh.  Maintaining coherence in quantum computers. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[65]  DiVincenzo.  Two-bit gates are universal for quantum computation. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[66]  Raymond Laflamme,et al.  Quantum Computers, Factoring, and Decoherence , 1995, Science.

[67]  X. Wen.  Topological Orders and Edge Excitations in FQH States , 1995 .

[68]  Knight,et al.  Realistic lower bounds for the factorization time of large numbers on a quantum computer. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[69]  Carlton M. Caves,et al.  Quantum information: How much information in a state vector? , 1996 .

[70]  Andrew M. Steane.  The ion trap quantum information processor , 1996 .

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

[72]  The Frobenius-Schur indicator in conformal field theory , 1996, hep-th/9610192.

[73]  Shor,et al.  Good quantum error-correcting codes exist. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[74]  Peter W. Shor,et al.  Fault-tolerant quantum computation , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[75]  Gottesman.  Class of quantum error-correcting codes saturating the quantum Hamming bound. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[76]  Knill,et al.  Decoherence Bounds on Quantum Computation with Trapped Ions. , 1996, Physical review letters.

[77]  R. Jozsa,et al.  Quantum Computation and Shor's Factoring Algorithm , 1996 .

[78]  Lov K. Grover.  A fast quantum mechanical algorithm for estimating the median , 1996, quant-ph/9607024.

[79]  Frank Wilczek,et al.  2n-quasihole states realize 2n−1-dimensional spinor braiding statistics in paired quantum Hall states , 1996 .

[80]  DiVincenzo,et al.  Fault-Tolerant Error Correction with Efficient Quantum Codes. , 1996, Physical review letters.

[81]  Laflamme,et al.  Perfect Quantum Error Correcting Code. , 1996, Physical review letters.

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

[83]  M. Finkelberg.  An equivalence of fusion categories , 1996 .

[84]  Steane,et al.  Simple quantum error-correcting codes. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[85]  Steane,et al.  Error Correcting Codes in Quantum Theory. , 1996, Physical review letters.

[86]  Seth Lloyd,et al.  Universal Quantum Simulators , 1996, Science.

[87]  Preskill,et al.  Efficient networks for quantum factoring. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[88]  Gilles Brassard,et al.  Tight bounds on quantum searching , 1996, quant-ph/9605034.

[89]  M. B. Plenio,et al.  Decoherence limits to quantum computation using trapped ions , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[90]  J. Preskill.  Fault-tolerant quantum computation , 1997, quant-ph/9712048.

[91]  염흥렬,et al.  [서평]「Applied Cryptography」 , 1997 .

[92]  E. Guitter,et al.  Meanders and the Temperley-Lieb algebra , 1997 .

[93]  А Е Китаев,et al.  Квантовые вычисления: алгоритмы и исправление ошибок@@@Quantum computations: algorithms and error correction , 1997 .

[94]  N. Gershenfeld,et al.  Bulk Spin-Resonance Quantum Computation , 1997, Science.

[95]  M. Plenio,et al.  Conditional generation of error syndromes in fault-tolerant error correction , 1997 .

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

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

[98]  A. Calderbank,et al.  Quantum Error Correction and Orthogonal Geometry , 1996, quant-ph/9605005.

[99]  S. Sarma,et al.  Perspectives in Quantum Hall Effects: Novel Quantum Liquids in Low-Dimensional Semiconductor Structures , 1997 .

[100]  Ming Li,et al.  Reversible Simulation of Irreversible Computation by Pebble Games , 1997, ArXiv.

[101]  Andrew Pressley,et al.  QUANTUM GROUPS (Graduate Texts in Mathematics 155) , 1997 .

[102]  D. Gottesman.  Theory of fault-tolerant quantum computation , 1997, quant-ph/9702029.

[103]  F. Wilczek,et al.  A Chern-Simons effective field theory for the Pfaffian quantum Hall state , 1997, cond-mat/9711087.

[104]  John Preskill,et al.  Topological Quantum Computation , 1998, QCQC.

[105]  D. DiVincenzo,et al.  Quantum computation with quantum dots , 1997, cond-mat/9701055.

[106]  D. Tambara,et al.  Tensor Categories with Fusion Rules of Self-Duality for Finite Abelian Groups , 1998 .

[107]  M H Freedman,et al.  P/NP, and the quantum field computer , 1998, Proc. Natl. Acad. Sci. USA.

[108]  X. Wen.  Topological Orders in Rigid States * , 2022 .