<p> COT6600 Quantum Computing – Fall 2010 Lecture: M – Wd 1:30 -2:45 Eng 0383</p><p>Instructor: Dan C. Marinescu Office HEC 439 B Office hours: M-Wd 2:45 – 3:45 PM</p><p>Class outline  Quantum Computing and Quantum Information Theory – The Big Picture  Mathematical Foundation  Algebraic structures (groups, fields, vector spaces)  Complex numbers  Complex vector spaces o Bases and Dimensions o Inner Products o Linear Operators o Hermitian and Unitary Matrices o Eigenvectors and Eigenvalues of Quantum Operators o Tensor Products of Vector Spaces o Spectral Decomposition of an Operator  Elements of Quantum Mechanics o Quantum State Postulate o Dynamics Postulate o Measurements Postulate o Uncertainty Principle o Density Operator o Pure and Mixed States o Entanglement; Monogamy of Entanglement  Qubits and their Physical Implementation o Quantum Gates and Quantum Circuits o One-qubit gates: X, Y, Z, Hadamard, Phase-shift o Two-qubit gates: CNOT o Three-qubit gates: Fredkin and Toffoli  Universality of Quantum Circuits, Solovay-Kitaev Theorem, Clifford Group  Quantum Computational Models  Introduction to Quantum Algorithms o Deutsch Algorithm o Deutsch-Jozsa Algorithm o Bernstein-Vazirani Algorithm o Simon's Algorithm  Algorithms with Superpolynomial Speed-up o Efficient Quantum Circuits for the Fourier Transform o Quantum Phase Estimation o Shor's Algorithm for Factoring Integers and Determining Discrete Logarithms  Chinese Reminder Theorem  Euclid’s Algorithm  Reduction of Factoring to Order Finding. o Algorithms for Hidden Subgroup Problems o Algorithms for Hidden Nonlinear Structures  Algorithms Based on Amplitude Amplification o Grover's quantum search algorithm o Amplitude Amplification</p><p> Quantum Complexity Theory. Complexity Classes BQP (Bounded Quantum Polynomial Time)  Quantum Teleportation and Dense Coding  Physical Implementation of Quantum Computers</p><p>References:  D. C. Marinescu and G. M. Marinescu, “Approaching Quantum Computing,” Prentice Hall, 2004.  D.C. Marinescu and G. M. Marinescu, “From Classical to Quantum Information,’’ Academic Press, 2010. Online version at http://www.cs.ucf.edu/~dcm/QCV2.pdf  M. Nielsen and I. Chuang, “Quantum Computing,” Cambridge University Press, 2000.  R.P. Feynman, “Lectures on Computation,” Addison-Wesley, Reading, MA, 1996.  N. D. Mermin, “Quantum Computer Science: An Introduction,” Cambridge University Press, 2007.  R. Penrose, “The Road to Reality: A Complete Guide to the Laws of the Universe,” Vintage Books, 2007.</p><p>Literature: Many research articles can be accessed through the quant-ph archive http://arxiv.org/abs/quant-ph/ maintained by Cornell University Library.</p><p> A.Barenco, C.H. Bennett, R. Cleve, D.P. DiVincenzo, N.Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter. `”Elementary Gates for Quantum Computation.” Physical Review A, 52:3457 - 3570, 1995.  P. Beniof. “The Computer as a Physical System: A Microscopic Quantum Mechanical Hamiltonian Model of Computers as Represented by Turing Machines.'' J. of Statistical Physics, 22:563 - 591, 1980.  P.Benioff. “`Quantum Mechanical Hamiltonian Models of Turing Machines.” J. of Statistical Physics, 29:515 - 546, 1982.  P.Benioff. “`Quantum Mechanical Hamiltonian Models of Computers.” Ann. NY Acad. Sciences, 480: 475, 1986.  P.Benioff. “Models of Quantum Turing Machines.” Fortschritte der Physik, 46: 423- 441, 1998.  C. H. Bennett, “Logical Reversibility of Computation,” IBM Journal of Research and Development, 17: 525 - 535, 1973.  C. H. Bennett and G. Brassard. ``Quantum Cryptography: Public Key Distribution and Coin Tossing.'' Proc. IEEE Conf. on Computers, Systems, and Signal Processing, IEEE Press, 175 - 179, 1984.  C. H. Bennett, G. Brassard, C. Cr\'epeau, R. Jozsa, A. Peres, and W. K. Wootters. ``Teleporting an Unknown State via Dual Classical and Einstein-Podolsky-Rosen Channels.'' Phys. Rev. Lett., 70(13):1895 - 1899, 1993.  C.H. Bennett. “Quantum Information and Computation.'' Physics Today, 24 - 30, October 1995.  C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. “Strengths and Weaknesses of Quantum Computation.” SIAM Journal on Computing, 26:1510 - 1523, 1997.  E. Bernstein and U.Vazirani, “Quantum Complexity Theory,” SIAM Journal of Computing, 26: 1411 - 1473, 1997.  A. Berthiaume and G.Brassard. “Oracle Quantum Computing.” Proc. Workshop on Physics of Computation, 195 - 199, IEEE Press, Los Alamitos, CA, 1992.  M. Born. “The Statistical Interpretations of Quantum Mechanics.” Nobel Lectures, Physics 1942 - 1962, 256 - 267. December 11, 1954. Also http://nobelprize.org/nobel\_prizes/physics/laureates/1954/born-lecture.pdf.  G.Brassard, P.Hoyer, M.Mosca, and A.Tapp. “Quantum Amplitude Amplification and Estimation.” Preprint, arxiv.org/quanth-ph/0005055 v1, 2000.  J. I. Cirac and P. Zoller. ``Quantum Computation with Cold Trapped Ions.'' Physical Review Letters, 74(20):4091 - 4094, 1995.  R.Cleve, A.Ekert, L.Henderson, C.Macchiavello, and M.Mosca. “On Quantum Algorithms.” Preprint, arxiv.org/quant-ph/9903061 v1, 1999.  P. A. M. Dirac. ``Theory of Electrons and Positrons.'' Nobel Lectures, Physics 1922- 1941, Elsevier Publishing Company, Amsterdam, 1965. See also http://nobelprize.org/nobel_prizes/physics/laureates/1933/dirac-lecture.pdf.  D. P. DiVincenzo, “Quantum Computation,” Science, 270: 255 - 261, 1995.  D.P. DiVincenzo.”Quantum Gates and Circuits.” Phil. Trans. Royal Soc. London A}, 454:261 - 276 1998, Also, Proc.: Mathematical, Physical and Engineering Sciences, Vol. 454, No. 1969, Quantum Coherence and Decoherence (January 1998), pp. 261- 276. Also, Preprint, arxiv.org/quanth-ph/9705009, May, 1997.  D. P. DiVincenzo, “The Physical Implementation of Quantum Computation,” Fortschritte der Physik, 48(9-11): 771 - 783, 2000.  C. M. Dawson and M. A. Nielsen. “The Solovay-Kitaev Algorithm.” Preprint, arxiv.org/quanth-ph/0505030}, 2005.  D.Deutsch. “Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer.” Proc. Royal Society London A, 400:97 - 117, 1985.  D. Deutsch, A. Barenco, and A. Ekert. “Universality in Quantum Computation.” Proc. Royal Society London, 449:669 - 677, 1995.  A.K.Ekert and R.Jozsa. “Quantum Algorithms: Entanglement Enhanced Information Processing.” Proc. Royal Society London A, 356(1743):1769 - 1782, 1998.  A.K. Ekert, P. Hayden, and H. Inamori. “Basic Concepts in Quantum Computing.'' Preprint, arxiv.org/quant-ph/0011013 v1, 2000.  E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. “Quantum Computation by Adiabatic Evolution.” Preprint, arxiv.org/quant-ph/0001106v1 v1}, 2000.  R. P. Feynman, “Simulating Physics with Computers,” International Journal of Theoretical Physics, 21: 467 - 488, 1982. See also http://www.scribd.com/doc/15244327/Simulating-Physics-with-Computers-by-R-P- Feynman  B. Furrow. “A Panoply of Quantum Algorithms.” Preprint, arxiv.org/quant- ph/0606127, 2006  L. K. Grover, “A Fast Quantum Algorithm for Database Search,” Proceedings, ACM Symposium on Theory of Computing, ACM Press, NY, 212 - 219, 1996. Also updated version: Preprint, arxiv.org/quanth-ph/9605043, 1996.  L. K. Grover, “From Schrödinger's Equation to the Quantum Search Algorithm,” American Journal of Physics, 69(7): 769 - 777, 2001.  W. Heisenberg, ``The Development of Quantum Mechanics.'' Nobel Lectures, Physics 1922-1941, Elsevier Publishing Company, Amsterdam, 1965, http://nobelprize.org/nobel_prizes/physics/laureates/1932/heisenberg-lecture.pdf.  C. K. Hong and L. Mandel. ``Experimental Realization of a Localized One-Photon State.'' Physical Review Letters, 56(1):58 - 60, 1986.  P.Hoyer. “Arbitrary Phases in Quantum Amplitude Purification.” Physical Review A, 62:052304, 2000.  R. Jozsa, “Entanglement and Quantum Computation,” Geometric Issues in the Foundations of Science. S. Hugget, L. Mason, K.P. Tod, S.T. Tsou, and Woodhouse, Eds. Oxford University Press, 1997.Also: Preprint, arxiv.org/quant-ph/9707034 v1,1997.  R. Jozsa. “Quantum Algorithms and the Fourier Transform.”. Preprint, arxiv.org/quant-ph/9707033 v1}, 1997.  R. Jozsa, “Quantum Factoring, Discrete Logarithms, and the Hidden Subgroup Problem,” Preprint, arxiv.org/quant-ph/0012084 v1, 2000.  \bibitem{Kielpinski02}  D. Kielpinski, C. Monroe, and D. J. Wineland. ``Architecture for a Large-Scale Ion- Trap Quantum Computer.'' Nature, 417:709 - 711, 2002.  T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O'Brien. ``Quantum Computers.'' Nature, 464:4552, 2010. (Overview of implementations).  R. Landauer, “Irreversibility and Heat Generation in the Computing Process,” IBM Journal of Research and Development, 5: 182 - 192, 1961.  L. B. Levitin and T. Toffoli. ``Thermodynamic Cost of Reversible Computing.'' Physical Review Letters, 99: 110502, 2007.  S. Lloyd, “A Potentially Realizable Quantum Computer,” Science, 261: 1569 - 1571, 1993.  S. Lloyd, “Almost any Quantum Logic Gate is Universal,” Physical Review Letters, 75, 346 - 349 1995.  G. J. Milburn. ``Quantum Optical Fredkin Gate.'' Physical Review Letters, 62:2124 - 2127, 1988.  W. Pauli. ``Exclusion Principle and Quantum Mechanics.'' Nobel Lectures, Physics 1942 - 1962, 27 - 43, December 13, 1946. Also: http://nobelprize.org/nobel_prizes/physics/laureates/1945/pauli-lecture.pdf.  M. K. E. L. Planck. ``The Genesis and Present State of Development of the Quantum Theory.'' Nobel Lectures, Physics 1901 - 1922. Elsevier Publishing Company, Amsterdam. Also: http://nobelprize.org/nobel_prizes/physics/laureates/1918/planck- lecture.pdf.  J. Preskill, “Lecture Notes for Physics 229: Quantum Information and Computing,” California Institute of Technology, http://www.theory.caltech.edu/~preskill/ph229/  R. Raussendorf and H. J. Briegel. ``One-Way Quantum Computer.'' Physical Review Letters, 86(22):5188 - 5191, 2001.  E.Rieffel and W.Polak, “An Introduction to Quantum Computing for Non- Physicists,” ACM Computing Surveys, 32(3): 300 - 335, 2000.  E. Schr\"odinger. ``The Fundamental Idea of Wave Mechanics.'' Nobel Lectures, Physics 1922-1941, Elsevier Publishing Company, Amsterdam, 1965. See also http://nobelprize.org/nobel_prizes/physics/laureates/1933/schrodinger-lecture.pdf.  P.W.Shor, “Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer,'' SIAM Journal of Computing,” 26: 1484 - 1509, 1997.  P.W.Shor, “Introduction to Quantum Algorithms,” Preprint, arxiv.org/quant- ph/0005003, July 2001.  P.W.Shor, “Why Haven't More Quantum Algorithms Been Found?” Journal of the ACM, 50(1): 87 - 90, 2003.  A.M.Steane, “Quantum Computing,” Reports on Progress in Physics, 61: 117, 1998.Also: Preprint, arxiv.org/quant-ph/97080222 v2, September 1997.  T. Tulsi, L. K. Grover, and A. D. Patel. ``A New Algorithm for Directed Quantum Search.'' Preprint, arxiv.org/quanth-ph/0505007 v2, 2005.  W. K. Wootters and W. H. Zurek. ``A Single Quantum Cannot Be Cloned.'' Nature, 299:802 - 803, 1982. See also http://pm1.bu.edu/~tt/qcl/pdf/wootterw198276687665.pdf.  W. K. Wootters. ``Entanglement of Formation of an Arbitrary State of Two Qubits.'' Physical Review Letters, 80(10): 2245 - 2248, 1998.  W. H. Zurek. ``Decoherence and the Transition from Quantum to  Classical.'' Physics Today, 44(10):36 - 44, 1991.</p><p>Grading policy: Homework 30% Class Presentations 35% Final exam 35% </p>