COT6600 Quantum Computing – Fall 2010
Lecture: M – Wd 1:30 -2:45 Eng 0383
Instructor: Dan C. Marinescu
Office HEC 439 B
Office hours: M-Wd 2:45 – 3:45 PM
Class outline
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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
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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
Quantum Complexity Theory. Complexity Classes BQP (Bounded Quantum
Polynomial Time)
Quantum Teleportation and Dense Coding
Physical Implementation of Quantum Computers
References:
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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.
Literature:
Many research articles can be accessed through the quant-ph archive
http://arxiv.org/abs/quant-ph/ maintained by Cornell University Library.
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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: 423441, 1998.
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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 19221941, 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. 261276. 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.''
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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-PFeynman
B. Furrow. “A Panoply of Quantum Algorithms.” Preprint, arxiv.org/quantph/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 IonTrap 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.
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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/plancklecture.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 NonPhysicists,” 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/quantph/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.
Grading policy:
Homework 30%
Class Presentations 35%
Final exam 35%