# COT6600 - Computer Science

```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.
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.
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Computation.” Physical Review A, 52:3457 - 3570, 1995.
P. Beniof. “The Computer as a Physical System: A Microscopic Quantum
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P.Benioff. “`Quantum Mechanical Hamiltonian Models of Turing Machines.” J. of
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P.Benioff. “`Quantum Mechanical Hamiltonian Models of Computers.” Ann. NY
P.Benioff. “Models of Quantum Turing Machines.” Fortschritte der Physik, 46: 423441, 1998.
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C.H. Bennett. “Quantum Information and Computation.'' Physics Today, 24 - 30,
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E. Bernstein and U.Vazirani, “Quantum Complexity Theory,” SIAM Journal of
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D. P. DiVincenzo, “Quantum Computation,” Science, 270: 255 - 261, 1995.
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D. P. DiVincenzo, “The Physical Implementation of Quantum Computation,”
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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
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D. Deutsch, A. Barenco, and A. Ekert. “Universality in Quantum Computation.” Proc.
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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
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R. P. Feynman, “Simulating Physics with Computers,” International Journal of
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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,
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C. K. Hong and L. Mandel. ``Experimental Realization of a Localized One-Photon
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P.Hoyer. “Arbitrary Phases in Quantum Amplitude Purification.” Physical Review
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v1,1997.
R. Jozsa. “Quantum Algorithms and the Fourier Transform.”. Preprint,
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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.
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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
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http://nobelprize.org/nobel_prizes/physics/laureates/1945/pauli-lecture.pdf.
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P.W.Shor, “Polynomial-Time Algorithms for Prime Factorization and Discrete
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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
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A.M.Steane, “Quantum Computing,” Reports on Progress in Physics, 61: 117,
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