COT6600 Quantum Computing
Credit: 3 units
Offered: Fall semester
Instructors: Dan Marinescu and Pawel Wocjan
Class outline
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Linear Algebra & Dirac Notation
o Hilbert Spaces and Linear Operators
o Eigenvectors and Eigenvalues of Quantum Operators
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 Algorithm
o Shor's Algorithm for Factoring Integers and Determining Discrete Logarithms
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
o Quantum Counting
o Quantum Walks
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Quantum Computational Complexity Theory & Lower Bounds
o Quantum Complexity Classes BQP (Bounded Quantum Polynomial Time)
and QMA (Quantum Arthur Merlin)
o Complete Problems for QMA: local Hamiltonian problem, Compatibility of
density matrices
o Polynomial method
o Adversary methods
References:
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D. C. Marinescu and G. M. Marinescu, “Approaching Quantum Computing,”
Prentice Hall, 2004.
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 Los Alamos National Laboratory.
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C. H. Bennett, “Logical Reversibility of Computation,” IBM Journal of Research and
Development, 17: 525 - 535, 1973.
E. Bernstein and U.Vazirani, “Quantum Complexity Theory,” SIAM Journal of
Computing, 26: 1411 - 1473, 1997.
D. P. DiVincenzo, “Quantum Computation,” Science, 270: 255 - 261, 1995.
D. P. DiVincenzo, “The Physical Implementation of Quantum Computation,”
Fortschritte der Physik, 48(9-11): 771 - 783, 2000.
R. P. Feynman, “Simulating Physics with Computers,” International Journal of
Theoretical Physics, 21: 467 - 488, 1982.
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.
R. Jozsa, “Entanglement and Quantum Computation,” Geometric Issues in the
Foundations of Science. S. Hugget, L. Mason, K.P. Tod, S.T. Tsou, and N. M. J.
Woodhouse, Oxford University Press, 1997.Also: Preprint, arxiv.org/quantph/9707034 v1,1997.
R. Jozsa, “Quantum Factoring, Discrete Logarithms, and the Hidden Subgroup
Problem,” Preprint, arxiv.org/quant-ph/0012084 v1, 2000.
R. Landauer, “Irreversibility and Heat Generation in the Computing Process,” IBM
Journal of Research and Development, 5: 182 - 192, 1961.
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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.
J. Preskill, “Lecture Notes for Physics 229: Quantum Information and Computing,”
California Institute of Technology, http://www.theory.caltech.edu/~preskill/ph229/
E.Rieffel and W.Polak, “An Introduction to Quantum Computing for NonPhysicists,” ACM Computing Surveys, 32(3): 300 - 335, 2000.
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.
Grading policy:
Homework 35%
Midterm
25%
Final exam 40%