Quantum Computing

• Text: Quantum Computing
• Author: Mika Hirvensalo
• Publisher: Springer-Verlag
• ISBN: 3540667830
• Web page for purchasing this from Amazon

This is a course on Quantum Computing. Its prerequisites include an extensive knowledge of linear algebra (eigenvalues,unitary matrices, spectrum). The course will also deal with other mathematical topics such as number theory and partial differential equations (so some knowledge of these topics will be helpful).

Exams will be administered on web pages that will use MathML. To browse such pages, it will be necessary to use the Mozilla web browser
Page for obtaining the Mozilla browser
It will also be necessary to install math fonts for this browser. Information on how to install these can be found on
The MathML Fonts Page
For linux users, the necessary fonts can be found in: mozilla-math.tar.gz (these should be installed in X windows).

Note: Exam pages and other web pages for this course will not necessarily even show up in other browsers! Once you have installed Mozilla and the fonts, you can check your installation by viewing:

The MathML Start Page

Installing the Jave Plugin on Windows

Exam 2 (due 4/29/2002

The course will cover topics including:

1. Introduction to Quantum Mechanics (or fifty years of Quantum Mechanics in fifty minutes).
   1. Black-body radiation and Planck's constant
   2. Einstein's Photoelectric Effect
   3. Particle-wave duality. The Feynmann Double Slit
   4. Probability waves and the Copenhagen School. The Schrödinger Wave Equation.
   5. Paradoxes: Schrödinger's much-abused cat.
   6. The Stern-Gerlach Experiment --- Discrete values for physical quantities.
   7. Everett's Relative-State Formulation of Quantum Mechanics
   8. Phase space in classical physics: A mathematical convention.
   9. How phase space in Quantum Mechanics becomes interesting.
   10. Quantum entanglement. (Bell's inequality)
2. A review of Linear Algebra
   1. Vector spaces bases, linear transformations, eigenvalues
   2. normed spaces
   3. Hilbert spaces and unitary transformations
   4. Tensor products of vector spaces. Kronecker products of matrices.
   5. Spectra and eigenvalues
   6. Dirac bra-ket notation.
3. Qubits.
   1. A Quantum RAM computer
   2. Abstract definition and physical realization.
   3. Measuring the value of a qubit.
   4. Multi-qubit systems as tensor products of single qubits.
4. Basic quantum computing elements.
   1. Algorithms as unitary transformations, reversible computations.
   2. Controlled-NOT
   3. Toffoli gates
   4. AND, OR gates.
   5. The square root of NOT
5. Shor's Algorithm for factoring large numbers
   1. Overview. Implications for cryptography.
   2. Review of group theory
      (Material from Chapter 8 of the text)
      • Abelian and nonabelian groups
      • Subgroups and cosets
      • Quotient groups
      • Fermat's Little Theorem
      • Multiplicative groups of residue class fields: the Euler phi-function, order of an element.
      • Fourier transforms and characters of finite groups.
   3. Review of number theory: Euclidean Algorithm for greatest common divisor and continued fractions.
   4. Order-computation as a large Fourier Transform calculation.
6. Grover's algorithm for searching an unordered list.
   1. Paradigm of Quantum Computing: it's easy to get answers but it's hard to locate the right one.
   2. Search as a problem involving geometric transformations.

Students will be required to give to read a research paper and give a 10 minute summary of it at the end of the semester. Use the following page to sign up for a paper:

Sign up sheet for presentations

Look up your grade

Links

• The Oxford Institute of Quantum Computing
• Quantum Computation/Cryptography at Los Alamos
• Quantum Information at IBM.
• Quantum Computing Simulators
• Quantum Computing in Perl (there's more than one universe in which to do it)
• IBM's 7-qubit Quantum Computer
  (In a single molecule)
• Silicon buckyballs (as potential qubits)
• Dave's Short Course on Complex Numbers
  This covers Complex numbers. It introduces the standard notation for conjugates. The text uses a slightly nonstandard notation (as a *-superscript).
• Maxima Home Page
  Maxima is a free computer algebra system for Unix and Windows.
• Maple Home Page
  Maple is a computer algebra system for Unix and Windows to which Drexel has a license.
• Quantum Computing tutorial
• Quantum Teleportation

Articles

• A. Barenco, "Quantum physics and computers," Contemporary Physics, 37, 375-89 (1996).
• C. H. Bennett, "Quantum information and computation," Physics Today, October 1995, 24-30 (1995).
• A. Ekert, P. Hayden, H. Inamori, "Basic concepts in quantum computation," LANL preprint quant-ph/0011013 (2000).
• D. Gottesman and H-K. Lo, " From Quantum Cheating to Quantum Security," Physics Today, Nov. 2000, 22 (2000).
• J. Preskill, " Quantum computing: pro and con," Proc. R. Soc. Lond. A, 454, 469-86 (1998).
• J. Preskill, "Battling decoherence: the fault-tolerant quantum computer," Physics Today, June 1999, 24-30 (1999).
• A. Steane, " Quantum computing," Rep. Prog. Phys. 61, 117-173 (1998).
• A. M. Steane and W. van Dam, "Physicists triumph at guess my number," PhysicsToday, Feb 2000, 35 - 39 (2000).
• V. Vedral and M. B. Plenio, "Basics of quantum computation," Prog. Quant. Electron 22, 1-39 (1998).
• A. Zeilinger, " Fundamentals of quantum information," Physics World, 11, March 1998.