Prerequisites:
Basic linear algebra and analysis. Some knowledge of
quantum mechanics would be helpful, but I will spend a few lectures
reviewing the basics.
Recommended Literature:
There will be no textbook. The following books
contain part of what I plan to cover in the course:
1) Michael Nielsen, Isaac Chuang, Quantum Computation and Quantum
Information, Cambridge University Press, 2000.
2) A. Kitaev, A.H. Shen, M.N. Vyali, Classical and quantum computation
American Mathematical Society 2002.
3) R.F. Werner, Quantum Information Theory - An Invitation,
quant-ph/0101061.
Further references to research articles will be given throughout the course.
Syllabus:
I will discuss in this course some of the fundamental
aspects of quantum computing and
quantum information theory, that is the theory of information
processing using a quantum physical system.
After a short introduction to quantum mechanics (observables, states,
measurements, density matrices and all that), I plan to discuss the
notion of entanglement of quantum systems. Entanglement is a feature
of quantum mechanics, which does not exist in classical physics
(Einstein called it the ``spooky action at a distance''). It
expresses a correlation of subsystems of a quantum physical system which
appears naturally as soon as the commutative algebras of functions in
classical physics are replaced by non-commutative algebras of
operators (matrices) in quantum physics. If correlated quantum systems
contain ``enough'' entanglement, then they can be used to transmit quantum
information on a classical channel (so-called quantum teleportation).
Mathematically, noncommutative structures described by
operator theory, the theory of operator algebras and
the theory of completely positive maps are at the heart of these
phenomena.
In the second part of the course I will present the basics of quantum
computation and quantum algorithms. In particular I will describe
Peter Shor's famous factoring algorithm, which can be used to factor
integers on a (hypothetical) quantum computer in polynomial time. Shor's
result was quite a surprise since the best known algorithms on a classical
computer to date are exponential in the number of digits of the integer to
be factored. Current encryption schemes that insure network and data
security are based on the assumption that factoring an integer (or equivalent
problems) is a hard problem in the sense of complexity theory. Quantum
computing has triggered the need for new encryption schemes, which has led
to the field of quantum cryptography, currently a very active
research area.
This course will concentrate on the more theoretical aspects of quantum
computing and quantum information theory. For instance, I will not have
time to discuss the numerous ideas currently on the market for actually
building a quantum computer.
Grading: There will be no exams. The course grade will be based on
attendance. I will assign (optional) problems during the lectures.