Fall Semester 2014, Quantum Information Theory
(Math 390A)



Instructor: Dietmar Bisch
Lecture: TuTh, 9:35am-10:50am, SC 1308
Office: SC 1405, (615) 322-4168
Office hours: TuTh 10:50am-11:50am & Fr 3:00pm-4:00pm
Mailbox: SC 1326


Prerequisites: Linear algebra, real analysis and basic functional 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, 2011 (first edition: 2000).
2) A. Kitaev, A.H. Shen, M.N. Vyali, Classical and quantum computation , American Mathematical Society 2002.
3) Zhenghan Wang, Topological Quantum Computation, American Mathematical Society 2010.

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. If there is time, I will spend the third part of the course on topological quantum computing. I will describe what the advantages of a topological quantum computer would be.

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.