QUANTUM THEORY & QUANTUM COMPUTERS

Dr Hannabuss 16 lectures HT 2009 Monday and Wednesday 11.00 Suitable for third and fourth year students who have done a b7 quantum course.

Entangled Beryllium ions: NIST teleportation experiment

Aims and Objectives

This course builds directly on the first course in quantum mechanics and covers a series of important topics, particularly features of systems containing several particles. The behaviour of identical particles in quantum theory is more subtle than in classical mechanics, and an understanding of these features allows one to understand the periodic table of elements and the rigidity of matter. It also introduces a new property of entanglement linking particles which can be quite widely dispersed.

There are rarely neat solutions to problems involving several particles, so usually one needs some approximation methods. In very complicated systems, such as the molecules of gas in a container, quantum mechanical uncertainty is compounded by ignorance about other details of the system and requires tools of quantum statistical mechanics.

Two state quantum systems enable one to encode binary information in a new way which permits superpositions. This leads to a quantum theory of information processing, and by exploiting entanglement to other ideas such as quantum teleportation.

Synopsis

Identical particles, symmetric and anti-symmetric states, Fermi-Dirac and Bose-Einstein statistics and atomic structure.

Heisenberg representation, interaction representation, time dependent perturbation theory and Feynman-Dyson expansion. Approximation methods, Rayleigh-Schrödinger time-independent perturbation theory and variation principles. The virial theorem. Helium.

Mixed states, density operators. The example of spin systems. Purification. Gibbs states and the KMS condition.

Entanglement. The EPR paradox, Bell's inequalities, Aspect's experiment. GHZ states

Quantum information processing, qubits and quantum computing. The no-cloning theorem, quantum teleportation. Quantum logic gates. Quantum operations. The quantum Fourier transform. Grover's search algorithm.

Reading

• Hannabuss, Introduction to quantum mechanics, OUP (1997). Chapters 10-12 and 14, 16, supplemented by lecture notes on quantum computers on the web.
• M A Nielsen and I L Chuang: Quantum Computation, Cambridge University Press, (2000).
• S Stenholm and K-A Suominen: Quantum approach to informatics, Wiley (2005).

Further reading:

• A popular non-technical account of the subject: A Hey and P Walters, The new Quantum Universe, Cambridge (2003).
• Other accounts of the first part of the course:
  • I P Grant, Classical and Quantum Mechanics, Mathematical Institute Notes (1991).
  • L I Schiff, Quantum Mechanics, 3rd edition, Mc Graw Hill (1968).
  • B J Bransden and C J Joachain, Introduction to quantum mechanics, Longman (1995)
  • A I M Rae, Quantum Mechanics, 4th edition, Institute of Physics (1993)
• Additional background material on quantum computing and paradoxes:
  • Y Aharanov and D Rohrlich: Quantum Paradoxes, Wiley-VCH (2005).
  • J Audretsch: Verschränkte Systeme, Wiley-VCH (2005).
  • P Kaye, R Laflamme, M Mosca: An introduction to quantum computing, Oxford UP (2007).