Quantum Error-correcting Codes

These notes are a record of proceedings in the QMW Combinat-orics Study Group in November and December 1998. Since we are discrete mathematicians and know little quantum theory, the notes are not strong on the physics background (but we give references to several sources for this). We have tried to compare quantum with classical error correction where possible, and to provide enough information to see what are the workpoints for discrete mathematicians in this area. An exciting recent development on the border of computer science and physics has been the realisation that some important problems can be solved much more eeciently on a quantum computer than on a classical computer. Most notably, Shor 16] gave a randomised algorithm for factorising an integer in polynomial time on a quantum computer. Since factorisation of large integers is the hard problem which underpins public-key encryption systems such as RSA 14, 19], the importance of this result is obvious, and a large amount of research into the possibility of building a quantum computer is going on. (We discuss this in the Appendix.) The eeciency of a quantum computer derives from the fact that a quantum state is the superposition of a number of pure states, all of which evolve according to the laws of physics (the Schrr odinger equation); in eeect, a quantum computer is highly parallel. Various schemes for a quantum computer have been proposed. The limitation derives from the fact that a bit of information is stored by a single atom 1