Project page: www.math.wm.edu/~ckli/quantum1.html
Topic description: One may see http://en.wikipedia.org/wiki/Quantum_computing for a basic introduction of quantum computing.
To construct a quantum error correcting codes one has to solve a matrix problem, namely, given an n-by-n complex matrix A, can one apply a unitary similarity to A so that the leading k-by-k matrix is a scalar matrix? The set of these scalar matrices is called the rank-k numerical range of A. For example, given a n-by-n matrix A, it is important to know whether the rank-k numerical range is empty, and what kind of scalar matrices will lie in the rank-k numerical ranges.
By some recent results of Li and his collaborators, the rank-k numerical range of a single matrix A is quite well understood. See
Using these results, one can construct binary channel with quantum error correcting capability.
Research opportunities: To construct a general error correcting codes, one has to study the joint rank-k numerical range of several matrices A1, ..., Am; i.e., applying the same unitary similarity so that all the leading k-by-k matrices are scalar matrices. This is a wide open area, and offers many opportunities for research.
Prerequisities: Math 211 Linear Algebra.
Suggested prerequisities: Programming skills in Matlab and Maple will be very helpful. One can write computer program to test the non-emptyness of the joint rank-k numerical range.
Students with more advanced mathematics courses such as Math 405, 408 may be able to study some theoretical aspects of the problem.
Contact: Chi-Kwong Li
Last updated at Wednesday, 09-Apr-2008 16:31:25 EDT.