Transmitting quantum information losslessly through a network of particles is an important problem in quantum computing. Mathematically this amounts to studying solutions of the discrete Schrödinger equation $d \phi /dt = i H \phi$, where $H$ is typically the adjacency or Laplace matrix of the graph. This in turn leads to questions about subtle number-theoretic behavior of the eigenvalues of $H$.
It has proven to be difficult to find graphs which support such information transfer. I will talk about recent progress in understanding what happens when one is allowed to apply magnetic fields (that is, adding a diagonal matrix to H) to the system of particles.
Joint work with Mark Kempton and S-T Yau