Random Unitaries Give Quantum Expanders
arXiv:0706.0556 · doi:10.1103/PhysRevA.76.032315
Abstract
We show that randomly choosing the matrices in a completely positive map from the unitary group gives a quantum expander. We consider Hermitian and non-Hermitian cases, and we provide asymptotically tight bounds in the Hermitian case on the typical value of the second largest eigenvalue. The key idea is the use of Schwinger-Dyson equations from lattice gauge theory to efficiently compute averages over the unitary group.
14 pages, 1 figure