Random Unitaries Give Quantum Expanders

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

References in corpus (3)

• Lieb-Schultz-Mattis in Higher Dimensions
• Entropy and Entanglement in Quantum Ground States
• Locality in Quantum and Markov Dynamics on Lattices and Networks

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