Quantum nonexpander problem is quantum-Merlin-Arthur-complete

A quantum expander is a unital quantum channel that is rapidly mixing, has only a few Kraus operators, and can be implemented efficiently on a quantum computer. We consider the problem of estimating the mixing time (i.e., the spectral gap) of a quantum expander. We show that the problem of deciding whether a quantum channel is not rapidly mixing is a complete problem for the quantum Merlin-Arthur complexity class. This has applications to testing randomized constructions of quantum expanders and studying thermalization of open quantum systems.United States. Dept. of Energy (Cooperative Research Agreement DE-FG02-05ER41360)National Science Foundation (U.S.). Center for Science of Information (Grant CCF-0939370)National Science Foundation (U.S.) (CAREER Award CCF-0746600