Classical and quantum tensor product expanders

Authors:
M. B. Hastings;A. W. Harrow
Affiliations:
Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM;Department of Computer Science, University of Bristol, Bristol, U.K.
Venue:
Quantum Information & Computation
Year:
2009

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Abstract

We introduce the concept of quantum tensor product expanders. These generalize theconcept of quantum expanders, which are quantum maps that are efficient randomizersand use only a small number of Kraus operators. Quantum tensor product expandersact on several copies of a given system, where the Kraus operators are tensor products ofthe Kraus operator on a single system. We begin with the classical case, and show thata classical two-copy expander can be used to produce a quantum expander. We thendiscuss the quantum case and give applications to the Solovay-Kitaev problem. We giveprobabilistic constructions in both classical and quantum cases, giving tight bounds onthe expectation value of the largest nontrivial eigenvalue in the quantum case.