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Unitary designs and codes

By Aidan Roy and A. J. Scott

Abstract

A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. We also introduce the notion of a unitary code — a subset of U(d) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct values — and give an upper bound for the size of a code of degree s in U(d) for any d and s. These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary t-design in U(d), and we catalogue some t-designs that arise from finite groups

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.311.9043
Provided by: CiteSeerX
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