Designs, Codes and Cryptography

, Volume 53, Issue 1, pp 13–31 | Cite as

Unitary designs and codes

Article

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 absolute values—and give an upper bound for the size of a code with s inner product values 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.

Keywords

Unitary design Unitary code Unitary group Rank bound Linear programming bound Delsarte bound Spherical design Spherical code Quantum process tomography Zonal polynomial 

Mathematics Subject Classification (2000)

05B30 41A55 81P15 94A20 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Institute for Quantum Information ScienceUniversity of CalgaryCalgaryCanada
  2. 2.Centre for Quantum Dynamics, Centre for Quantum Computer TechnologyGriffith UniversityBrisbaneAustralia

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