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Unitary designs and codes

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.

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Correspondence to Aidan Roy.

Additional information

Communicated by W. H. Haemers.

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Roy, A., Scott, A.J. Unitary designs and codes. Des. Codes Cryptogr. 53, 13–31 (2009). https://doi.org/10.1007/s10623-009-9290-2

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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