2008/09/22 by Aidan Roy, A. J. Scott, A.J. Scott · 1 citation
Decision Sciences · Engineering · Mathematics · Physics and Astronomy · #Circular ensemble #Code (set theory) #Combinatorics #Computer science #Discrete mathematics #Geometry #Group (periodic table) #Inner product space #Mathematical Approximation and Integration #Mathematical analysis #Mathematical physics #Mathematics #Optimal Experimental Design Methods #Physics #Product (mathematics) #Pure mathematics #Quantum mechanics #Special unitary group #TRACE (psycholinguistics) #Unit (ring theory) #Unitary group #Unitary matrix #Unitary state #Upper and lower bounds #graph theory and CDMA systems #math.CO #msc:05B30 #msc:41A55 #msc:81P15 #msc:94A20 #quant-ph
paper · pdf · doi:10.1007/s10623-009-9290-2
published as Des. Codes Cryptogr. 53, 13-31 (2009) · 25 pages, no figures
arxiv created 2008/09/22 · openalex publication_date 2009/04/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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.