Unitary designs and codes

TL;DR

This paper establishes bounds on the size of unitary t-designs and codes in the unitary group, introduces the concept of unitary codes, and provides constructions and catalogues of specific designs.

Contribution

It provides general lower and upper bounds for unitary t-designs and codes, introduces the notion of unitary codes, and offers new constructions and catalogues of designs from finite groups.

Findings

Derived a lower bound on the size of unitary t-designs.

Established an upper bound for the size of unitary codes.

Catalogued some t-designs arising from finite groups.

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.