Explicit construction of exact unitary designs

TL;DR

This paper presents explicit constructions of unitary t-designs for all t and dimensions, using Gelfand pairs and representation theory, providing new methods for designing unitary groups and spheres.

Contribution

It introduces an inductive construction method for unitary t-designs using Gelfand pairs and representation theory, applicable to both unitary and orthogonal groups.

Findings

Explicit constructions for all t and d in unitary groups.

Application of Gelfand pairs to construct designs on compact groups.

Extension of methods to orthogonal groups and spheres.

Abstract

The purpose of this paper is to give explicit constructions of unitary $t$-designs in the unitary group $U(d)$ for all $t$ and $d$. It seems that the explicit constructions were so far known only for very special cases. Here explicit construction means that the entries of the unitary matrices are given by the values of elementary functions at the root of some given polynomials. We will discuss what are the best such unitary $4$-designs in $U(4)$ obtained by these methods. Indeed we give an inductive construction of designs on compact groups by using Gelfand pairs $(G,K)$. Note that $(U(n),U(m) \times U(n-m))$ is a Gelfand pair. By using the zonal spherical functions for $(G,K)$, we can construct designs on $G$ from designs on $K$. We remark that our proofs use the representation theory of compact groups crucially. We also remark that this method can be applied to the orthogonal groups $O(d)$, and thus provides another explicit construction of spherical $t$-designs on the $d$ dimensional sphere $S^{d-1}$ by the induction on $d$.