Explicit construction of exact unitary designs

Abstract

In this paper, we first provide an inductive construction of designs on compact groups G from those on subgroups K when (G,K) is a Gelfand pair. We then use the construction to provide explicit constructions of unitary t-designs in the unitary group U(d) for all t and d. To the best of our knowledge, 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 given polynomials. We will compare our constructions with previously known ones in the case of unitary 4-designs in U(4). We finally remark that this method can be applied to the orthogonal groups O(d) and thus provides another explicit construction of spherical t-design on the sphere Sd−1 in the d dimensional Euclidean space by the induction on d.