Structure of exact and approximate unitary t-designs

When studying “random” operators it is essential to be able to integrate over the Haar measure, both analytically and algorithmically. Unitary t-designs provide a method to simplify integrating polynomials of degree less than t over U(d). In particular, by replacing averages over the Haar measure by averages over a finite set, they allow applications in algorithms. We provide three equivalent definitions for unitary t-designs and introduce group and approximate designs. The main tool in this note is our generalization of an important result the trace double sum inequality into the trace 2p-sum inequality. We use the trace double sum inequality to produce a correspondence between minimal designs and unique minimal weight functions. We culminate our exploration of the structure of t-designs by showing that t-designs span {U⊗t|U ∈ U(d)}. This result produces two conjectures which we believe are an important step in the classification of minimum unitary t-designs.