Properties of unitary t-designs
Artem Kaznatcheev , McGill University

Abstract:

Unitary t-designs provide a method to simplify integrating polynomials of degree less than t over U(d). Designs allows us to replace the average over U(d) (an integral) by an average over the design (a finite sum). We prove the trace double sum inequality and use it to provide a metric definition of designs. The new definition provides an easier way of checking in a set of unitaries forms a design. In the search for small designs, we classify minimal designs according to their weight function. As an alternative approach to deriving lower bounds on the size of t-designs, we introduce a greedy ‘algorithm’ for constructing designs. We provide a construction for minimum 1-designs based on orthonormal bases of the space of d-by-d matrices. The constructions provides a simple way to evaluate the average of U*V*UV for fixed V. This allows us to prove that t-designs are non-commuting sets, supporting our intuition that the elements of a design are well ‘spread out’.