Unitary design: bounds on their size

          @misc{ pirsa_PIRSA:08100071,
            doi = {10.48660/08100071},
            url = {https://pirsa.org/08100071},
            author = {Scott, Andrew},
            keywords = {Quantum Information, Quantum Foundations},
            language = {en},
            title = {Unitary design: bounds on their size},
            publisher = {Perimeter Institute},
            year = {2008},
            month = {oct},
            note = {PIRSA:08100071 see, \url{https://pirsa.org}}
          }
          

Abstract

As a means of exactly derandomizing certain quantum information processing tasks, unitary designs have become an important concept in quantum information theory. A unitary design is a collection of unitary matrices that approximates the entire unitary group, much like a spherical design approximates the entire unit sphere. 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. The tightness of these bounds is then considered, where specific unitary 2-designs are introduced that are analogous to SIC-POVMs and complete sets of MUBs in the complex projective case. Additionally, we catalogue the known constructions of unitary t-designs and give an upper bound on the size of the smallest weighted unitary t-design in U(d). This is joint work with Aidan Roy (Calgary): \'Unitary designs and codes,\' arXiv:0809.3813.