The representation theory of the Clifford group, with applications to resource theories

          @misc{ scivideos_PIRSA:19110141,
            doi = {10.48660/19110141},
            url = {https://pirsa.org/19110141},
            author = {Gross, David},
            keywords = {Quantum Information},
            language = {en},
            title = {The representation theory of the Clifford group, with applications to resource theories},
            publisher = {Perimeter Institute for Theoretical Physics},
            year = {2019},
            month = {nov},
            note = {PIRSA:19110141 see, \url{https://scivideos.org/index.php/pirsa/19110141}}
          }
          

Abstract

I will report on an ongoing project to work out and exploit an analogue of Schur-Weyl duality for the Clifford group. Schur-Weyl establishes a one-one correspondence between irreps of the unitary group and those of the symmetric group. A similar program can be carried out for Cliffords. The permutations are then replaced by certain discrete orthogonal maps. As is the case for Schur-Weyl, this duality has many applications for problems in quantum information. It can be used, e.g., to derive quantum property tests for stabilizerness and Cliffordness, a new direct interpretation of the sum-negativity of Wigner functions, bounds on stabilizer rank, the construction of designs using few non-Clifford resources, etc. [arXiv:1609.08172, arXiv:1712.08628, arXiv:1906.07230, arXiv:out.soon].