PIRSA:14090030

An invariant of topologically ordered states under local unitary transformations

Haah, J. (2014). An invariant of topologically ordered states under local unitary transformations. Perimeter Institute. https://pirsa.org/14090030

Haah, Jeongwan. An invariant of topologically ordered states under local unitary transformations. Perimeter Institute, Sep. 10, 2014, https://pirsa.org/14090030 

          @misc{ pirsa_PIRSA:14090030,
            doi = {10.48660/14090030},
            url = {https://pirsa.org/14090030},
            author = {Haah, Jeongwan},
            keywords = {Quantum Information},
            language = {en},
            title = {An invariant of topologically ordered states under local unitary transformations},
            publisher = {Perimeter Institute},
            year = {2014},
            month = {sep},
            note = {PIRSA:14090030 see, \url{https://pirsa.org}}
          }

Jeongwan Haah Massachusetts Institute of Technology (MIT) - Department of Physics

Talk numberPIRSA:14090030
DOI10.48660/14090030
Collection
Talk Type Scientific Series
Subject

Abstract

For an anyon model in two spatial dimensions described by a modular tensor category, the topological S-matrix encodes the mutual braiding statistics, the quantum dimensions, and the fusion rules of anyons. It is nontrivial whether one can compute the topological S-matrix from a single ground state wave function. In this talk, I will show that, for a class of Hamiltonians, it is possible to define the S-matrix regardless of the degeneracy of the ground state. The definition manifests invariance of the S-matrix under local unitary transformations (quantum circuits). The defined S-matrix depends only on the ground state, in the sense that it can be computed by any Hamiltonian in the class of which the state is a ground state. This property, together with the local unitary invariance implies that any quantum circuit that connects two ground states of distinct topological S-matrices must have depth that is at least linear in the diameter of the system. A higher dimensional analog is straightforward. [arXiv:1407.2926]