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An invariant of topologically ordered states under local unitary transformations

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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]