2014/07/31 by Jeongwan Haah
Mathematics · Physics and Astronomy · #Anyon #Combinatorics #Ground state #Hamiltonian (control theory) #Invariant (physics) #Mathematical Dynamics and Fractals #Mathematical physics #Mathematics #Matrix product state #Physics #Pure mathematics #Quantum #Quantum computer #Quantum entanglement #Quantum mechanics #Spectral Theory in Mathematical Physics #Topological quantum computer #Topology (electrical circuits) #Toric code #Unitary state #advanced mathematical theories #cond-mat.str-el #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1007/s00220-016-2594-y
published as Communications in Mathematical Physics, 342(3), 771-801 (2016) · revtex 11pt, 43 pages, (v2) minor change (v3) ref. added. To appear in Commun. Math. Phys
arxiv created 2015/12/30 · openalex publication_date 2016/02/22 · arxiv updated 2016/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 S-matrix from a single ground state wave function. Here, we define a class of Hamiltonians consisting of local commuting projectors and an associated matrix that is invariant under local unitary transformations. We argue that the invariant is equivalent to the topological S-matrix. The definition does not require degeneracy of the ground state. We prove that the invariant depends on the state only, in the sense that it can be computed by any Hamiltonian in the class of which the state is a ground state. As a corollary, we prove that any local quantum circuit that connects two ground states of quantum double models (discrete gauge theories) with non-isomorphic abelian groups, must have depth that is at least linear in the system's diameter. As a tool for the proof, a manifestly Hamiltonian-independent notion of locally invisible operators is introduced. This gives a sufficient condition for a many-body state not to be generated from a product state by any small depth quantum circuit; this is a many-body entanglement witness.