Dominic Williamson: Anyons and matrix product operator algebras

Опубликовано 1 февраля 2017, 1:08

Quantum tensor network states provide a natural framework for the representation of ground states of gapped, topologically ordered systems. From the technological point of view, such systems could be instrumental in creating fault tolerant architectures for quantum computation. From the theoretical point of view, such systems are fascinating due to the fact that all the relevant physics is encoded in the entanglement structure of the corresponding many body wavefunction. This is captured in the tensor network framework by a matrix product operator symmetry of the underlying tensors. In our work we present a systematic study of those matrix product operators, and show how this relates entanglement properties of projected entangled-pair states to the formalism of fusion tensor categories. From the matrix product operators we construct a C*-algebra and find that emergent topological superselection sectors can be identified with the central idempotents of this algebra. This allows us to construct projected entangled-pair states containing an arbitrary number of anyons. Physical properties such as topological spin, the S matrix, fusion and braiding relations are readily extracted from the idempotents. As the matrix product operator symmetries are acting purely on the virtual level of the tensor network, the ensuing Wilson loops are not fattened when perturbing the system. This opens up the possibility of simulating topological theories away from commuting projector fixed point Hamiltonians and studying topological phase transitions due to anyon condensation. We explicitly describe how discrete gauge theories and string-net models fit into the general formalism. Our approach leads to a new description of topological quantum computation where the relevant information is carried by virtual degrees of freedom in a tensor network, reminiscent of the PEPS description of measurement-based quantum computation.