Realizing string-net condensation: Fibonacci anyon braiding for universal gates and sampling chromatic polynomials

Non-Abelian anyons are exotic quasiparticle excitations hosted by certain topological phases of matter. They break the fermion-boson dichotomy and obey non-Abelian braiding statistics: their interchanges yield unitary operations, rather than merely a phase factor, in a space spanned by topologically degenerate wavefunctions. They are the building blocks of topological quantum computing. However, experimental observation of non-Abelian anyons and their characterizing braiding ...

We examine the zero-temperature phase diagram of the two-dimensional Levin-Wen string-net model with Fibonacci anyons in the presence of competing interactions. Combining high-order series expansions around three exactly solvable points and exact diagonalizations, we find that the non-Abelian doubled Fibonacci topological phase is separated from two nontopological phases by different second-order quantum critical points, the positions of which are computed accurately. These t...

In three spatial dimensions, particles are limited to either bosonic or fermionic statistics. Two-dimensional systems, on the other hand, can support anyonic quasiparticles exhibiting richer statistical behaviours. An exciting proposal for quantum computation is to employ anyonic statistics to manipulate information. Since such statistical evolutions depend only on topological characteristics, the resulting computation is intrinsically resilient to errors. So-called non-Abeli...

We show that quantum systems of extended objects naturally give rise to a large class of exotic phases - namely topological phases. These phases occur when the extended objects, called ``string-nets'', become highly fluctuating and condense. We derive exactly soluble Hamiltonians for 2D local bosonic models whose ground states are string-net condensed states. Those ground states correspond to 2D parity invariant topological phases. These models reveal the mathematical framewo...

The topological model for quantum computation is an inherently fault-tolerant model built on anyons in topological phases of matter. A key role is played by the braid group, and in this survey we focus on a selection of ways that the mathematical study of braids is crucial for the theory. We provide some brief historical context as well, emphasizing ways that braiding appears in physical contexts. We also briefly discuss the 3-dimensional generalization of braiding: motions...

We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for its ground state and low-lying quasiparticle excitations to be in the DFib topological phase. Using the string...

String-net condensation can give rise to non-Abelian anyons whereas loop condensation usually gives rise to Abelian anyons. It has been proposed that generalized quantum loop gases with non-orthogonal inner products can produce non-Abelian anyons. We detail an exact mapping between the string-net and the generalized loop models and explain how the non-orthogonal products arise. We also introduce a loop model of double-stranded nets where quantum loops with an orthogonal inner...

An explicit lattice realization of a non-Abelian topological memory is presented. The correspondence between logical and physical states is seen directly by use of the stabilizer formalism. The resilience of the encoded states against errors is studied and compared to that of other memories. A set of non-topological operations are proposed to manipulate the encoded states, resulting in universal quantum computation. This work provides insight into the non-local encoding non-A...

We describe how to construct generalized string-net models, a class of exactly solvable lattice models that realize a large family of 2D topologically ordered phases of matter. The ground states of these models can be thought of as superpositions of different "string-net configurations", where each string-net configuration is a trivalent graph with labeled edges, drawn in the $xy$ plane. What makes this construction more general than the original string-net construction is th...

We remove the need to physically transport computational anyons around each other from the implementation of computational gates in topological quantum computing. By using an anyonic analog of quantum state teleportation, we show how the braiding transformations used to generate computational gates may be produced through a series of topological charge measurements.