Oliver Buerschaper

A hierarchy of topological tensor network states

We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the algebraic setting of finite-dimensional Hopf C*-algebras. At the top of the hierarchy we identify ground states of new topological lattice models extending Kitaevs quantum double models [Ann. Phys. 303, 2 (2003)10.1016/S0003-4916(02)00018-0]. For these states we exhibit the mechanism responsible for their non-zero topological entanglement entropy by constructing an entanglement renormalization flow. Furthermore, we argue that the hierarchy states are related to each other by the condensation of topological charges.

Explicit tensor network representation for the ground states of string-net models

We provide a simple expression for the ground states of arbitrary string-net models in the form of local tensor networks. These tensor networks encode the data of the fusion category underlying a string-net model and thus represent all doubled topological phases of matter in the infrared limit according to Levin and Wen [Phys. Rev. B 71, 045110 (2005)]. Furthermore, our construction highlights the importance of the fat lattice equivalence between lattice and continuum descriptions of string-net models.

Mapping Kitaev’s quantum double lattice models to Levin and Wen’s string-net models

We exhibit a mapping identifying Kitaevs quantum double lattice models explicitly as a subclass of Levin and Wens string net models via a completion of the local Hilbert spaces with auxiliary degrees of freedom. This identification allows to carry over to these string net models the representation-theoretic classification of the excitations in quantum double models, as well as define them in arbitrary lattices, and provides an illustration of the abstract notion of Morita equivalence. The possibility of generalising the map to broader classes of string nets is considered.

Electric–magnetic duality of lattice systems with topological order

Abstract We investigate the duality structure of quantum lattice systems with topological order, a collective order also appearing in fractional quantum Hall systems. We define electromagnetic (EM) duality for all of Kitaevʼs quantum double models based on discrete gauge theories with Abelian and non-Abelian groups, and identify its natural habitat as a new class of topological models based on Hopf algebras. We interpret these as extended string-net models, whereupon Levin and Wenʼs string-nets, which describe all intrinsic topological orders on the lattice with parity and time-reversal invariance, arise as magnetic and electric projections of the extended models. We conjecture that all string-net models can be extended in an analogous way, using more general algebraic and tensor-categorical structures, such that EM duality continues to hold. We also identify this EM duality with an invertible domain wall. Physical applications include topology measurements in the form of pairs of dual tensor networks.

Simulation of anyons with tensor network algorithms

Interacting systems of anyons pose a unique challenge to condensed-matter simulations due to their nontrivial exchange statistics. These systems are of great interest as they have the potential for robust universal quantum computation but numerical tools for studying them are as yet limited. We show how existing tensor network algorithms may be adapted for use with systems of anyons and demonstrate this process for the one-dimensional multiscale entanglement renormalization ansatz (MERA). We apply the MERA to infinite chains of interacting Fibonacci anyons, computing their scaling dimensions and local scaling operators. The scaling dimensions obtained are seen to be in agreement with conformal field theory. The techniques developed are applicable to any tensor network algorithm, and the ability to adapt these ansatze for use on anyonic systems opens the door for numerical simulation of large systems of free and interacting anyons in one and two dimensions.

Protected gates for topological quantum field theories

We study restrictions on locality-preserving unitary logical gates for topological quantum codes in two spatial dimensions. A locality-preserving operation is one which maps local operators to local operators --- for example, a constant-depth quantum circuit of geometrically local gates, or evolution for a constant time governed by a geometrically-local bounded-strength Hamiltonian. Locality-preserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized, and hence sufficiently dilute errors remain correctable. By invoking general properties of two-dimensional topological field theories, we find that the locality-preserving logical gates are severely limited for codes which admit non-abelian anyons; in particular, there are no locality-preserving logical gates on the torus or the sphere with M punctures if the braiding of anyons is computationally universal. Furthermore, for Ising anyons on the M-punctured sphere, locality-preserving gates must be elements of the logical Pauli group. We derive these results by relating logical gates of a topological code to automorphisms of the Verlinde algebra of the corresponding anyon model, and by requiring the logical gates to be compatible with basis changes in the logical Hilbert space arising from local F-moves and the mapping class group.

Topological Transitions from Multipartite Entanglement with Tensor Networks: A Procedure for Sharper and Faster Characterization

Topological order in two-dimensional (2D) quantum matter can be determined by the topological contribution to the entanglement Rényi entropies. However, when close to a quantum phase transition, its calculation becomes cumbersome. Here, we show how topological phase transitions in 2D systems can be much better assessed by multipartite entanglement, as measured by the topological geometric entanglement of blocks. Specifically, we present an efficient tensor network algorithm based on projected entangled pair states to compute this quantity for a torus partitioned into cylinders and then use this method to find sharp evidence of topological phase transitions in 2D systems with a string-tension perturbation. When compared to tensor network methods for Rényi entropies, our approach produces almost perfect accuracies close to criticality and, additionally, is orders of magnitude faster. The method can be adapted to deal with any topological state of the system, including minimally entangled ground states. It also allows us to extract the critical exponent of the correlation length and shows that there is no continuous entanglement loss along renormalization group flows in topological phases.

A non-commuting stabilizer formalism

We propose a non-commutative extension of the Pauli stabilizer formalism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our framework, stabilizer operators are tensor products of single-qubit operators drawn from the group 〈αI, X, S〉, where α = eiπ/4 and S = diag(1, i). We provide techniques to efficiently compute various properties related to bipartite entanglement, expectation values of local observables, preparation by means of quantum circuits, parent Hamiltonians, etc. We also highlight significant differences compared to the Pauli stabilizer formalism. In particular, we give examples of states in our formalism which cannot arise in the Pauli stabilizer formalism, such as topological models that support non-Abelian anyons.

Translation invariance, topology, and protection of criticality in chains of interacting anyons

Using finite-size scaling arguments, the critical properties of a chain of interacting anyons can be extracted from the low-energy spectrum of a finite system. Feiguin showed that an antiferromagnetic chain of Fibonacci anyons on a torus is in the same universality class as the tricritical Ising model and that criticality is protected by a topological symmetry. In the present paper we first review the graphical formalism for the study of anyons on the disk and demonstrate how this formalism may be consistently extended to the study of systems on surfaces of higher genus. We then employ this graphical formalism to study finite rings of interacting anyons on both the disk and the torus and show that analysis on the disk necessarily yields an energy spectrum which is a subset of that which is obtained on the torus. For a critical Hamiltonian, one may extract from this subset the scaling dimensions of the local scaling operators which respect the topological symmetry of the system. Related considerations are also shown to apply for open chains.

Geometric entanglement in topologically ordered states

Here we investigate the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. We do this for a variety of topologically ordered systems such as the toric code, double semion, colour code and quantum double models. As happens for the entanglement entropy, we find that for sufficiently large block sizes the geometric entanglement is, up to possible sub-leading corrections, the sum of two contributions: a bulk contribution obeying a boundary law times the number of blocks and a contribution quantifying the underlying pattern of long-range entanglement of the topologically ordered state. This topological contribution is also present in the case of single-spin blocks in most cases, and constitutes an alternative characterization of topological order for these quantum states based on a multipartite entanglement measure. In particular, we see that the topological term for the two-dimensional colour code is twice as much as the one for the toric code, in accordance with recent renormalization group arguments (Bombin et al 2012 New J. Phys. 14 073048). Motivated by these results, we also derive a general formalism to obtain upper- and lower-bounds to the geometric entanglement of states with a non-Abelian group symmetry, and which we explicitly use to analyse quantum double models. Furthermore, we also provide an analysis of the robustness of the topological contribution in terms of renormalization and perturbation theory arguments, as well as a numerical estimation for small systems. Some of the results in this paper rely on the ability to disentangle single sites from the quantum state, which is always possible for the systems that we consider. Additionally we relate our results to the behaviour of the relative entropy of entanglement in topologically ordered systems, and discuss a number of numerical approaches based on tensor networks that could be employed to extract this topological contribution for large systems beyond exactly solvable models.