Establishing Robustness of a Spatial Dataset in a Tolerance-Based Vector Model

Spatial data are usually described through a vector model in which geometries are rep- resented by a set of coordinates embedded into an Euclidean space. The use of a finite representation, instead of the real numbers theoretically required, causes many robustness problems which are well-known in literature. Such problems are made even worst in a distributed context, where data is exchanged between different systems and several perturbations can be introduced in the data representation. In this context, a spatial dataset is said to be robust if the evaluation of the spatial relations existing among its objects can be performed in different systems, producing always the same result.In order to discuss the robustness of a spatial dataset, two implementation models have to be distinguished, since they determine different ways to evaluate the relations existing among geometric objects: the identity and the tolerance model. The robustness of a dataset in the identity model has been widely discussed in [Belussi et al., 2012, Belussi et al., 2013, Belussi et al., 2015a] and some algorithms of the Snap Rounding (SR) family [Hobby, 1999, Halperin and Packer, 2002, Packer, 2008, Belussi et al., 2015b] can be successfully applied in such context. Conversely, this problem has been less explored in the tolerance model. The aim of this paper is to propose an algorithm inspired by the ones of SR family for establishing or restoring the robustness of a vector dataset in the tolerance model. The main ideas are to introduce an additional operation which spreads instead of snapping geometries, in order to preserve the original relation between them, and to use a tolerance region for such operation instead of a single snapping location. Finally, some experiments on real-world datasets are presented, which confirms how the proposed algorithm can establish the robustness of a dataset

Generalized Toric Codes Coupled to Thermal Baths

We have studied the dynamics of a generalized toric code based on qudits at finite temperature by finding the master equation coupling the code's degrees of freedom to a thermal bath. As a consequence, we find that for qutrits new types of anyons and thermal processes appear that are forbidden for qubits. These include creation, annihilation and diffusion throughout the system code. It is possible to solve the master equation in a short-time regime and find expressions for the decay rates as a function of the dimension $d$ of the qudits. Although we provide an explicit proof that the system relax to the Gibbs state for arbitrary qudits, we also prove that above a certain crossing temperature, qutrits initial decay rate is smaller than the original case for qubits. Surprisingly this behavior only happens with qutrits and not with other qudits with $d>3$.Comment: Revtex4 file, color figures. New Journal of Physics' versio

Robustness of Spatial Relation Evaluation

In the last few years the amount of spatial data available through the network has increased both in volume and in heterogeneity, so that dealing with this huge amount of information has become an interesting new research challenge. In particular, spatial data is usually represented through a vector model upon which several spatial relations have been defined. Such relations represent the basic tools for querying spatial data and their robust evaluation in a distributed heterogeneous environment is an important issue to consider, in order to allow an effective usage of this kind of data. Among all possible spatial relations, this report considers the topological ones, since they are the most widely available in existing systems and represent the building blocks for the implementation of other spatial relations. The conditions and the operations needed to make a dataset robust w.r.t. topological interpretations strictly depends on the adopted evaluation model. In particular, this report considers an environment where two different eval- uation models for topological relations exist, one in which equality is based on identity of geometric primitives, and the other one where a tolerance in equality evaluation is introduced. Given such premises, the report proposes a set of rules for guaranteeing the robustness in both models, and discusses the applicability of available algorithms of the Snap Rounding family, in order to preserve robustness in case of perturbations

Symmetry protected topological order at nonzero temperature

We address the question of whether symmetry-protected topological (SPT) order can persist at nonzero temperature, with a focus on understanding the thermal stability of several models studied in the theory of quantum computation. We present three results in this direction. First, we prove that nontrivial SPT order protected by a global on-site symmetry cannot persist at nonzero temperature, demonstrating that several quantum computational structures protected by such on-site symmetries are not thermally stable. Second, we prove that the 3D cluster state model used in the formulation of topological measurement-based quantum computation possesses a nontrivial SPT-ordered thermal phase when protected by a global generalized (1-form) symmetry. The SPT order in this model is detected by long-range localizable entanglement in the thermal state, which compares with related results characterizing SPT order at zero temperature in spin chains using localizable entanglement as an order parameter. Our third result is to demonstrate that the high error tolerance of this 3D cluster state model for quantum computation, even without a protecting symmetry, can be understood as an application of quantum error correction to effectively enforce a 1-form symmetry.Comment: 42 pages, 10 figures, comments welcome; v2 published versio

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.Comment: 50 pages, many figures, v3: updated to match published versio