Related papers: A graph-based formalism for surface codes and twis…
In this article, we define homological quantum codes in arbitrary qudit dimensions $D\geq 2$ by directly defining CSS operators on a 2-Complex $\Sigma$. If the 2-Complex is constructed from a surface, we obtain a qudit surface code. We then…
Twists are defects in the lattice that can be used to perform encoded computations. Three basic types of twists can be introduced in color codes, namely, twists that permute color, charge of anyons and domino twists that permute the charge…
We introduce in this article a new method to estimate the minimum distance of codes from algebraic surfaces. This lower bound is generic, i.e. can be applied to any surface, and turns out to be ``liftable'' under finite morphisms, paving…
This thesis captures the ongoing development of twisted cubes, which is a modification of cubes (in a topological sense) where its homotopy type theory does not require paths or higher paths to be invertible. My original motivation to…
The purpose of this paper is twofold: 1. we prove the triangulability of smooth orbifolds with corners, generalizing the same statement for orbifolds. 2. based on 1, we propose a new homology theory. We call it geometric homology theory…
Surface codes describe quantum memory stored as a global property of interacting spins on a surface. The state space is fixed by a complete set of quasi-local stabilizer operators and the code dimension depends on the first homology group…
Quantum error correction is a crucial technology for fault tolerant quantum computing. On superconducting platforms, hardware defects in large scale quantum processors can disrupt the regular lattice structure of topological codes and…
Graphs are commonly used to encode relationships among entities, yet their abstractness makes them difficult to analyze. Node-link diagrams are popular for drawing graphs, and force-directed layouts provide a flexible method for node…
We present a general framework for constructing high rate error correcting codes that are locally correctable (and hence locally decodable if linear) with a sublinear number of queries, based on lifting codes with respect to functions on…
In recent years, surface codes have become a leading method for quantum error correction in theoretical large scale computational and communications architecture designs. Their comparatively high fault-tolerant thresholds and their natural…
Topological subsystem codes proposed recently by Bombin are quantum error correcting codes defined on a two-dimensional grid of qubits that permit reliable quantum information storage with a constant error threshold. These codes require…
A remarkable characteristic of quantum computing is the potential for reliable computation despite faulty qubits. This can be achieved through quantum error correction, which is typically implemented by repeatedly applying static syndrome…
Geometric modeling by constraints, whose applications are of interest to communities from various fields such as mechanical engineering, computer aided design, symbolic computation or molecular chemistry, is now integrated into standard…
Higher dimensional graphs can be used to colour two-dimensional geometric graphs. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colourable with 4 colours. The later goal is achieved if…
Simplicial surfaces describe the incidence relations between vertices, edges and faces of triangulated 2-dimensional manifolds in a purely combinatorial way. By considering only the incidences of edges and faces, simplicial surfaces are…
The surface code is a powerful quantum error correcting code that can be defined on a 2-D square lattice of qubits with only nearest neighbor interactions. Syndrome and data qubits form a checkerboard pattern. Information about errors is…
In this paper we define homological stabilizer codes which encompass codes such as Kitaev's toric code and the topological color codes. These codes are defined solely by the graphs they reside on. This feature allows us to use properties of…
The evaluation of graphs on 2-spheres is a central ingredient of the Turaev-Viro construction of three-dimensional topological field theories. In this article, we introduce a class of graphs, called extruded graphs, that is relevant for the…
We introduce graphcodes, a novel multi-scale summary of the topological properties of a dataset that is based on the well-established theory of persistent homology. Graphcodes handle datasets that are filtered along two real-valued scale…
A topology is defined on the mapping class group of a compact connected orientable surface. It is shown that a notion of "genericity" on subsets of the mapping class group arises from this definition. Many plausible results follow from this…