Related papers: Groupoid Toric Codes
In this paper, a construction of constant weight codes based on the unique decomposition of elements in lattices is presented. The conditions for unique primary decomposition and unique irreducible decomposition in lattices are discussed…
Is gauge symmetry merely a redundancy in our description, or does it carry a deeper information-theoretic significance? Quantum error-correcting codes (QECCs) show that redundancy can serve as a resource for protecting information against…
The stable torsion length in a group is the stable word length with respect to the set of all torsion elements. We show that the stable torsion length vanishes in crystallographic groups. We then give a linear programming algorithm to…
We consider quantum two-block group algebra (2BGA) codes, a previously unstudied family of smallest lifted-product (LP) codes. These codes are related to generalized-bicycle (GB) codes, except a cyclic group is replaced with an arbitrary…
We study a simple lattice model with local symmetry, whose construction is based on a crossed module of finite groups. Its dynamical degrees of freedom are associated both to links and faces of a four-dimensional lattice. In special limits…
The paper presents a comprehensive study of group codes from non-abelian split metacyclic group algebras. We derive an explicit Wedderburn-like decomposition of finite split metacyclic group algebras over fields with characteristic coprime…
We study the phase transition from two different topological phases to the ferromagnetic phase by focusing on points of the phase transition. To this end, we present a detailed mapping from such models to the Ising model in a transverse…
In this paper a wide family of identifying codes over regular Cayley graphs of degree four which are built over finite Abelian groups is presented. Some of the codes in this construction are also perfect. The graphs considered include some…
Let $X$ be a complete simplicial toric variety over a finite field with a split torus $T_X$. For any matrix $Q$, we are interested in the subgroup $Y_Q$ of $T_X$ parameterized by the columns of $Q$. We give an algorithm for obtaining a…
A general method for constructing convolutional codes from units in Laurent series over matrix rings is presented. Using group ring as matrix rings, this forms a basis for in-depth exploration of convolutional codes from group ring…
A group code structure of a linear code is a description of the code as one-sided or two-sided ideal of a group algebra of a finite group. In these realizations, the group algebra is identified with the ambient space, and the group elements…
Topologically ordered quantum spin systems have become an area of great interest, as they may provide a fault-tolerant means of quantum computation. One of the simplest examples of such a spin system is Kitaev's toric code. Naaijkens made…
This thesis addresses whether it is possible to build a robust memory device for quantum information. A three-dimensional gapped lattice spin model is found which demonstrates for the first time that a reliable quantum memory at finite…
We introduce a two-body quantum Hamiltonian model with spins-$\half$ located on the vertices of a 2D spatial lattice. The model exhibits an exact topological degeneracy in all coupling regimes. This is a remarkable non-perturbative effect.…
We present a generic framework for the emergence of the $\mathbb{Z}_2$ toric-code and the double-semion topological order in a wide class of hardcore Bose-Hubbard-type models governed by density-density interaction and in the…
The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same multigrading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like…
The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the…
We study an abstract group of reversible Turing machines. In our model, each machine is interpreted as a homeomorphism over a space which represents a tape filled with symbols and a head carrying a state. These homeomorphisms can only…
A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper we show how such gauge theories possess a higher-group global symmetry, which we study in detail. We derive the $d$-group…
The effect of decoherence on topological order (TO) has been most deeply understood for the toric code, the paragon of Abelian TOs. We show that certain non-Abelian TOs can be analyzed and understood to a similar degree, despite being…