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The construction of the topologically protected code space of Kitaev's model for fault-tolerant quantum computation is extended from complex semisimple to arbitrary finite-dimensional Hopf algebras admitting pairs in involution. One input…

We prove that Kitaev's lattice model for a finite-dimensional semisimple Hopf algebra H is equivalent to the combinatorial quantisation of Chern-Simons theory for the Drinfeld double D(H). This shows that Kitaev models are a special case of…

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…

Kitaev's quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this…

Kitaev's toric code is constructed using a finite gauge group from gauge theory. Such gauge theories can be generalized with the gauge group generalized to any finite-dimensional semisimple Hopf algebra. This also leads to generalizations…

We define an exactly solvable model for 2+1D topological phases of matter on a triangulated surface derived from a crossed module of semisimple finite-dimensional Hopf algebras, the `Hopf-algebraic higher Kitaev model'. This model…

Two topological phases are equivalent if they are connected by a local unitary transformation. In this sense, classifying topological phases amounts to classifying long-range entanglement patterns. We show that all 2D topological stabilizer…

We define and study parafermion stabilizer codes which can be viewed as generalizations of Kitaev's one dimensional model of unpaired Majorana fermions. Parafermion stabilizer codes can protect against low-weight errors acting on a small…

We construct a Kitaev model, consisting of a Hamiltonian which is the sum of commuting local projectors, for surfaces with boundaries and defects of dimension 0 and 1. More specifically, we show that one can consider cell decompositions of…

In this paper, we examine Kitaev's lattice model for an arbitrary complex, semisimple Hopf algebra. We prove that this model gives the same topological invariants as Turaev-Viro theory. Using the description of Turaev-Viro theory as an…

Local decoders provide a promising approach to real-time quantum error-correction by replacing centralized classical decoding, with significant hardware constraints, by a fully distributed architecture based on a simple, local update rule.…

We develop a framework for the classification of invertible translation-invariant stabilizer codes modulo condensation and stabilization with simple codes. We introduce generalizations of the Pauli groups of local unitaries for quantum…

We provide a detailed study of the general structure of two-dimensional topological stabilizer quantum error correcting codes, including subsystem codes. Under the sole assumption of translational invariance, we show that all such codes can…

We report the existence of \emph{flat bands} in a p-wave superconducting Kitaev ladder. We identify two sets of parameters for which the Kitaev ladder sustains flat bands. These flat bands are accompanied by highly localized eigenstates…

Kitaev's quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. Originally it is based on finite groups, and is later generalized to semi-simple Hopf algebras. We…

Since the long range entanglement is a universal characteristic of topological quantum states belonging to the same class, a suitable mathematical representation of the long range entanglement has to be also universal. In this Letter, we…

Using the completed inductive, projective and injective tensor products of Grothendieck for locally convex topological vector spaces, we develop a systematic theory of locally convex Hopf algebras with an emphasis on Pontryagin-type…

We study the Twisted Kitaev Quantum Double model within the framework of Local Topological Order (LTO). We extend its definition to arbitrary 2D lattices, enabling an explicit characterization of the ground state space through the invariant…

We present a solution of Kitaev's spin model on the honeycomb lattice and of related topologically ordered spin models. We employ a Jordan-Wigner type fermionization and find that the Hamiltonian takes a BCS type form, allowing the system…

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for…