Related papers: Homological Stabilizer Codes
Locality-preserving logical operators in topological codes are naturally fault-tolerant, since they preserve the correctability of local errors. Using a correspondence between such operators and gapped domain walls, we describe a procedure…
Topological subsystem codes were proposed by Bombin based on 3-face-colorable cubic graphs. Suchara, Bravyi and Terhal generalized this construction and proposed a method to construct topological subsystem codes using 3-valent hypergraphs…
We study the resources needed to construct topological 2D stabilizer codes as a way to estimate in part their efficiency and this leads us to perform a comparative study of surface codes and color codes. This study clarifies the…
Graphs and recently hypergraphs have been known as an important tool for considering different properties of quantum many-body systems. In this paper, we study a mapping between an important class of quantum systems namely quantum…
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…
Graphs are closely related to quantum error-correcting codes: every stabilizer code is locally equivalent to a graph code, and every codeword stabilized code can be described by a graph and a classical code. For the construction of good…
Deciding if a given family of quantum states is topologically ordered is an important but nontrivial problem in condensed matter physics and quantum information theory. We derive necessary and sufficient conditions for a family of graph…
The topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. Here we show that the color code on a $d$-dimensional closed manifold is equivalent to multiple decoupled copies of…
Lattice-based cryptography is not only for thwarting future quantum computers, and is also the basis of Fully Homomorphic Encryption. Motivated from the advantage of graph homomorphisms we combine graph homomorphisms with graph total…
Topological Coding consists of two different kinds of mathematics: topological structure and mathematical relation. The colorings and labelings of graph theory are main techniques in topological coding applied in asymmetric encryption…
We consider the problem of a generic stabilizer Hamiltonian under local, incoherent Pauli errors. Using two different approaches -- (i) Haah's polynomial formalism arXiv:1204.1063 and (ii) the homological perspective on CSS codes -- we…
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 show that any stabilizer code over a finite field is equivalent to a graphical quantum code. Furthermore we prove that a graphical quantum code over a finite field is a stabilizer code. The technique used in the proof establishes a new…
While stabilizer tableaus have proven useful as a descriptive tool for additive quantum codes, they otherwise offer little guidance for concrete constructions or algorithm analysis. We introduce a representation of stabilizer codes as…
Spatially-coupled (SC) codes is a class of convolutional LDPC codes that has been well investigated in classical coding theory thanks to their high performance and compatibility with low-latency decoders. We describe toric codes as quantum…
We generalize the proof of stability of topological order, due to Bravyi, Hastings and Michalakis, to stabilizer Hamiltonians corresponding to low-density parity check (LDPC) codes without the restriction of geometric locality in Euclidean…
The discovery of holographic codes established a surprising connection between quantum error correction and the anti-de Sitter-conformal field theory correspondence. Recent technological progress in artificial quantum systems renders the…
Twist defects in surface codes can be used to encode more logical qubits, improve the code rate, and implement logical gates. In this work we provide a rigorous formalism for constructing surface codes with twists generalizing the…
The color code is a topological quantum error-correcting code supporting a variety of valuable fault-tolerant logical gates. Its two-dimensional version, the triangular color code, may soon be realized with currently available…
We construct a family of two-dimensional topological stabilizer codes on continuous variable (CV) degrees of freedom, which generalize homological rotor codes and the toric-GKP code. Our topological codes are built using the concept of…