Related papers: Quantum Error Correction with Goppa Codes from Max…
In this paper, we study some codes of algebraic geometry related to certain maximal curves. Quantum stabilizer codes obtained through the self orthogonality of Hermitian codes of this error correcting do not always have good parameters.…
In this paper, we examine algebraic geometric (AG) codes associated with curves generated by separated polynomials, and we create AG codes and quantum stabilizer codes from these curves by varying their parameters. Our research involves a…
This is an expository article aiming to introduce the reader to the underlying mathematics and geometry of quantum error correction. Information stored on quantum particles is subject to noise and interference from the environment. Quantum…
The following open problems, which concern a fundamental limit on coding properties of quantum codes with realistic physical constraints, are analyzed and partially answered here: (a) the upper bound on code distances of quantum…
Quantum error correction and the use of quantum error correction codes is likely to be essential for the realisation of practical quantum computing. Because the error models of quantum devices vary widely, quantum codes which are tailored…
Quantum algorithms have the potential to provide exponential speedups over some of the best known classical algorithms. These speedups may enable quantum devices to solve currently intractable problems such as those in the fields of…
A curve attaining the Hasse-Weil bound is called a maximal curve. Usually classical error-correcting codes obtained from a maximal curve have good parameters. However, the quantum stabilizer codes obtained from such classical…
We establish the connection between a recent new construction technique for quantum error correcting codes, based on graphs, and the so-called stabilizer codes: Each stabilizer code can be realized as a graph code and vice versa.
We investigate the performance of a quantum error-correcting code when pushed beyond its intended capacity to protect information against errors, presenting formulae for the probability of failure when the errors affect more qudits than…
Series of maximum distance quantum error-correcting codes are developed and analysed. For a given rate and given error-correction capability, quantum error-correcting codes with these specifications are constructed. The codes are explicit…
This Diplom thesis provides an explicit construction of a quantum Goppa code for any hyperelliptic curve over a non-binary field. Hyperelliptic curves have conjugate pairs of rational places. We use these pairs to construct self-orthogonal…
We introduce a purely graph-theoretical object, namely the coding clique, to construct quantum errorcorrecting codes. Almost all quantum codes constructed so far are stabilizer (additive) codes and the construction of nonadditive codes,…
We give a new construction of nonlinear error-correcting codes over suitable finite fields k from the geometry of modular curves with many rational points over k, combining two recent improvements on Goppa's construction. The resulting…
Controlling operational errors and decoherence is one of the major challenges facing the field of quantum computation and other attempts to create specified many-particle entangled states. The field of quantum error correction has developed…
Quantum error-correcting codes aim to protect information in quantum systems to enable fault-tolerant quantum computations. The most prevalent method, stabilizer codes, has been well developed for many varieties of systems, however, largely…
We investigate a novel class of quantum error correcting codes to correct errors on both qubits and higher-state quantum systems represented as qudits. These codes arise from an original graph-theoretic representation of sets of quantum…
We give an introduction to the theory of quantum error correction using stabilizer codes that is geared towards the working computer scientists and mathematicians with an interest in exploring this area. To this end, we begin with an…
We describe a quantum error correction scheme aimed at protecting a flow of quantum information over long distance communication. It is largely inspired by the theory of classical convolutional codes which are used in similar circumstances…
We introduce a new construction of error-correcting codes from algebraic curves over finite fields. Modular curves of genus g -> infty over a field of size q0^2 yield nonlinear codes more efficient than the linear Goppa codes obtained from…
The stabilizer code is the most general algebraic construction of quantum error-correcting codes proposed so far. A stabilizer code can be constructed from a self-orthogonal subspace of a symplectic space over a finite field. We propose a…