29 research outputs found
Algebraic techniques in designing quantum synchronizable codes
Quantum synchronizable codes are quantum error-correcting codes that can
correct the effects of quantum noise as well as block synchronization errors.
We improve the previously known general framework for designing quantum
synchronizable codes through more extensive use of the theory of finite fields.
This makes it possible to widen the range of tolerable magnitude of block
synchronization errors while giving mathematical insight into the algebraic
mechanism of synchronization recovery. Also given are families of quantum
synchronizable codes based on punctured Reed-Muller codes and their ambient
spaces.Comment: 9 pages, no figures. The framework presented in this article
supersedes the one given in arXiv:1206.0260 by the first autho
Algebraic techniques in designing quantum synchronizable codes
Quantum synchronizable codes are quantum error-correcting codes that can correct the effects of quantum noise as well as block synchronization errors. We improve the known general framework for designing quantum synchronizable codes through more extensive use of the theory of finite fields. This makes it possible to widen the range of tolerable magnitude of block synchronization errors while giving mathematical insight into the algebraic mechanism of synchronization recovery. Also given are families of quantum synchronizable codes based on punctured Reed-Muller codes and their ambient spaces
Block synchronization for quantum information
Locating the boundaries of consecutive blocks of quantum information is a
fundamental building block for advanced quantum computation and quantum
communication systems. We develop a coding theoretic method for properly
locating boundaries of quantum information without relying on external
synchronization when block synchronization is lost. The method also protects
qubits from decoherence in a manner similar to conventional quantum
error-correcting codes, seamlessly achieving synchronization recovery and error
correction. A family of quantum codes that are simultaneously synchronizable
and error-correcting is given through this approach.Comment: 7 pages, no figures, final accepted version for publication in
Physical Review
Quantum Synchronizable Codes From Quadratic Residue Codes and Their Supercodes
Quantum synchronizable codes are quantum error-correcting codes designed to
correct the effects of both quantum noise and block synchronization errors.
While it is known that quantum synchronizable codes can be constructed from
cyclic codes that satisfy special properties, only a few classes of cyclic
codes have been proved to give promising quantum synchronizable codes. In this
paper, using quadratic residue codes and their supercodes, we give a simple
construction for quantum synchronizable codes whose synchronization
capabilities attain the upper bound. The method is applicable to cyclic codes
of prime length
Quantum Synchronizable Codes From Finite Geometries
Quantum synchronizable error-correcting codes are special quantum error-correcting codes that are designed to correct both the effect of quantum noise on qubits and misalignment in block synchronization. It is known that, in principle, such a code can be constructed through a combination of a classical linear code and its subcode if the two are both cyclic and dual-containing. However, finding such classical codes that lead to promising quantum synchronizable error-correcting codes is not a trivial task. In fact, although there are two families of classical codes that are proved to produce quantum synchronizable codes with good minimum distances and highest possible tolerance against misalignment, their code lengths have been restricted to primes and Mersenne numbers. In this paper, examining the incidence vectors of projective spaces over the finite fields of characteristic 2, we give quantum synchronizable codes from cyclic codes whose lengths are not primes or Mersenne numbers. These projective geometric codes achieve good performance in quantum error correction and possess the best possible ability to recover synchronization, thereby enriching the variety of good quantum synchronizable codes. We also extend the current knowledge of cyclic codes in classical coding theory by explicitly giving generator polynomials of the finite geometric codes and completely characterizing the minimum weight nonzero codewords. In addition to the codes based on projective spaces, we carry out a similar analysis on the well-known cyclic codes from Euclidean spaces that are known to be majority logic decodable and determine their exact minimum distances