108 research outputs found
Covering radius and the chromatic number of Kneser graphs
AbstractLet C be a binary linear code with covering radius R and let C0 be a subcode of C with codimension i. We prove that the covering radius R0 of C satisfies R0 ⩽ 2R + 2i − 1, by setting up a graph coloring problem involving Kneser graphs
Quantum error-correcting codes associated with graphs
We present a construction scheme for quantum error correcting codes. The
basic ingredients are a graph and a finite abelian group, from which the code
can explicitly be obtained. We prove necessary and sufficient conditions for
the graph such that the resulting code corrects a certain number of errors.
This allows a simple verification of the 1-error correcting property of
fivefold codes in any dimension. As new examples we construct a large class of
codes saturating the singleton bound, as well as a tenfold code detecting 3
errors.Comment: 8 pages revtex, 5 figure
Efficient Computations of Encodings for Quantum Error Correction
We show how, given any set of generators of the stabilizer of a quantum code,
an efficient gate array that computes the codewords can be constructed. For an
n-qubit code whose stabilizer has d generators, the resulting gate array
consists of O(n d) operations, and converts k-qubit data (where k = n - d) into
n-qubit codewords.Comment: 16 pages, REVTeX, 3 figures within the tex
Overhead and noise threshold of fault-tolerant quantum error correction
Fault tolerant quantum error correction (QEC) networks are studied by a
combination of numerical and approximate analytical treatments. The probability
of failure of the recovery operation is calculated for a variety of CSS codes,
including large block codes and concatenated codes. Recent insights into the
syndrome extraction process, which render the whole process more efficient and
more noise-tolerant, are incorporated. The average number of recoveries which
can be completed without failure is thus estimated as a function of various
parameters. The main parameters are the gate (gamma) and memory (epsilon)
failure rates, the physical scale-up of the computer size, and the time t_m
required for measurements and classical processing. The achievable computation
size is given as a surface in parameter space. This indicates the noise
threshold as well as other information. It is found that concatenated codes
based on the [[23,1,7]] Golay code give higher thresholds than those based on
the [[7,1,3]] Hamming code under most conditions. The threshold gate noise
gamma_0 is a function of epsilon/gamma and t_m; example values are
{epsilon/gamma, t_m, gamma_0} = {1, 1, 0.001}, {0.01, 1, 0.003}, {1, 100,
0.0001}, {0.01, 100, 0.002}, assuming zero cost for information transport. This
represents an order of magnitude increase in tolerated memory noise, compared
with previous calculations, which is made possible by recent insights into the
fault-tolerant QEC process.Comment: 21 pages, 12 figures, minor mistakes corrected and layout improved,
ref added; v4: clarification of assumption re logic gate
Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with respect to the {I,H,N}^n Transform
We enumerate the inequivalent self-dual additive codes over GF(4) of
blocklength n, thereby extending the sequence A090899 in The On-Line
Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a
well-known interpretation as quantum codes. They can also be represented by
graphs, where a simple graph operation generates the orbits of equivalent
codes. We highlight the regularity and structure of some graphs that correspond
to codes with high distance. The codes can also be interpreted as quadratic
Boolean functions, where inequivalence takes on a spectral meaning. In this
context we define PAR_IHN, peak-to-average power ratio with respect to the
{I,H,N}^n transform set. We prove that PAR_IHN of a Boolean function is
equivalent to the the size of the maximum independent set over the associated
orbit of graphs. Finally we propose a construction technique to generate
Boolean functions with low PAR_IHN and algebraic degree higher than 2.Comment: Presented at Sequences and Their Applications, SETA'04, Seoul, South
Korea, October 2004. 17 pages, 10 figure
Achievable rates for the Gaussian quantum channel
We study the properties of quantum stabilizer codes that embed a
finite-dimensional protected code space in an infinite-dimensional Hilbert
space. The stabilizer group of such a code is associated with a symplectically
integral lattice in the phase space of 2N canonical variables. From the
existence of symplectically integral lattices with suitable properties, we
infer a lower bound on the quantum capacity of the Gaussian quantum channel
that matches the one-shot coherent information optimized over Gaussian input
states.Comment: 12 pages, 4 eps figures, REVTe
Protecting Quantum Information with Entanglement and Noisy Optical Modes
We incorporate active and passive quantum error-correcting techniques to
protect a set of optical information modes of a continuous-variable quantum
information system. Our method uses ancilla modes, entangled modes, and gauge
modes (modes in a mixed state) to help correct errors on a set of information
modes. A linear-optical encoding circuit consisting of offline squeezers,
passive optical devices, feedforward control, conditional modulation, and
homodyne measurements performs the encoding. The result is that we extend the
entanglement-assisted operator stabilizer formalism for discrete variables to
continuous-variable quantum information processing.Comment: 7 pages, 1 figur
Quantum Stabilizer Codes and Classical Linear Codes
We show that within any quantum stabilizer code there lurks a classical
binary linear code with similar error-correcting capabilities, thereby
demonstrating new connections between quantum codes and classical codes. Using
this result -- which applies to degenerate as well as nondegenerate codes --
previously established necessary conditions for classical linear codes can be
easily translated into necessary conditions for quantum stabilizer codes.
Examples of specific consequences are: for a quantum channel subject to a
delta-fraction of errors, the best asymptotic capacity attainable by any
stabilizer code cannot exceed H(1/2 + sqrt(2*delta*(1-2*delta))); and, for the
depolarizing channel with fidelity parameter delta, the best asymptotic
capacity attainable by any stabilizer code cannot exceed 1-H(delta).Comment: 17 pages, ReVTeX, with two figure
The twistor spinors of generic 2- and 3-distributions
Generic distributions on 5- and 6-manifolds give rise to conformal structures
that were discovered by P. Nurowski resp. R. Bryant. We describe both as
Fefferman-type constructions and show that for orientable distributions one
obtains conformal spin structures. The resulting conformal spin geometries are
then characterized by their conformal holonomy and equivalently by the
existence of a twistor spinor which satisfies a genericity condition. Moreover,
we show that given such a twistor spinor we can decompose a conformal Killing
field of the structure. We obtain explicit formulas relating conformal Killing
fields, almost Einstein structures and twistor spinors.Comment: 26 page
- …