312 research outputs found
A Theory of Fault-Tolerant Quantum Computation
In order to use quantum error-correcting codes to actually improve the
performance of a quantum computer, it is necessary to be able to perform
operations fault-tolerantly on encoded states. I present a general theory of
fault-tolerant operations based on symmetries of the code stabilizer. This
allows a straightforward determination of which operations can be performed
fault-tolerantly on a given code. I demonstrate that fault-tolerant universal
computation is possible for any stabilizer code. I discuss a number of examples
in more detail, including the five-qubit code.Comment: 30 pages, REVTeX, universal swapping operation added to allow
universal computation on any stabilizer cod
Resilient Quantum Computation: Error Models and Thresholds
Recent research has demonstrated that quantum computers can solve certain
types of problems substantially faster than the known classical algorithms.
These problems include factoring integers and certain physics simulations.
Practical quantum computation requires overcoming the problems of environmental
noise and operational errors, problems which appear to be much more severe than
in classical computation due to the inherent fragility of quantum
superpositions involving many degrees of freedom. Here we show that arbitrarily
accurate quantum computations are possible provided that the error per
operation is below a threshold value. The result is obtained by combining
quantum error-correction, fault tolerant state recovery, fault tolerant
encoding of operations and concatenation. It holds under physically realistic
assumptions on the errors.Comment: 19 pages in RevTex, many figures, the paper is also avalaible at
http://qso.lanl.gov/qc
Compact Einstein-Weyl four-dimensional manifolds
We look for four dimensional Einstein-Weyl spaces equipped with a regular
Bianchi metric. Using the explicit 4-parameters expression of the distance
obtained in a previous work for non-conformally-Einstein Einstein-Weyl
structures, we show that only four 1-parameter families of regular metrics
exist on orientable manifolds : they are all of Bianchi type and
conformally K\"ahler ; moreover, in agreement with general results, they have a
positive definite conformal scalar curvature. In a Gauduchon's gauge, they are
compact and we obtain their topological invariants. Finally, we compare our
results to the general analyses of Madsen, Pedersen, Poon and Swann : our
simpler parametrisation allows us to correct some of their assertions.Comment: Latex file, 13 pages, an important reference added and a critical
discussion of its claims offered, others minor modification
Fault-Tolerant Error Correction with Efficient Quantum Codes
We exhibit a simple, systematic procedure for detecting and correcting errors
using any of the recently reported quantum error-correcting codes. The
procedure is shown explicitly for a code in which one qubit is mapped into
five. The quantum networks obtained are fault tolerant, that is, they can
function successfully even if errors occur during the error correction. Our
construction is derived using a recently introduced group-theoretic framework
for unifying all known quantum codes.Comment: 12 pages REVTeX, 1 ps figure included. Minor additions and revision
Quantum Convolutional Error Correcting Codes
I report two general methods to construct quantum convolutional codes for
-state quantum systems. Using these general methods, I construct a quantum
convolutional code of rate 1/4, which can correct one quantum error for every
eight consecutive quantum registers.Comment: Minor revisions and clarifications. To appear in Phys. Rev.
Active stabilisation, quantum computation and quantum state synthesis
Active stabilisation of a quantum system is the active suppression of noise
(such as decoherence) in the system, without disrupting its unitary evolution.
Quantum error correction suggests the possibility of achieving this, but only
if the recovery network can suppress more noise than it introduces. A general
method of constructing such networks is proposed, which gives a substantial
improvement over previous fault tolerant designs. The construction permits
quantum error correction to be understood as essentially quantum state
synthesis. An approximate analysis implies that algorithms involving very many
computational steps on a quantum computer can thus be made possible.Comment: 8 pages LaTeX plus 4 figures. Submitted to Phys. Rev. Let
Weighing matrices and spherical codes
Mutually unbiased weighing matrices (MUWM) are closely related to an
antipodal spherical code with 4 angles. In the present paper, we clarify the
relationship between MUWM and the spherical sets, and give the complete
solution about the maximum size of a set of MUWM of weight 4 for any order.
Moreover we describe some natural generalization of a set of MUWM from the
viewpoint of spherical codes, and determine several maximum sizes of the
generalized sets. They include an affirmative answer of the problem of Best,
Kharaghani, and Ramp.Comment: Title is changed from "Association schemes related to weighing
matrices
The capacity of the noisy quantum channel
An upper limit is given to the amount of quantum information that can be
transmitted reliably down a noisy, decoherent quantum channel. A class of
quantum error-correcting codes is presented that allow the information
transmitted to attain this limit. The result is the quantum analog of Shannon's
bound and code for the noisy classical channel.Comment: 19 pages, Submitted to Science. Replaced give correct references to
work of Schumacher, to add a figure and an appendix, and to correct minor
mistake
Limits to error correction in quantum chaos
We study the correction of errors that have accumulated in an entangled state
of spins as a result of unknown local variations in the Zeeman energy (B) and
spin-spin interaction energy (J). A non-degenerate code with error rate kappa
can recover the original state with high fidelity within a time kappa^1/2 /
max(B,J) -- independent of the number of encoded qubits. Whether the
Hamiltonian is chaotic or not does not affect this time scale, but it does
affect the complexity of the error-correcting code.Comment: 4 pages including 1 figur
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