Scalar--Flat Lorentzian Einstein--Weyl Spaces

We find all three-dimensional Einstein--Weyl spaces with the vanishing scalar curvatureComment: 4 page

Quantum Reed-Solomon Codes

After a brief introduction to both quantum computation and quantum error correction, we show how to construct quantum error-correcting codes based on classical BCH codes. With these codes, decoding can exploit additional information about the position of errors. This error model - the quantum erasure channel - is discussed. Finally, parameters of quantum BCH codes are provided.Comment: Summary only (2 pages), for the full version see: Proceedings Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC-13), Lecture Notes in Computer Science 1719, Springer, 199

An efficient quantum algorithm for the hidden subgroup problem in extraspecial groups

Extraspecial groups form a remarkable subclass of p-groups. They are also present in quantum information theory, in particular in quantum error correction. We give here a polynomial time quantum algorithm for finding hidden subgroups in extraspecial groups. Our approach is quite different from the recent algorithms presented in [17] and [2] for the Heisenberg group, the extraspecial p-group of size p3 and exponent p. Exploiting certain nice automorphisms of the extraspecial groups we define specific group actions which are used to reduce the problem to hidden subgroup instances in abelian groups that can be dealt with directly.Comment: 10 page

Self-dual Einstein Spaces, Heavenly Metrics and Twistors

Four-dimensional quaternion-Kahler metrics, or equivalently self-dual Einstein spaces M, are known to be encoded locally into one real function h subject to Przanowski's Heavenly equation. We elucidate the relation between this description and the usual twistor description for quaternion-Kahler spaces. In particular, we show that the same space M can be described by infinitely many different solutions h, associated to different complex (local) submanifolds on the twistor space, and therefore to different (local) integrable complex structures on M. We also study quaternion-Kahler deformations of M and, in the special case where M has a Killing vector field, show that the corresponding variations of h are related to eigenmodes of the conformal Laplacian on M. We exemplify our findings on the four-sphere S^4, the hyperbolic plane H^4 and on the "universal hypermultiplet", i.e. the hypermultiplet moduli space in type IIA string compactified on a rigid Calabi-Yau threefold.Comment: 44 pages, 1 figure; misprints correcte

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 $IX$ 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

String loop corrections to the universal hypermultiplet

We study loop corrections to the universal dilaton supermultiplet for type IIA strings compactified on Calabi-Yau threefolds. We show that the corresponding quaternionic kinetic terms receive non-trivial one-loop contributions proportional to the Euler number of the Calabi-Yau manifold, while the higher-loop corrections can be absorbed by field redefinitions. The corrected metric is no longer Kahler. Our analysis implies in particular that the Calabi-Yau volume is renormalized by loop effects which are present even in higher orders, while there are also one-loop corrections to the Bianchi identities for the NS and RR field strengths.Comment: 30 pages, harvmac, 1 figure. v2: minor typos corrected. Version to appear in Classical and Quantum Gravit

Quantum codewords contradict local realism

Quantum codewords are highly entangled combinations of two-state systems. The standard assumptions of local realism lead to logical contradictions similar to those found by Bell, Kochen and Specker, Greenberger, Horne and Zeilinger, and Mermin. The new contradictions have some noteworthy features that did not appear in the older ones.Comment: 9 pages LaTeX, 1 figur

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

Efficient fault-tolerant quantum computing

Fault tolerant quantum computing methods which work with efficient quantum error correcting codes are discussed. Several new techniques are introduced to restrict accumulation of errors before or during the recovery. Classes of eligible quantum codes are obtained, and good candidates exhibited. This permits a new analysis of the permissible error rates and minimum overheads for robust quantum computing. It is found that, under the standard noise model of ubiquitous stochastic, uncorrelated errors, a quantum computer need be only an order of magnitude larger than the logical machine contained within it in order to be reliable. For example, a scale-up by a factor of 22, with gate error rate of order $10^{-5}$, is sufficient to permit large quantum algorithms such as factorization of thousand-digit numbers.Comment: 21 pages plus 5 figures. Replaced with figures in new format to avoid problem

Prolongations of Geometric Overdetermined Systems

We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on the dimension of the solution space.Comment: 22 pages. In the second version, a comparison with the classical theory of prolongations was added. In this third version more details were added concerning our construction and especially the use of Kostant's computation of Lie algebra cohomolog