Fault-ignorant Quantum Search

We investigate the problem of quantum searching on a noisy quantum computer. Taking a 'fault-ignorant' approach, we analyze quantum algorithms that solve the task for various different noise strengths, which are possibly unknown beforehand. We prove lower bounds on the runtime of such algorithms and thereby find that the quadratic speedup is necessarily lost (in our noise models). However, for low but constant noise levels the algorithms we provide (based on Grover's algorithm) still outperform the best noiseless classical search algorithm.Comment: v1: 15+8 pages, 4 figures; v2: 19+8 pages, 4 figures, published version (Introduction section significantly expanded, presentation clarified, results and order unchanged

Hilbert's projective metric in quantum information theory

We introduce and apply Hilbert's projective metric in the context of quantum information theory. The metric is induced by convex cones such as the sets of positive, separable or PPT operators. It provides bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under LOCC protocols or other cone-preserving operations. The results are formulated in terms of general cones and base norms and lead to contractivity bounds for quantum channels, for instance improving Ruskai's trace-norm contraction inequality. A new duality between distinguishability measures and base norms is provided. For two given pairs of quantum states we show that the contraction of Hilbert's projective metric is necessary and sufficient for the existence of a probabilistic quantum operation that maps one pair onto the other. Inequalities between Hilbert's projective metric and the Chernoff bound, the fidelity and various norms are proven.Comment: 32 pages including 3 appendices and 3 figures; v2: minor changes, published versio

Grand unification through gravitational effects

We systematically study the unification of gauge couplings in the presence of (one or more) effective dimension-5 operators cHGG/4MPl, induced into the grand unified theory by gravitational interactions at the Planck scale MPl. These operators alter the usual condition for gauge coupling unification, which can, depending on the Higgs content H and vacuum expectation value, result in unification at scales MX significantly different than naively expected. We find non-supersymmetric models of SU(5) and SO(10) unification, with natural Wilson coefficients c, that easily satisfy the constraints from proton decay. Furthermore, gauge coupling unification at scales as high as the Planck scale seems feasible, possibly hinting at simultaneous unification of gauge and gravitational interactions. In the Appendix we work out the group theoretical aspects of this scenario for SU(5) and SO(10) unified groups in detail; this material is also relevant in the analysis of non-universal gaugino masses obtained from supergravity.Comment: 27 pages, 5 figures, 8 tables, 1 appendix, revtex; v2: introduction and conclusion expanded, references added, minor changes, version published in PR

Phylogenetic analyses suggest reverse splicing spread of group I introns in fungal ribosomal DNA

BACKGROUND: Group I introns have spread into over 90 different sites in nuclear ribosomal DNA (rDNA) with greater than 1700 introns reported in these genes. These ribozymes generally spread through endonuclease-mediated intron homing. Another putative pathway is reverse splicing whereby a free group I intron inserts into a homologous or heterologous RNA through complementary base-pairing between the intron and exon RNA. Reverse-transcription of the RNA followed by general recombination results in intron spread. Here we used phylogenetics to test for reverse splicing spread in a taxonomically broadly sampled data set of fungal group I introns including 9 putatively ancient group I introns in the rDNA of the yeast-like symbiont Symbiotaphrina buchneri. RESULTS: Our analyses reveal a complex evolutionary history of the fungal introns with many cases of vertical inheritance (putatively for the 9 introns in S. buchneri) and intron lateral transfer. There are several examples in which introns, many of which are still present in S. buchneri, may have spread through reverse splicing into heterologous rDNA sites. If the S. buchneri introns are ancient as we postulate, then group I intron loss was widespread in fungal rDNA evolution. CONCLUSION: On the basis of these results, we suggest that the extensive distribution of fungal group I introns is at least partially explained by the reverse splicing movement of existing introns into ectopic rDNA sites

Monsters, black holes and the statistical mechanics of gravity

We review the construction of monsters in classical general relativity. Monsters have finite ADM mass and surface area, but potentially unbounded entropy. From the curved space perspective they are objects with large proper volume that can be glued on to an asymptotically flat space. At no point is the curvature or energy density required to be large in Planck units, and quantum gravitational effects are, in the conventional effective field theory framework, small everywhere. Since they can have more entropy than a black hole of equal mass, monsters are problematic for certain interpretations of black hole entropy and the AdS/CFT duality. In the second part of the paper we review recent developments in the foundations of statistical mechanics which make use of properties of high-dimensional (Hilbert) spaces. These results primarily depend on kinematics -- essentially, the geometry of Hilbert space -- and are relatively insensitive to dynamics. We discuss how this approach might be adopted as a basis for the statistical mechanics of gravity. Interestingly, monsters and other highly entropic configurations play an important role.Comment: 9 pages, 4 figures, revtex; invited Brief Review to be published in Modern Physics Letters