Limitations of Quantum Coset States for Graph Isomorphism

It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup. An open question has been how hard it is to use these states to solve graph isomorphism. It was recently shown by Moore, Russell, and Schulman that only an exponentially small amount of information is available from one, or a pair of coset states. A potential source of power to exploit are entangled quantum measurements that act jointly on many states at once. We show that entangled quantum measurements on at least \Omega(n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound. This may be viewed as a negative result because highly entangled measurements seem hard to implement in general. Our main theorem is very general and also rules out using joint measurements on few coset states for some other groups, such as GL(n, F_{p^m}) and G^n where G is finite and satisfies a suitable property.Comment: 25 page

Computer Simulation of Particle Suspensions

Particle suspensions are ubiquitous in our daily life, but are not well understood due to their complexity. During the last twenty years, various simulation methods have been developed in order to model these systems. Due to varying properties of the solved particles and the solvents, one has to choose the simulation method properly in order to use the available compute resources most effectively with resolving the system as well as needed. Various techniques for the simulation of particle suspensions have been implemented at the Institute for Computational Physics allowing us to study the properties of clay-like systems, where Brownian motion is important, more macroscopic particles like glass spheres or fibers solved in liquids, or even the pneumatic transport of powders in pipes. In this paper we will present the various methods we applied and developed and discuss their individual advantages.Comment: 31 pages, 11 figures, to appear in Lecture Notes in Applied and Computational Mechanics, Springer (2006

A Convex Reconstruction Model for X-ray Tomographic Imaging with Uncertain Flat-fields

Classical methods for X-ray computed tomography are based on the assumption that the X-ray source intensity is known, but in practice, the intensity is measured and hence uncertain. Under normal operating conditions, when the exposure time is sufficiently high, this kind of uncertainty typically has a negligible effect on the reconstruction quality. However, in time- or dose-limited applications such as dynamic CT, this uncertainty may cause severe and systematic artifacts known as ring artifacts. By carefully modeling the measurement process and by taking uncertainties into account, we derive a new convex model that leads to improved reconstructions despite poor quality measurements. We demonstrate the effectiveness of the methodology based on simulated and real data sets.Comment: Accepted at IEEE Transactions on Computational Imagin

Ideal Projections and Forcing Projections

It is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman-Magidor-Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove: (1) If I is a normal ideal on ω2 which satisfies stationary antichain catching, then there is an inner model with a Woodin cardinal; (2) For any n ∈ ω, it is consistent relative to large cardinals that there is a normal ideal I on ωn which satisfies projective antichain catching, yet I is not saturated (or even strong). This provides a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory ([7])

Minisuperspace results for causal dynamical triangulations

Detailed applications of minisuperspace methods are presented and compared with results obtained in recent years by means of causal dynamical triangulations (CDTs), mainly in the form of effective actions. The analysis sheds light on conceptual questions such as the treatment of time or the role and scaling behavior of statistical and quantum fluctuations. In the case of fluctuations, several analytical and numerical results show agreement between the two approaches and offer possible explanations of effects that have been seen in causal dynamical triangulations but whose origin remained unclear. The new approach followed here suggests `CDT experiments' in the form of new simulations or evaluations motivated by theoretical predictions, testing CDTs as well as the minisuperspace approximation.Comment: 51 pages, 16 figure