Improved hardness results for the guided local Hamiltonian problem

Estimating the ground state energy of a local Hamiltonian is a central problem in quantum chemistry. In order to further investigate its complexity and the potential of quantum algorithms for quantum chemistry, Gharibian and Le Gall (STOC 2022) recently introduced the guided local Hamiltonian problem (GLH), which is a variant of the local Hamiltonian problem where an approximation of a ground state (which is called a guiding state) is given as an additional input. Gharibian and Le Gall showed quantum advantage (more precisely, BQP-completeness) for GLH with 6-local Hamiltonians when the guiding state has fidelity (inverse-polynomially) close to 1/2 with a ground state. In this paper, we optimally improve both the locality and the fidelity parameter: we show that the BQP-completeness persists even with 2-local Hamiltonians, and even when the guiding state has fidelity (inverse-polynomially) close to 1 with a ground state. Moreover, we show that the BQP-completeness also holds for 2-local physically motivated Hamiltonians on a 2D square lattice or a 2D triangular lattice. Beyond the hardness of estimating the ground state energy, we also show BQP-hardness persists when considering estimating energies of excited states of these Hamiltonians instead. Those make further steps towards establishing practical quantum advantage in quantum chemistry

Random Convex Hulls and Extreme Value Statistics

In this paper we study the statistical properties of convex hulls of $N$ random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy's formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of $n$ independent random walks. In the continuum time limit this reduces to $n$ independent planar Brownian trajectories for which we compute exactly, for all $n$, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting

Titan’s surface properties inferred from the seasonal brightness variation at 2-cm wavelength

International audienceA comprehensive calibration and mapping of the thermal microwave emission from Titan’s surface at 2.2-cm wavelength has been completed by the passive radiometer included in the Cassini RADAR instrument. A seasonal brightness temperature variation has been determined that is comparable to but slightly smaller than that obtained by Cassini's Composite Infrared Spectrometer (CIRS). This difference has implications for the composition and structure of Titan’s surface; namely, that most of Titan’s surface is covered by the deposition and possible redistribution of tholin-like atmospheric photochemical products to a depth of at least a meter

Titan's Landscapes Explained: Smothered, Covered, and Thin

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Composition, seasonal change and bathymetry of Ligeia Mare, Titan, derived from its 2.2-cm thermal emission

International audienceLigeia Mare is the second largest sea of Saturn's moon Titan. It is also the first extraterritorial sea for which a bathymetry profile was obtained. In this paper, we analyze all data acquired up to July 2013 in the passive mode of the RADAR (i.e., radiometry mode) on board the Cassini probe in order to constrain its composition, seasonal change and bathymetry

Structure and biosynthesis of two exopolysaccharides produced by lactobacillus johnsonii FI9785

Exopolysaccharides were isolated and purified from Lactobacillus johnsonii FI9785, which has previously been shown to act as a competitive exclusion agent to control Clostridium perfringens in poultry. Structural analysis by NMR spectroscopy revealed that L. johnsonii FI9785 can produce two types of exopolysaccharide: EPS-1 is a branched dextran with the unusual feature that every backbone residue is substituted with a 2-linked glucose unit, and EPS-2 was shown to have a repeating unit with the following structure: -6)-?-Glcp-(1-3)-?-Glcp-(1-5)-?-Galf-(1-6)-?-Glcp-(1-4) -?-Gal p-(1-4)-?-Glcp-(1-. Sites on both polysaccharides were partially occupied by substituent groups: 1-phosphoglycerol and O-acetyl groups in EPS-1 and a single O-acetyl group in EPS-2. Analysis of a deletion mutant (?epsE) lacking the putative priming glycosyltransferase gene located within a predicted eps gene cluster revealed that the mutant could produce EPS-1 but not EPS-2, indicating that epsE is essential for the biosynthesis of EPS-2. Atomic force microscopy confirmed the localization of galactose residues on the exterior of wild type cells and their absence in the ?epsE mutant. EPS2 was found to adopt a random coil structural conformation. Deletion of the entire 14-kb eps cluster resulted in an acapsular mutant phenotype that was not able to produce either EPS-2 or EPS-1. Alterations in the cell surface properties of the EPS-specific mutants were demonstrated by differences in binding of an anti-wild type L. johnsonii antibody. These findings provide insights into the biosynthesis and structures of novel exopolysaccharides produced by L. johnsonii FI9785, which are likely to play an important role in biofilm formation, protection against harsh environment of the gut, and colonization of the host. © 2013 by The American Society for Biochemistry and Molecular Biology, Inc