Sum of squares lower bounds for refuting any CSP

Let $P:\{0,1\}^k \to \{0,1\}$ be a nontrivial $k$-ary predicate. Consider a random instance of the constraint satisfaction problem $\mathrm{CSP}(P)$ on $n$ variables with $\Delta n$ constraints, each being $P$ applied to $k$ randomly chosen literals. Provided the constraint density satisfies $\Delta \gg 1$, such an instance is unsatisfiable with high probability. The \emph{refutation} problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate $P$ supports a $t$-\emph{wise uniform} probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree $d = \Theta(\frac{n}{\Delta^{2/(t-1)} \log \Delta})$ (which runs in time $n^{O(d)}$) \emph{cannot} refute a random instance of $\mathrm{CSP}(P)$. In particular, the polynomial-time SOS algorithm requires $\widetilde{\Omega}(n^{(t+1)/2})$ constraints to refute random instances of CSP$(P)$ when $P$ supports a $t$-wise uniform distribution on its satisfying assignments. Together with recent work of Lee et al. [LRS15], our result also implies that \emph{any} polynomial-size semidefinite programming relaxation for refutation requires at least $\widetilde{\Omega}(n^{(t+1)/2})$ constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate~$P$, they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of Allen et al. [AOW15] and Raghavendra et al. [RRS16], this full three-way tradeoff is \emph{tight}, up to lower-order factors.Comment: 39 pages, 1 figur

Phase estimation by photon counting measurements in the output of a linear Mach Zehnder (MZI) interferometer

Photon counting measurements are analyzed for obtaining a classical phase parameter in linear Mach Zehnder interferometer (MZI), by the use of phase estimation theories. The detailed analysis is made for four cases: a) Coherent states inserted into the interferometer. b) Fock number state inserted in one input port of the interferometer and the vacuum into the other input port. c) Coherent state inserted into one input port of the interferometer and squeezed-vacuum state in the other input port. d) Exchanging the first beam-splitter (BS1) of a MZI by a non-linear system which inserts a NOON state into the interferometer and by using photon counting for parity measurements. The properties of photon counting for obtaining minimal phase uncertainties for the above special cases and for the general case are discussed.Comment: 27 page

Quantum fast Fourier transform and quantum computation by linear optics

Using the quantum fast Fourier transform in linear optics the input mode annihilation operators ͕â 0 , â 1 , . . . ,â s−1 ͖ are transformed into output mode annihilation operators ͕b 0 , b 1 , . . . ,b s−1 ͖. We show how to implement experimentally such transformations based on the Cooley-Tukey algorithm, by the use of beam splitters and phase shifters in a linear optical system. Optical systems implementing 1,2, and 3 qubits discrete Fourier transform (DFT) are described, and a general method for implementing the n-qubit DFT is analyzed. These transformations are used on various input radiation states by which phase estimation and order finding can be computed

Bacterially mediated removal of phosphorus and cycling of nitrate and sulfate in the waste stream of a "zero-discharge" recirculating mariculture system

Simultaneous removal of nitrogen and phosphorus by microbial biofilters has been used in a variety of water treatment systems including treatment systems in aquaculture. In this study, phosphorus, nitrate and sulfate cycling in the anaerobic loop of a zero-discharge, recirculating mariculture system was investigated using detailed geochemical measurements in the sludge layer of the digestion basin. High concentrations of nitrate and sulfate, circulating in the overlying water (~15 mM), were removed by microbial respiration in the sludge resulting in a sulfide accumulation of up to 3 mM. Modelling of the observed S and O isotopic ratios in the surface sludge suggested that, with time, major respiration processes shifted from heterotrophic nitrate and sulfate reduction to autotrophic nitrate reduction. The much higher inorganic P content of the sludge relative to the fish feces is attributed to conversion of organic P to authigenic apatite. This conclusion is supported by: (a) X-ray diffraction analyses, which pointed to an accumulation of a calcium phosphate mineral phase that was different from P phases found in the feces, (b) the calculation that the pore waters of the sludge were highly oversaturated with respect to hydroxyapatite (saturation index = 4.87) and (c) there was a decrease in phosphate (and in the Ca/Na molar ratio) in the pore waters simultaneous with an increase in ammonia showing there had to be an additional P removal process at the same time as the heterotrophic breakdown of organic matter

ClaPIM: Scalable Sequence CLAssification using Processing-In-Memory

DNA sequence classification is a fundamental task in computational biology with vast implications for applications such as disease prevention and drug design. Therefore, fast high-quality sequence classifiers are significantly important. This paper introduces ClaPIM, a scalable DNA sequence classification architecture based on the emerging concept of hybrid in-crossbar and near-crossbar memristive processing-in-memory (PIM). We enable efficient and high-quality classification by uniting the filter and search stages within a single algorithm. Specifically, we propose a custom filtering technique that drastically narrows the search space and a search approach that facilitates approximate string matching through a distance function. ClaPIM is the first PIM architecture for scalable approximate string matching that benefits from the high density of memristive crossbar arrays and the massive computational parallelism of PIM. Compared with Kraken2, a state-of-the-art software classifier, ClaPIM provides significantly higher classification quality (up to 20x improvement in F1 score) and also demonstrates a 1.8x throughput improvement. Compared with EDAM, a recently-proposed SRAM-based accelerator that is restricted to small datasets, we observe both a 30.4x improvement in normalized throughput per area and a 7% increase in classification precision

A synthetic biology approach for evaluating the functional contribution of designer cellulosome components to deconstruction of cellulosic substrates

BACKGROUND: Select cellulolytic bacteria produce multi-enzymatic cellulosome complexes that bind to the plant cell wall and catalyze its efficient degradation. The multi-modular interconnecting cellulosomal subunits comprise dockerin-containing enzymes that bind cohesively to cohesin-containing scaffoldins. The organization of the modules into functional polypeptides is achieved by intermodular linkers of different lengths and composition, which provide flexibility to the complex and determine its overall architecture. RESULTS: Using a synthetic biology approach, we systematically investigated the spatial organization of the scaffoldin subunit and its effect on cellulose hydrolysis by designing a combinatorial library of recombinant trivalent designer scaffoldins, which contain a carbohydrate-binding module (CBM) and 3 divergent cohesin modules. The positions of the individual modules were shuffled into 24 different arrangements of chimaeric scaffoldins. This basic set was further extended into three sub-sets for each arrangement with intermodular linkers ranging from zero (no linkers), 5 (short linkers) and native linkers of 27–35 amino acids (long linkers). Of the 72 possible scaffoldins, 56 were successfully cloned and 45 of them expressed, representing 14 full sets of chimaeric scaffoldins. The resultant 42-component scaffoldin library was used to assemble designer cellulosomes, comprising three model C. thermocellum cellulases. Activities were examined using Avicel as a pure microcrystalline cellulose substrate and pretreated cellulose-enriched wheat straw as a model substrate derived from a native source. All scaffoldin combinations yielded active trivalent designer cellulosome assemblies on both substrates that exceeded the levels of the free enzyme systems. A preferred modular arrangement for the trivalent designer scaffoldin was not observed for the three enzymes used in this study, indicating that they could be integrated at any position in the designer cellulosome without significant effect on cellulose-degrading activity. Designer cellulosomes assembled with the long-linker scaffoldins achieved higher levels of activity, compared to those assembled with short-and no-linker scaffoldins. CONCLUSIONS: The results demonstrate the robustness of the cellulosome system. Long intermodular scaffoldin linkers are preferable, thus leading to enhanced degradation of cellulosic substrates, presumably due to the increased flexibility and spatial positioning of the attached enzymes in the complex. These findings provide a general basis for improved designer cellulosome systems as a platform for bioethanol production

Symmetry Breaking and Order in the Age of Quasicrystals

The discovery of quasicrystals has changed our view of some of the most basic notions related to the condensed state of matter. Before the age of quasicrystals, it was believed that crystals break the continuous translation and rotation symmetries of the liquid-phase into a discrete lattice of translations, and a finite group of rotations. Quasicrystals, on the other hand, possess no such symmetries-there are no translations, nor, in general, are there any rotations, leaving them invariant. Does this imply that no symmetry is left, or that the meaning of symmetry should be revised? We review this and other questions related to the liquid-to-crystal symmetry-breaking transition using the notion of indistinguishability. We characterize the order-parameter space, describe the different elementary excitations, phonons and phasons, and discuss the nature of dislocations-keeping in mind that we are now living in the age of quasicrystals.Comment: To appear in a special issue on quasicrystals of The Israel Journal of Chemistry, in celebration of the 2011 Nobel Prize in Chemistr