Hexagons and Correlators in the Fishnet Theory

We investigate the hexagon formalism in the planar 4d conformal fishnet theory. This theory arises from N=4 SYM by a deformation that preserves both conformal symmetry and integrability. Based on this relation, we obtain the hexagon form factors for a large class of states, including the BMN vacuum, some excited states, and the Lagrangian density. We apply these form factors to the computation of several correlators and match the results with direct Feynman diagrammatic calculations. We also study the renormalisation of the hexagon form factor expansion for a family of diagonal structure constants and test the procedure at higher orders through comparison with a known universal formula for the Lagrangian insertion.Comment: 63 page

Performance and limitations of the QAOA at constant levels on large sparse hypergraphs and spin glass models

The Quantum Approximate Optimization Algorithm (QAOA) is a general purpose quantum algorithm designed for combinatorial optimization. We analyze its expected performance and prove concentration properties at any constant level (number of layers) on ensembles of random combinatorial optimization problems in the infinite size limit. These ensembles include mixed spin models and Max-$q$-XORSAT on sparse random hypergraphs. To enable our analysis, we prove a generalization of the multinomial theorem which is a technical result of independent interest. We then show that the performance of the QAOA at constant levels for the pure $q$-spin model matches asymptotically the ones for Max-$q$-XORSAT on random sparse Erd\H{o}s-R\'{e}nyi hypergraphs and every large-girth regular hypergraph. Through this correspondence, we establish that the average-case value produced by the QAOA at constant levels is bounded away from optimality for pure $q$-spin models when $q\ge 4$ is even. This limitation gives a hardness of approximation result for quantum algorithms in a new regime where the whole graph is seen.Comment: 12+46 page

OPE for all Helicity Amplitudes

We extend the Operator Product Expansion (OPE) for scattering amplitudes in planar N=4 SYM to account for all possible helicities of the external states. This is done by constructing a simple map between helicity configurations and so-called charged pentagon transitions. These OPE building blocks are generalizations of the bosonic pentagons entering MHV amplitudes and they can be bootstrapped at finite coupling from the integrable dynamics of the color flux tube. A byproduct of our map is a simple realization of parity in the super Wilson loop picture.Comment: 30 page

Anomalous diffusion for neuronal growth on surfaces with controlled geometries

Geometrical cues are known to play a very important role in neuronal growth and the formation of neuronal networks. Here, we present a detailed analysis of axonal growth and dynamics for neuronal cells cultured on patterned polydimethylsiloxane surfaces. We use fluorescence microscopy to image neurons, quantify their dynamics, and demonstrate that the substrate geometrical patterns cause strong directional alignment of axons. We quantify axonal growth and report a general stochastic approach that quantitatively describes the motion of growth cones. The growth cone dynamics is described by Langevin and Fokker-Planck equations with both deterministic and stochastic contributions. We show that the deterministic terms contain both the angular and speed dependence of axonal growth, and that these two contributions can be separated. Growth alignment is determined by surface geometry, and it is quantified by the deterministic part of the Langevin equation. We combine experimental data with theoretical analysis to measure the key parameters of the growth cone motion: speed and angular distributions, correlation functions, diffusion coefficients, characteristics speeds and damping coefficients. We demonstrate that axonal dynamics displays a cross-over from Brownian motion (Ornstein-Uhlenbeck process) at earlier times to anomalous dynamics (superdiffusion) at later times. The superdiffusive regime is characterized by non-Gaussian speed distributions and power law dependence of the axonal mean square length and the velocity correlation functions. These results demonstrate the importance of geometrical cues in guiding axonal growth, and could lead to new methods for bioengineering novel substrates for controlling neuronal growth and regeneration.Comment: 27 pages, 13 figures. arXiv admin note: substantial text overlap with arXiv:1903.0133

Audiovisual integration increases the intentional step synchronization of side-by-side walkers

When people walk side-by-side, they often synchronize their steps. To achieve this, individuals might cross-modally match audiovisual signals from the movements of the partner and kinesthetic, cutaneous, visual and auditory signals from their own movements. Because signals from different sensory systems are processed with noise and asynchronously, the challenge of the CNS is to derive the best estimate based on this conflicting information. This is currently thought to be done by a mechanism operating as a Maximum Likelihood Estimator (MLE). The present work investigated whether audiovisual signals from the partner are integrated according to MLE in order to synchronize steps during walking. Three experiments were conducted in which the sensory cues from a walking partner were virtually simulated. In Experiment 1 seven participants were instructed to synchronize with human-sized Point Light Walkers and/or footstep sounds. Results revealed highest synchronization performance with auditory and audiovisual cues. This was quantified by the time to achieve synchronization and by synchronization variability. However, this auditory dominance effect might have been due to artifacts of the setup. Therefore, in Experiment 2 human-sized virtual mannequins were implemented. Also, audiovisual stimuli were rendered in real-time and thus were synchronous and co-localized. All four participants synchronized best with audiovisual cues. For three of the four participants results point toward their optimal integration consistent with the MLE model. Experiment 3 yielded performance decrements for all three participants when the cues were incongruent. Overall, these findings suggest that individuals might optimally integrate audiovisual cues to synchronize steps during side-by-side walking.info:eu-repo/semantics/publishedVersio

Parameter Setting in Quantum Approximate Optimization of Weighted Problems

Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate algorithm for solving combinatorial optimization problems on quantum computers. However, in many cases QAOA requires computationally intensive parameter optimization. The challenge of parameter optimization is particularly acute in the case of weighted problems, for which the eigenvalues of the phase operator are non-integer and the QAOA energy landscape is not periodic. In this work, we develop parameter setting heuristics for QAOA applied to a general class of weighted problems. First, we derive optimal parameters for QAOA with depth $p=1$ applied to the weighted MaxCut problem under different assumptions on the weights. In particular, we rigorously prove the conventional wisdom that in the average case the first local optimum near zero gives globally-optimal QAOA parameters. Second, for $p\geq 1$ we prove that the QAOA energy landscape for weighted MaxCut approaches that for the unweighted case under a simple rescaling of parameters. Therefore, we can use parameters previously obtained for unweighted MaxCut for weighted problems. Finally, we prove that for $p=1$ the QAOA objective sharply concentrates around its expectation, which means that our parameter setting rules hold with high probability for a random weighted instance. We numerically validate this approach on general weighted graphs and show that on average the QAOA energy with the proposed fixed parameters is only $1.1$ percentage points away from that with optimized parameters. Third, we propose a general heuristic rescaling scheme inspired by the analytical results for weighted MaxCut and demonstrate its effectiveness using QAOA with the XY Hamming-weight-preserving mixer applied to the portfolio optimization problem. Our heuristic improves the convergence of local optimizers, reducing the number of iterations by 7.2x on average

Analyzing Prospects for Quantum Advantage in Topological Data Analysis

Lloyd et al. were first to demonstrate the promise of quantum algorithms for computing Betti numbers, a way to characterize topological features of data sets. Here, we propose, analyze, and optimize an improved quantum algorithm for topological data analysis (TDA) with reduced scaling, including a method for preparing Dicke states based on inequality testing, a more efficient amplitude estimation algorithm using Kaiser windows, and an optimal implementation of eigenvalue projectors based on Chebyshev polynomials. We compile our approach to a fault-tolerant gate set and estimate constant factors in the Toffoli complexity. Our analysis reveals that super-quadratic quantum speedups are only possible for this problem when targeting a multiplicative error approximation and the Betti number grows asymptotically. Further, we propose a dequantization of the quantum TDA algorithm that shows that having exponentially large dimension and Betti number are necessary, but insufficient conditions, for super-polynomial advantage. We then introduce and analyze specific problem examples which have parameters in the regime where super-polynomial advantages may be achieved, and argue that quantum circuits with tens of billions of Toffoli gates can solve seemingly classically intractable instances.Comment: 54 pages, 7 figures. Added a number of theorems and lemmas to clarify findings and also a discussion in the main text and new appendix about variants of our problems with high Betti numbers that are challenging for recent classical algorithm

Effect of MTA-based sealer on the healing of periapical lesions

Some manufacturers have recently added specific components to improve the ease of handling and insertion material properties of MTA in order to create MTA-based sealers. Objective The aim of this study was to evaluate the healing of periapical lesions in canine teeth after a single session of endodontic treatment with MTA Fillapex® compared with Sealapex® or Endo-CPM-Sealer®. Material and Methods Sixty-two root canals were performed on two 1-year-old male dogs. After coronal access and pulp extirpation, the canals were exposed to the oral cavity for 6 months in order to induce periapical lesions. The root canals were prepared, irrigated with a solution of 2.5% sodium hypochlorite and filled with gutta-percha and different sealers, according to the following groups: 1) Sealapex®; 2) Endo-CPM-Sealer®; and 3) MTA Fillapex®. Some teeth with periapical lesions were left untreated for use as positive controls. Healthy teeth were used as negative controls. After 6 months, the animals were sacrificed and serial sections from the roots were prepared for histomorphologic analysis and stained with hematoxylin and eosin and the Brown and Brenn technique. The lesions were scored according to pre-established histomorphologic parameters and the scores statistically analyzed using the Kruskal-Wallis test. Results All 3 materials produced similar patterns of healing (p>;0.05); in particular, persistent inflammation and absence of complete periapical tissue healing were consistently noted. Conclusions Preparation of the infected root canals followed by filling with the materials studied was insufficient to provide complete healing of the periapical tissues