Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure

Two types of Gaussian processes, namely the Gaussian field with generalized Cauchy covariance (GFGCC) and the Gaussian sheet with generalized Cauchy covariance (GSGCC) are considered. Some of the basic properties and the asymptotic properties of the spectral densities of these random fields are studied. The associated self-similar random fields obtained by applying the Lamperti transformation to GFGCC and GSGCC are studied.Comment: 32 pages, 6 figure

Optimized numerical gradient and Hessian estimation for variational quantum algorithms

Sampling noisy intermediate-scale quantum devices is a fundamental step that converts coherent quantum-circuit outputs to measurement data for running variational quantum algorithms that utilize gradient and Hessian methods in cost-function optimization tasks. This step, however, introduces estimation errors in the resulting gradient or Hessian computations. To minimize these errors, we discuss tunable numerical estimators, which are the finite-difference (including their generalized versions) and scaled parameter-shift estimators [introduced in Phys. Rev. A 103, 012405 (2021)], and propose operational circuit-averaged methods to optimize them. We show that these optimized numerical estimators offer estimation errors that drop exponentially with the number of circuit qubits for a given sampling-copy number, revealing a direct compatibility with the barren-plateau phenomenon. In particular, there exists a critical sampling-copy number below which an optimized difference estimator gives a smaller average estimation error in contrast to the standard (analytical) parameter-shift estimator, which exactly computes gradient and Hessian components. Moreover, this critical number grows exponentially with the circuit-qubit number. Finally, by forsaking analyticity, we demonstrate that the scaled parameter-shift estimators beat the standard unscaled ones in estimation accuracy under any situation, with comparable performances to those of the difference estimators within significant copy-number ranges, and are the best ones if larger copy numbers are affordable.Comment: 24 pages, 7 figures (updated Fig. 4, new Fig. 6, new Secs. IV C, V C, VII and Appendix C5 since last version

Effective response theory for zero energy Majorana bound states in three spatial dimensions

We propose a gravitational response theory for point defects (hedgehogs) binding Majorana zero modes in (3+1)-dimensional superconductors. Starting in 4+1 dimensions, where the point defect is extended into a line, a coupling of the bulk defect texture with the gravitational field is introduced. Diffeomorphism invariance then leads to an $SU(2)_2$ Kac-Moody current running along the defect line. The $SU(2)_2$ Kac-Moody algebra accounts for the non-Abelian nature of the zero modes in 3+1 dimensions. It is then shown to also encode the angular momentum density which permeates throughout the bulk between hedgehog-anti-hedgehog pairs.Comment: 7 pages, 3 figure

Wnt-Frizzled Signaling Regulates Activity-Mediated Synapse Formation

The formation of synapses is a tightly regulated process that requires the coordinated assembly of the presynaptic and postsynaptic sides. Defects in synaptogenesis during development or in the adult can lead to neurodevelopmental disorders, neurological disorders, and neurodegenerative diseases. In order to develop therapeutic approaches for these neurological conditions, we must first understand the molecular mechanisms that regulate synapse formation. The Wnt family of secreted glycoproteins are key regulators of synapse formation in different model systems from invertebrates to mammals. In this review, we will discuss the role of Wnt signaling in the formation of excitatory synapses in the mammalian brain by focusing on Wnt7a and Wnt5a, two Wnt ligands that play an in vivo role in this process. We will also discuss how changes in neuronal activity modulate the expression and/or release of Wnts, resulting in changes in the localization of surface levels of Frizzled, key Wnt receptors, at the synapse. Thus, changes in neuronal activity influence the magnitude of Wnt signaling, which in turn contributes to activity-mediated synapse formation

Optimal integral force feedback and structured PI tracking control : application for objective lens positioner

Peer reviewedPostprin