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The prosperous development of both hardware and algorithms for quantum computing (QC) potentially prompts a paradigm shift in scientific computing in various fields. As an increasingly active topic in QC, the variational quantum algorithm (VQA) leads a promising direction for solving partial differential equations on Noisy Intermediate Scale Quantum (NISQ) devices. Although a clear perspective on the advantages of QC over classical computing techniques for specific mathematical and physical problems exists, applications of QC in computational fluid dynamics to solve practical flow problems, though promising, are still in an early stage of development. To explore QC in practical simulation of flow problems, this work applies a variational hybrid quantum-classical algorithm, namely the variational quantum linear solver (VQLS), to resolve the heat conduction equation through finite difference discretization of the Laplacian operator. Details of VQLS implementation are discussed by various test instances of linear systems. Finally, the successful statevector simulations of the heat conduction equation in one and two dimensions demonstrate the validity of the present algorithm by proof-of-concept results. In addition, the heuristic scaling for the heat conduction problem indicates that the time complexity of the present approach is logarithmically dependent on the precision {\epsilon} and linearly dependent on the number of qubits n., Comment: 36 pages, 19 figures

Bi-quadratic programming over unit spheres is a fundamental problem in quantum mechanics introduced by pioneer work of Einstein, Schr\"odinger, and others. It has been shown to be NP-hard; so it must be solve by efficient heuristic algorithms such as the block improvement method (BIM). This paper focuses on the maximization of bi-quadratic forms, which leads to a rank-one approximation problem that is equivalent to computing the M-spectral radius and its corresponding eigenvectors. Specifically, we provide a tight upper bound of the M-spectral radius for nonnegative fourth-order partially symmetric (PS) tensors, which can be considered as an approximation of the M-spectral radius. Furthermore, we showed that the proposed upper bound can be obtained more efficiently, if the nonnegative fourth-order PS-tensors is a member of certain monoid semigroups. Furthermore, as an extension of the proposed upper bound, we derive the exact solutions of the M-spectral radius and its corresponding M-eigenvectors for certain classes of fourth-order PS-tensors. Lastly, as an application of the proposed bound, we obtain a practically testable sufficient condition for nonsingular elasticity M-tensors with strong ellipticity condition. We conduct several numerical experiments to demonstrate the utility of the proposed results. The results show that: (a) our proposed method can attain a tight upper bound of the M-spectral radius with little computational burden, and (b) such tight and efficient upper bounds greatly enhance the convergence speed of the BIM-algorithm, allowing it to be applicable for large-scale problems in applications., Comment: 26 pages, 6 figures

We consider a natural class of long range random walks on torsion free nilpotent groups and develop limit theorems for these walks. Given the original discrete group $\Gamma$ and a random walk $(S_n)_ {n\ge1}$ driven by a certain type of symmetric probability measure $\mu$, we construct a homogeneous nilpotent Lie group $G_\bullet(\Gamma,\mu)$ which carries an adapted dilation structure and a stable-like process $(X_t)_{ t\ge0}$ which appears in a Donsker-type functional limit theorem as the limit of a rescaled version of the random walk. Both the limit group and the limit process on that group depend on the measure $\mu$. In addition, the functional limit theorem is complemented by a local limit theorem.

The transfer matrix formalism is widely used in modeling heat diffusion in layered structures.Due to an intrinsic numerical instability issue, which has not yet drawn enough attention to the heat transfer community,this formalism fails at high heating frequencies and/or in thick structures. Inspired by its success in modeling wave propagation, we develop a numerically-stable scattering matrix framework to model periodic heat diffusion in stratified solid media.As a concreate example, we apply this scattering matrix methodology to the three omega method.We first validate our framework using various well-known solutions.Next, we demonstrate the numerical stability of the framework using a configuration that resembles the three-dimensional stacked architecture for chip packing. Last, we propose synthetic experiments to exhibit, under certain circumstances, the merits of the scattering matrix formalism in extracting thermal properties.

Electron ptychography provides new opportunities to resolve atomic structures with deep sub-angstrom spatial resolution and studying electron-beam sensitive materials with high dose efficiency. In practice, obtaining accurate ptychography images requires simultaneously optimizing multiple parameters that are often selected based on trial-and-error, resulting in low-throughput experiments and preventing wider adoption. Here, we develop an automatic parameter selection framework to circumvent this problem using Bayesian optimization with Gaussian processes. With minimal prior knowledge, the workflow efficiently produces ptychographic reconstructions that are superior than the ones processed by experienced experts. The method also facilitates better experimental designs by exploring optimized experimental parameters from simulated data.

The Zero-Shot Sketch-based Image Retrieval (ZS-SBIR) is a challenging task because of the large domain gap between sketches and natural images as well as the semantic inconsistency between seen and unseen categories. Previous literature bridges seen and unseen categories by semantic embedding, which requires prior knowledge of the exact class names and additional extraction efforts. And most works reduce domain gap by mapping sketches and natural images into a common high-level space using constructed sketch-image pairs, which ignore the unpaired information between images and sketches. To address these issues, in this paper, we propose a novel Three-Stream Joint Training Network (3JOIN) for the ZS-SBIR task. To narrow the domain differences between sketches and images, we extract edge maps for natural images and treat them as a bridge between images and sketches, which have similar content to images and similar style to sketches. For exploiting a sufficient combination of sketches, natural images, and edge maps, a novel three-stream joint training network is proposed. In addition, we use a teacher network to extract the implicit semantics of the samples without the aid of other semantics and transfer the learned knowledge to unseen classes. Extensive experiments conducted on two real-world datasets demonstrate the superiority of our proposed method.

Recently, several Vision Transformer (ViT) based methods have been proposed for Fine-Grained Visual Classification (FGVC).These methods significantly surpass existing CNN-based ones, demonstrating the effectiveness of ViT in FGVC tasks.However, there are some limitations when applying ViT directly to FGVC.First, ViT needs to split images into patches and calculate the attention of every pair, which may result in heavy redundant calculation and unsatisfying performance when handling fine-grained images with complex background and small objects.Second, a standard ViT only utilizes the class token in the final layer for classification, which is not enough to extract comprehensive fine-grained information. To address these issues, we propose a novel ViT based fine-grained object discriminator for FGVC tasks, ViT-FOD for short. Specifically, besides a ViT backbone, it further introduces three novel components, i.e, Attention Patch Combination (APC), Critical Regions Filter (CRF), and Complementary Tokens Integration (CTI). Thereinto, APC pieces informative patches from two images to generate a new image so that the redundant calculation can be reduced. CRF emphasizes tokens corresponding to discriminative regions to generate a new class token for subtle feature learning. To extract comprehensive information, CTI integrates complementary information captured by class tokens in different ViT layers. We conduct comprehensive experiments on widely used datasets and the results demonstrate that ViT-FOD is able to achieve state-of-the-art performance.

The cavity electromechanical devices with radiation-pressure-interaction induced Kerr-like nonlinearity are promising candidates to generate microwave frequency combs. We construct a silicon-nitride-membrane-based superconducting cavity electromechanical device and study two mechanical modes mediated synergistic frequency combs. Around the threshold of intracavity field instability, we firstly show independent frequency combs with tooth spacing equalling to each mechanical mode frequency. At the overlap boundaries of these two individual mechanical mode mediated instability thresholds, we observe hybridization of frequency combs based on the cavity field mediated indirect coupling between these two mechanical modes. The spectrum lines turn to be unequally spaced but can be recognized into combinations of the coexisting frequency combs. Beyond the boundary, the comb reverts to the single-mode case, and which mechanical mode frequency will the tooth spacing depend on the mode competition. Our work demonstrates mechanical mode competition enabled switchability of frequency comb tooth spacing and can be extended to other devices with multiple nonlinearities., Comment: 6 pages, 4 figures

Recent work has focused on data-driven learning of the evolution of unknown systems via deep neural networks (DNNs), with the goal of conducting long time prediction of the evolution of the unknown system. Training a DNN with low generalization error is a particularly important task in this case as error is accumulated over time. Because of the inherent randomness in DNN training, chiefly in stochastic optimization, there is uncertainty in the resulting prediction, and therefore in the generalization error. Hence, the generalization error can be viewed as a random variable with some probability distribution. Well-trained DNNs, particularly those with many hyperparameters, typically result in probability distributions for generalization error with low bias but high variance. High variance causes variability and unpredictably in the results of a trained DNN. This paper presents a computational technique which decreases the variance of the generalization error, thereby improving the reliability of the DNN model to generalize consistently. In the proposed ensemble averaging method, multiple models are independently trained and model predictions are averaged at each time step. A mathematical foundation for the method is presented, including results regarding the distribution of the local truncation error. In addition, three time-dependent differential equation problems are considered as numerical examples, demonstrating the effectiveness of the method to decrease variance of DNN predictions generally.

We study the kick dynamics of periodically driven quantum systems, and provide a timeindependent effective Hamiltonian with the analytical form to reasonably describe the effective dynamics in a long timescale. It is shown that the effective coupling strength can be much larger than the coupling strength of the original system in some parameter regions, which stems from the zero time duration of kicks. Furthermore, different regimes can be transformed from and to each other in the same three-level system by only modulating the period of periodic kicks. In particular, the population of excited states can be selectively suppressed in periodic kicks, benefiting from the large detuning regime of the original system. Finally, some applications and physical implementation of periodic kicks are demonstrated in quantum systems. Those unique features would make periodic kicks becoming a powerful tool for quantum state engineering.