Plateau Problem Research Articles

On the singular planar Plateau problem

On the singular planar Plateau problem

Towards determining the presence of barren plateaus in some chemically inspired variational quantum algorithms

In quantum chemistry, the variational quantum eigensolver (VQE) is a promising algorithm for molecular simulations on near-term quantum computers. However, VQEs using hardware-efficient circuits face scaling challenges due to the barren plateau problem. This raises the question of whether chemically inspired circuits from unitary coupled cluster (UCC) methods can avoid this issue. Here we provide theoretical evidence indicating they may not. By examining alternated dUCC ansätzes and relaxed Trotterized UCC ansätzes, we find that in the infinite depth limit, a separation occurs between particle-hole one- and two-body unitary operators. While one-body terms yield a polynomially concentrated energy landscape, adding two-body terms leads to exponential concentration. Numerical simulations support these findings, suggesting that popular 1-step Trotterized unitary coupled-cluster with singles and doubles (UCCSD) ansätze may not scale. Our results emphasize the link between trainability and circuit expressiveness, raising doubts about VQEs’ ability to surpass classical methods.

On Busemann--Hausdorff densities of dimension two and of codimension two, with an application to Plateau Problem

On Busemann--Hausdorff densities of dimension two and of codimension two, with an application to Plateau Problem

A non-parametric Plateau problem with partial free boundary

A non-parametric Plateau problem with partial free boundary

Equivalence of cost concentration and gradient vanishing for quantum circuits: an elementary proof in the Riemannian formulation

Abstract The optimization of quantum circuits can be hampered by a decay of average gradient amplitudes with increasing system size. When the decay is exponential, this is called the barren plateau problem. Considering explicit circuit parametrizations (in terms of rotation angles), it has been shown in Arrasmith et al (2022 Quantum Sci. Technol. 7 045015) that barren plateaus are equivalent to an exponential decay of the variance of cost-function differences. We show that the issue is particularly simple in the (parametrization-free) Riemannian formulation of such optimization problems and obtain a tighter bound for the cost-function variance. An elementary derivation shows that the single-gate variance of the cost function is strictly equal to half the variance of the Riemannian single-gate gradient, where we sample variable gates according to the uniform Haar measure. The total variances of the cost function and its gradient are then both bounded from above by the sum of single-gate variances and, conversely, bound single-gate variances from above. So, decays of gradients and cost-function variations go hand in hand, and barren plateau problems cannot be resolved by avoiding gradient-based in favor of gradient-free optimization methods.

Nonlinear dynamics as a ground-state solution on quantum computers

For the solution of time-dependent nonlinear differential equations, we present variational quantum algorithms (VQAs) that encode both space and time in qubit registers. The spacetime encoding enables us to obtain the entire time evolution from a single ground-state computation. We describe a general procedure to construct efficient quantum circuits for the cost function evaluation required by VQAs. To mitigate the barren plateau problem during the optimization, we propose an adaptive multigrid strategy. The approach is illustrated for the nonlinear Burgers equation. We classically optimize quantum circuits to represent the desired ground-state solutions, run them on IBM Q System One and Quantinuum System Model H1, and demonstrate that current quantum computers are capable of accurately reproducing the exact results. Published by the American Physical Society 2024

A Bayesian network model integrating organizational, individual and psychological factors for strengthening construction worker safety behavior

Objectives. Construction worker safety behavior (CWSB) may be affected by a confluence of multilevel and interrelated factors. Cultivating and maintaining CWSB is vital for improving construction safety. Current studies focus on organization-level or individual-level CWSB antecedents. However, few studies have examined the influence of psychological factors on CWSB, thereby reducing the joint effects of multilevel factors on CWSB. Methods. To determine effective strategies for strengthening CWSB, this study adopted the Bayesian network technique to explore the interrelationships between CWSB and its antecedent factors. A Bayesian belief network model was developed and trained with data collected from Chinese construction workers, which connected organizational, individual and psychological factors with CWSB. Results. According to the sensitivity analysis, safety knowledge, safety climate and psychological capital are the three most significant influencing factors for CWSB. A combined strategy that enhances safety knowledge, safety climate and communication competence simultaneously is the most effective option for strengthening CWSB. The validation and robustness of the network showed good accuracy for safety behavior judgment. Conclusion. This study proposes an alternative way to improve safety behavior by identifying its interactive causes and illustrates the importance of initiating systematic safety measures, which may help to mitigate the problem of safety plateau.

Uniqueness of weak solutions of the Plateau flow

In this paper, we study the uniqueness of weak solutions of the heat flow of half-harmonic maps, which was first introduced by Wettstein as a half-Laplacian heat flow and recently studied by Struwe using more classical techniques. On top of its similarity with the two dimensional harmonic map flow, this geometric gradient flow is of interest due to its links with free boundary minimal surfaces and the Plateau problem, leading Struwe to propose the name Plateau flow, which we adopt throughout. We obtain uniqueness of weak solutions of this flow under a natural condition on the energy, which answers positively a question raised by Struwe.

Numerical Analysis of the Plateau Problem by the Method of Fundamental Solutions

Toward identifying the number of minimal surfaces sharing the same boundary from the geometry of the boundary, we propose a numerical scheme with high speed and high accuracy. Our numerical scheme is based on the method of fundamental solutions. We establish the convergence analysis for Dirichlet energy and L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document}-error analysis for mean curvature. Each of the approximate solutions in our scheme is a smooth surface, a significant difference from previous studies that required mesh generation.

Hybrid quantum neural network structures for image multi-classification

Image classification is a fundamental problem in computer vision, and neural networks provide an effective solution. With the advancement of quantum technology, quantum neural networks have attracted a lot of attention. However, they are only suitable for low-dimensional data and require dimensionality reduction and quantum encoding. Two image classification methods have recently emerged: one employs PCA for dimensionality reduction and angular encoding, and the other integrates QNN into a CNN to improve performance. Despite numerous algorithms, the differences between them remain unclear. This study explores these algorithms’ performance in multi-class image classification and proposes an optimized hybrid quantum neural network suitable for the current environment. As the number of classes increases, research on PCA-based quantum algorithms reveals the barren plateau problem of QNN, which is not suitable for multiple classes in a hybrid setting. Our proposed model combining traditional CNN with QNN addresses QNN’s multi-class training difficulties to some extent and achieves satisfactory classification results. Nevertheless, its accuracy remains below that of the top-performing CNN models. Furthermore, we investigate transfer learning in hybrid quantum neural network models and assess the performance of our models on the quantum hardware from IBM. In conclusion, quantum neural networks show promise but require further research and optimization, facing challenges ahead.

Minimal Diffeomorphisms with L1 Hopf Differentials

Abstract We prove that for any two Riemannian metrics $\sigma _{1}, \sigma _{2}$ on the unit disk, a homeomorphism $\partial \mathbb{D}\to \partial \mathbb{D}$ extends to at most one quasiconformal minimal diffeomorphism $(\mathbb{D},\sigma _{1})\to (\mathbb{D},\sigma _{2})$ with $L^{1}$ Hopf differential. For minimal Lagrangian diffeomorphisms between hyperbolic disks, the result is known, but this is the first proof that does not use anti-de Sitter geometry. We show that the result fails without the $L^{1}$ assumption in variable curvature. The key input for our proof is the uniqueness of solutions for a certain Plateau problem in a product of trees.

Analyzing variational quantum landscapes with information content

The parameters of the quantum circuit in a variational quantum algorithm induce a landscape that contains the relevant information regarding its optimization hardness. In this work, we investigate such landscapes through the lens of information content, a measure of the variability between points in parameter space. Our major contribution connects the information content to the average norm of the gradient, for which we provide robust analytical bounds on its estimators. This result holds for any (classical or quantum) variational landscape. We validate the analytical understating by numerically studying the scaling of the gradient in an instance of the barren plateau problem. In such instance, we are able to estimate the scaling pre-factors in the gradient. Our work provides a way to analyze variational quantum algorithms in a data-driven fashion well-suited for near-term quantum computers.

ResQNets: a residual approach for mitigating barren plateaus in quantum neural networks

The barren plateau problem in quantum neural networks (QNNs) is a significant challenge that hinders the practical success of QNNs. In this paper, we introduce residual quantum neural networks (ResQNets) as a solution to address this problem. ResQNets are inspired by classical residual neural networks and involve splitting the conventional QNN architecture into multiple quantum nodes, each containing its own parameterized quantum circuit, and introducing residual connections between these nodes. Our study demonstrates the efficacy of ResQNets by comparing their performance with that of conventional QNNs and plain quantum neural networks through multiple training experiments and analyzing the cost function landscapes. Our results show that the incorporation of residual connections results in improved training performance. Therefore, we conclude that ResQNets offer a promising solution to overcome the barren plateau problem in QNNs and provide a potential direction for future research in the field of quantum machine learning.

Minimal Surfaces and the Plateau Problem: Numerical Methods and Applications

This article offers a comprehensive overview of the results obtained through numerical methods in solving the minimal surface equation, along with exploring the applications of minimal surfaces in science, technology, and architecture. The content is enriched with practical examples highlighting the diverse applications of minimal surfaces.

QTN-VQC: an end-to-end learning framework for quantum neural networks

This work focuses on investigating an end-to-end learning approach for quantum neural networks (QNN) on noisy intermediate-scale quantum devices. The proposed model combines a quantum tensor network (QTN) with a variational quantum circuit (VQC), resulting in a QTN-VQC architecture. This architecture integrates a QTN with a horizontal or vertical structure related to the implementation of quantum circuits for a tensor-train network. The study provides theoretical insights into the quantum advantages of the end-to-end learning pipeline based on QTN-VQC from two perspectives. The first perspective refers to the theoretical understanding of QTN-VQC with upper bounds on the empirical error, examining its learnability and generalization powers; The second perspective focuses on using the QTN-VQC architecture to alleviate the Barren Plateau problem in the training stage. Our experimental simulation on CPU/GPUs is performed on a handwritten digit classification dataset to corroborate our proposed methods in this work.

Generic uniqueness for the Plateau problem

Given a complete Riemannian manifold M⊂Rd which is a Lipschitz neighborhood retract of dimension m+n, of class Ch,β and an oriented, closed submanifold Γ⊂M of dimension m−1, which is a boundary in integral homology, we construct a complete metric space B of Ch,α-perturbations of Γ inside M, with α<β, enjoying the following property. For the typical element b∈B, in the sense of Baire categories, there exists a unique m-dimensional integral current in M which solves the corresponding Plateau problem and it has multiplicity one.

Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces

Abstract In this paper we consider metric fillings of boundaries of convex bodies. We show that convex bodies are the unique minimal fillings of their boundary metrics among all integral current spaces. To this end, we also prove that convex bodies enjoy the Lipschitz-volume rigidity property within the category of integral current spaces, which is well known in the smooth category. As further applications of this result, we prove a variant of Lipschitz-volume rigidity for round spheres and answer a question of Perales concerning the intrinsic flat convergence of minimizing sequences for the Plateau problem.

Effects of surface tension and elasticity on critical points of the Kirchhoff–Plateau problem

AbstractWe introduce a modified Kirchhoff–Plateau problem adding an energy term to penalize shape modifications of the cross-sections appended to the elastic midline. In a specific setting, we characterize quantitatively some properties of minimizers. Indeed, choosing three different geometrical shapes for the cross-section, we derive Euler–Lagrange equations for a planar version of the Kirchhoff–Plateau problem. We show that in the physical range of the parameters, there exists a unique critical point satisfying the imposed constraints. Finally, we analyze the effects of the surface tension on the shape of the cross-sections at the equilibrium.

근속연수, 구조적 경력정체 및 대인 일탈행동간의 관계

Purpose - The purpose of this study was to examine the relationship between organizational tenure, structural career plateau, and interpersonal deviant behavior for nurses. Moreover, this study also investigated moderating role of proactivity on the effect of organizational tenure on structural career plateau.&#x0D; Design/methodology/approach - This study utilized 448 nurses working at three different types of hospital. The Exploratory Factor Analysis (EFA) and the multiple regression analysis were conducted to verify the hypotheses.&#x0D; Findings - The results showed that organizational tenure is positively related to structural career plateau and structural career plateau is positively related to interpersonal deviant behavior. However, moderating effect of proactivity is not significant.&#x0D; Research implications or Originality - Nurses are critical to the competitiveness of hospital organizations considering the proportion of manpower and the role of nurses. One of the important factors that decrease the motivation of nurses is structural career plateau. This study analyzed the antecedent and consequence of structural career plateau. In addition, practical implications for resolving the problem of structural career plateau were provided.

On the asymptotic Plateau problem in hyperbolic space

In this paper, we solve the asymptotic Plateau problem in hyperbolic space for constant σ n − 1 \sigma _{n-1} curvature, i.e. the existence of a complete hypersurface in H n + 1 \mathbb {H}^{n+1} satisfying σ n − 1 ( κ ) = σ ∈ ( 0 , n ) \sigma _{n-1}(\kappa )=\sigma \in (0,n) with a prescribed asymptotic boundary Γ \Gamma . The key ingredient is the curvature estimates. Previously, this was only known for σ 0 &gt; σ &gt; n \sigma _0&gt;\sigma &gt;n , where σ 0 \sigma _0 is a positive constant.