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Coherence fraction in Grover's search algorithm

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Physical Review A 2024年第6期110卷 062429-062429页

作者： Si-Qi Zhou Hai Jin Jin-Min Liang Shao-Ming Fei Yunlong Xiao Zhihao Ma School of Mathematical Sciences MOE-LSC Shanghai Jiao Tong University Shanghai 200240 China Shanghai Seres Information Technology Co. Ltd Shanghai 200040 China Shenzhen Institute for Quantum Science and Engineering Southern University of Science and Technology Shenzhen 518055 China State Key Laboratory for Mesoscopic Physics School of Physics Frontiers Science Center for Nano-Optoelectronics Peking University Beijing 100871 China School of Mathematical Sciences Capital Normal University Beijing 100048 China Max Planck Institute for Mathematics in the Sciences - 04103 Leipzig Germany Quantum Innovation Centre (Q.InC) Agency for Science Technology and Research (ASTAR) 2 Fusionopolis Way Innovis #08-03 Singapore 138634 Republic of Singapore Institute of High Performance Computing (IHPC) Agency for Science Technology and Research (ASTAR) 1 Fusionopolis Way #16-16 Connexis Singapore 138632 Republic of Singapore

The question of which resources drive the advantages in quantum algorithms has long been a fundamental challenge. While entanglement and coherence are critical to many quantum algorithms, our results indicate that they do not fully explain the quantum advantage achieved by the Grover search algorithm. By introducing a generalized Grover search algorithm, we demonstrate that the success probability depends not only on the querying number of oracles but also on the coherence fraction, which quantifies the fidelity between an arbitrary initial quantum state and the equal superposition state. Additionally, we explore the role of the coherence fraction in the quantum minimization algorithm, which offers a framework for solving complex problems in quantum machine learning. These findings offer insights into the origins of quantum advantage and open pathways for the development of new quantum algorithms.

关键词： Quantum entanglement

MAP-based Problem-Agnostic Diffusion Model for Inverse Problems

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arXiv 2025年

作者： Tao, Pingping Liu, Haixia Su, Jing Yang, Xiaochen Tan, Hongchen Shandong University Library No. 180 West Culture Road Shandong Weihai264209 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology No. 1037 Luoyu road Hubei Wuhan430074 China Dapartment of Basic Sciences Dalian University of Science and Technology No. 999-26 Harbor Road Liaoning Dalian116052 China School of Computer Science and Technology Harbin Institute of Technology No. 2 West Culture Road Shandong Weihai264209 China Institute of Future Technology Dalian University of Technology No. 2 Lingong Road Liaoning Dalian116081 China

Diffusion models have indeed shown great promise in solving inverse problems in image processing. In this paper, we propose a novel, problem-agnostic diffusion model called the maximum a posteriori (MAP)-based guided term estimation method for inverse problems. To leverage unconditionally pretrained diffusion models to address conditional generation tasks, we divide the conditional score function into two terms according to Bayes’ rule: an unconditional score function (approximated by a pretrained score network) and a guided term, which is estimated using a novel MAP-based method that incorporates a Gaussian-type prior of natural images. This innovation allows us to better capture the intrinsic properties of the data, leading to improved performance. Numerical results demonstrate that our method preserves contents more effectively compared to state-of-the-art methods—for example, maintaining the structure of glasses in super-resolution tasks and producing more coherent results in the neighborhood of masked regions during inpainting. Our numerical implementation is available at https://***/liuhaixias1/ MAP-DIFFUSION-IP. Copyright © 2025, The Authors. All rights reserved.

关键词： Inverse problems

Software trustworthiness evaluation using structural equation modeling

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International Journal of Performability engineering 2019年第10期15卷 2628-2635页

作者： Deng, Rumei Chen, Yixiang Wu, Hengyang Tao, Hongwei MoE Engineering Center for Software Hardware Co-Design Technology and Application East China Normal University Shanghai200062 China Shanghai Key Laboratory of Trustworthy Computing East China Normal University Shanghai200062 China School of Computer and Communication Engineering Zhengzhou University of Light Industry Zhengzhou450002 China

Software trustworthiness evaluation results can provide guidance for existing software development. Therefore, it is of great significance to study software trustworthiness evaluation. Weights and the relationship of trustworthy attributes have a huge effect on the results of software trustworthiness evaluation. However, few studies have considered weights and the quantitative relationship among trustworthy attributes at the same time. Structural equation modeling (SEM) provides a more scientific and reasonable method to obtain weights, where the objectification of subjective weights is guaranteed. In this paper, we propose a SEM-based method to evaluate software trustworthiness. Firstly, we establish the trustworthy evaluation indicator system for software. Based on the survey data, we construct the SEM for software trustworthiness evaluation to obtain the weights of trustworthy attributes and the relationship among them. Lastly, we apply the trustworthy measurement model to calculate the trustworthiness value of software to be surveyed. These trustworthiness values demonstrate the reasonability of our method. © 2019 Totem Publisher, Inc. All rights reserved.

关键词： Surveys

Quantum circuits for the heat equation with physical boundary conditions via Schrödingerization

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Journal of Computational Physics 2025年 538卷

作者： Shi Jin Nana Liu Yue Yu School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Shanghai Jiao Tong University Shanghai 200240 China University of Michigan-Shanghai Jiao Tong University Joint Institute Shanghai 200240 China School of Mathematics and Computational Science Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education National Center for Applied Mathematics in Hunan Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan 411105 Hunan China

This paper explores the explicit design of quantum circuits for quantum simulation of partial differential equations (PDEs) with physical boundary conditions. These equations and/or their discretized forms usually do not evolve via unitary dynamics, thus are not directly suitable for quantum simulation. Boundary conditions (either time-dependent or independent) make the problem more difficult. To tackle this challenge, the Schrödingerization method can be employed, which converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schrödinger-type equations, via the so-called warped phase transformation that maps the equation into one higher dimension. Despite advancements in Schrödingerization techniques, the explicit implementation of quantum circuits for solving general PDEs, especially with physical boundary conditions, remains underdeveloped. We present two methods for handling the inhomogeneous terms arising from time-dependent physical boundary conditions. One approach utilizes Duhamel’s principle to express the solution in integral form and employs linear combination of unitaries (LCU) for coherent state preparation. Another method applies an augmentation to transform the inhomogeneous problem into a homogeneous one. We then apply the quantum simulation technique from [1] to transform the resulting non-autonomous system to an autonomous system in one higher dimension. We provide detailed implementations of these two methods and conduct a comprehensive complexity analysis in terms of queries to the time evolution input oracle.

关键词：

Schrödingerization based quantum algorithms for the fractional Poisson equation

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arXiv 2025年

作者： Jin, Shi Liu, Nana Yu, Yue School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Shanghai Jiao Tong University Shanghai200240 China University of Michigan Shanghai Jiao Tong University Joint Institute Shanghai200240 China School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education National Center for Applied Mathematics in Hunan Xiangtan University Hunan Xiangtan411105 China

We develop a quantum algorithm for solving high-dimensional fractional Poisson equations. By applying the Caffarelli-Silvestre extension, the d-dimensional fractional equation is reformulated as a local partial differential equation in d + 1 dimensions. We propose a quantum algorithm for the finite element discretization of this local problem, by capturing the steady-state of the corresponding differential equations using the Schrödingerization approach from [29, 39, 40]. The Schrödingerization technique transforms general linear partial and ordinary differential equations into Schrödinger-type systems, making them suitable for quantum simulation. This is achieved through the warped phase transformation, which maps the equation into a higher-dimensional space. We provide detailed implementations of the method and conduct a comprehensive complexity analysis, which can show up to exponential advantage – with respect to the inverse of the mesh size in high dimensions – compared to its classical counterpart. Specifically, while the classical method requires Oe(d1/233d/2h−d−2) operations, the quantum counterpart requires Oe(d33d/2h−2.5) queries to the block-encoding input models, with the quantum complexity being independent of the dimension d in terms of the inverse mesh size h−1. Numerical experiments are conducted to verify the validity of our formulation. © 2025, CC BY-NC-ND.

关键词： Poisson equation

On the Schrödingerization method for linear non-unitary dynamics with optimal dependence on matrix queries

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arXiv 2025年

作者： Jin, Shi Liu, Nana Ma, Chuwen Yu, Yue School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Shanghai Jiao Tong University Shanghai200240 China University of Michigan Shanghai Jiao Tong University Joint Institute Shanghai200240 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education National Center for Applied Mathematics in Hunan School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

The Schrödingerization method converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schrödinger-type equations with unitary evolution. It does so via the so-called warped phase transformation that maps the original equation into a Schrödinger-type equation in one higher dimension [1,2]. We show that by employing a smooth initialization of the warped phase transform [3], Schrödingerization can in fact achieve optimal scaling in matrix queries. This paper presents the detailed implementation of three smooth initializations for the Schrödingerization method: (a) the cut-off function, (b) the higher-order polynomial interpolation, and (c) the Fourier transform methods, that achieve optimality for (a) and near-optimality for (b) and (c). A detailed analysis of key parameters affecting time complexity is conducted. Copyright © 2025, The Authors. All rights reserved.

关键词： Ordinary differential equations

RIGIDITY OF STEADY SOLUTIONS TO THE NAVIER-STOKES EQUATIONS IN HIGH DIMENSIONS AND ITS APPLICATIONS

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arXiv 2023年

作者： Bang, Jeaheang Gui, Changfeng Liu, Hao Wang, Yun Xie, Chunjing Department of Mathematics The University of Texas at San Antonio One UTSA Circle San AntonioTX78249 United States Department of Mathematics Faculty of Science and Technology University of Macau Taipa China School of Mathematical Sciences Institute of Natural Sciences Shanghai Jiao Tong University 800 Dongchuan Road Shanghai China School of Mathematical Sciences Center for dynamical systems and differential equations Soochow University Suzhou China School of Mathematical Sciences Institute of Natural Sciences Ministry of Education Key Laboratory of Scientific and Engineering Computing CMA-Shanghai Shanghai Jiao Tong University 800 Dongchuan Road Shanghai China

Solutions with scaling-invariant bounds such as self-similar solutions, play an important role in the understanding of the regularity and asymptotic structures of solutions to the Navier-Stokes equations. In this paper, we prove that any steady solution satisfying |u(x)| ≤ C/|x| for any constant C in n \ {0} with n ≥ 4, must be zero. Our main idea is to analyze the velocity field and the total head pressure via weighted energy estimates with suitable multipliers so that the proof is pretty elementary and short. These results not only give the Liouville-type theorem for steady solutions in higher dimensions with neither smallness nor self-similarity type assumptions, but also help to remove a class of singularities of solutions and give the optimal asymptotic behaviors of solutions at infinity in the exterior domains. Copyright © 2023, The Authors. All rights reserved.

关键词： Navier Stokes equations

Combined influences of Soret effect and horizontal magnetic field on dynamics of vertical convection in binary fluids

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Physical Review Fluids 2025年第5期10卷 053503-053503页

作者： Chao-Nan Zhang Lai-yun Zheng Xu-Long Li Juan-Juan Qin Bing-Xin Zhao Don Liu School of Mathematics and Statistics Ningxia University Yinchuan 750021 China School of Mechanical Engineering Ningxia University Yinchuan 750021 China Ningxia Key Laboratory of Interdisciplinary Mechanics and Scientific Computing Yinchuan 750021 China Ningxia Basic Science Research Center of Mathematics Yinchuan 750021 China Department of Mathematics and Mechanical Engineering Louisiana Tech University Ruston Louisiana 71270 USA

The influence of the Soret effect and a uniform magnetic field on the dynamic behavior of vertical convection are investigated using a high-accuracy numerical method. The effects of the Soret number (Sr, which quantifies the direction and magnitude of species migration relative to the temperature gradient; positive Sr denotes migration toward hotter regions, negative toward cooler regions) on flow, heat and mass transfer, and flow transition in the presence of a magnetic field are analyzed. The results reveal that, for a given magnetic field strength, positive and negative Soret effects have significantly different influences on heat and mass transfer as well as on flow structure. For a high Rayleigh number (Ra, vigor of the buoyancy) and a small magnetic field, the heat transfer initially decreases and then increases with increasing Soret number, while the mass transfer consistently decreases. In cases where temperature is dominant, heat transfer for Sr=−1 exceeds that for Sr=0 by 24%, while mass transfer with Sr=−1 exceeds that for Sr=0 by 85%. When concentration is dominant, heat transfer reaches a minimum at Sr=0, whereas mass transfer peaks at Sr=−1, corresponding to an approximate 71% increase over heat transfer at Sr=0, and around a 210% increase in mass transfer. Additionally, the flow exhibits greater complexity when the Soret number is negative than when it is positive. When the magnetic field is weak, the temperature-dominated flow changes from chaotic to periodic with increasing Soret number and eventually returns to a chaotic state. When the magnetic field is strong, the flow transitions from chaotic to periodic and finally to a steady state. The transition route involves two types of periodic flow: a simple periodic flow with an integer multiple of the fundamental frequency, and a complex periodic flow with multiple lower-frequency subharmonic waves. Furthermore, as the strength of the magnetic field increases, the flow in the rectangular enclosure is

关键词： Diffusion in solids

Analytic approximations of Von Krmn plate under arbitrary uniform pressure—equations in integral form

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Science China(Physics,Mechanics & Astronomy) 2018年第1期61卷 57-67页

作者： XiaoXu Zhong ShiJun Liao School of Naval Architecture Ocean and Civil Engineering Shanghai Jiao Tong University Collaborative Innovative Center for Advanced Ship and Deep-Sea Exploration State Key Laboratory of Ocean Engineering Ministry-of-Education Key Laboratory of Scientific and Engineering Computing

Analytic approximations of the Von Karman＇s plate equations in integral form for a circular plate under external uniform pressure to arbitrary magnitude are successfully obtained by means of the homotopy analysis method （HAM）, an analytic approximation technique for highly nonlinear problems. Two HAM-based approaches are proposed for either a given external uniform pressure Q or a given central deflection, respectively. Both of them are valid for uniform pressure to arbitrary magnitude by choosing proper values of the so-called convergence-control parameters c1 and c2 in the frame of the HAM. Besides, it is found that the HAM- based iteration approaches generally converge much faster than the interpolation iterative method. Furthermore, we prove that the interpolation iterative method is a special case of the first-order HAM iteration approach for a given external uniform pressure Q when c1= -0 and c2 = -1, where 0 denotes the interpolation iterative parameter. Therefore, according to the convergence theorem of Zheng and Zhou about the interpolation iterative method, the HAM-based approaches are valid for uniform pressure to arbitrary magnitude at least in the special case c1 = -0 and c2= -1. In addition, we prove that the HAM approach for the Von karman＇s plate equations in differential form is just a special case of the HAM for the Von karman＇s plate equations in integral form mentioned in this paper. All of these illustrate the validity and great potential of the HAM for highly nonlinear problems, and its superiority over perturbation techniques.

关键词： circular plate high nonlinearity homotopy analysis method