Related papers: Quantum algorithms and the finite element method
The finite element method (FEM) is a cornerstone numerical technique for solving partial differential equations (PDEs). Here, we present $\textbf{Qu-FEM}$, a fault-tolerant era quantum algorithm for the finite element method. In contrast to…
This paper presents a quantum algorithm for the solution of prototypical second-order linear elliptic partial differential equations discretized by $d$-linear finite elements on Cartesian grids of a bounded $d$-dimensional domain. An…
We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time.…
We present and experimentally realize a quantum algorithm for efficiently solving the following problem: given an $N\times N$ matrix $\mathcal{M}$, an $N$-dimensional vector $\textbf{\emph{b}}$, and an initial vector $\textbf{\emph{x}}(0)$,…
We develop a hybrid classical-quantum algorithm to solve a type of linear reaction-diffusion equation, the neutron diffusion (generalized) k-eigenvalue problem that establishes nuclear criticality. The algorithm handles an equation with…
In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The…
Quantum algorithm is an algorithm for solving mathematical problems using quantum systems encoded as information, which is found to outperform classical algorithms in some specific cases. The objective of this study is to develop a quantum…
Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space. Such algorithms can produce a quantum state proportional to the solution of a $d$-dimensional system…
This manuscript presents the Quantum Finite Element Method (Q-FEM) developed for use in noisy intermediate-scale quantum (NISQ) computers and employs the variational quantum linear solver (VQLS) algorithm. The proposed method leverages the…
Quantum computing holds significant promise for scientific computing due to its potential for polynomial to even exponential speedups over classical methods, which are often hindered by the curse of dimensionality. While neural networks…
Quantum algorithms are able to solve particular problems exponentially faster than conventional algorithms, when implemented on a quantum computer. However, all demonstrations to date have required already knowing the answer to construct…
We present classical and quantum algorithms based on spectral methods for a problem in tensor principal component analysis. The quantum algorithm achieves a quartic speedup while using exponentially smaller space than the fastest classical…
Quantum algorithms and complexity have recently been studied not only for discrete, but also for some numerical problems. Most attention has been paid so far to the integration problem, for which a speed-up is shown by quantum computers…
Though quantum algorithm acts as an important role in quantum computation science, not only for providing a great vision for solving classically unsolvable problems, but also due to the fact that it gives a potential way of understanding…
Solving linear systems of equations is ubiquitous in all areas of science and engineering. With rapidly growing data sets, such a task can be intractable for classical computers, as the best known classical algorithms require a time…
Partial differential equations frequently appear in the natural sciences and related disciplines. Solving them is often challenging, particularly in high dimensions, due to the "curse of dimensionality". In this work, we explore the…
Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for…
Quantum algorithms for both differential equation solving and for machine learning potentially offer an exponential speedup over all known classical algorithms. However, there also exist obstacles to obtaining this potential speedup in…
The finite element method can be viewed as a machine that automates the discretization of differential equations, taking as input a variational problem, a finite element and a mesh, and producing as output a system of discrete equations.…
We describe a quantum algorithm for preparing states that encode solutions of non-homogeneous linear partial differential equations. The algorithm is a continuous-variable version of matrix inversion: it efficiently inverts differential…