Related papers: High-precision quantum algorithms for partial diff…
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…
Partial differential equations (PDEs) are central to computational electromagnetics (CEM) and photonic design, but classical solvers face high costs for large or complex structures. Quantum Hamiltonian simulation provides a framework to…
Quantum computers are predicted to outperform classical ones for solving partial differential equations, perhaps exponentially. Here we consider a prototypical PDE - the heat equation in a rectangular region - and compare in detail the…
In contexts where relevant problems can easily attain configuration spaces of enormous sizes, solving Linear Differential Equations (LDEs) can become a hard achievement for classical computers; on the other hand, the rise of quantum…
Quantum algorithms offer an exponential advantage with respect to the number of dependent variables for solving certain nonlinear ordinary differential equations (ODEs). These algorithms typically begin by transforming the original…
We utilize extreme-learning machines for the prediction of partial differential equations (PDEs). Our method splits the state space into multiple windows that are predicted individually using a single model. Despite requiring only few data…
For the solution of partial differential equations (PDEs), we show that the quantum Fourier transform (QFT) can enable the design of quantum circuits that are particularly simple, both conceptually and with regard to hardware requirements.…
We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). Specifically, we show how the norm of the matrix exponential characterizes…
We study a quantum-algorithmic framework for parameterizing partial differential equations (PDEs). For a broad class of problems in which the discretized parameter field admits a diagonal representation, block-encodings of diagonal…
We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate…
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…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
Nonlinear differential equations exhibit rich phenomena in many fields but are notoriously challenging to solve. Recently, Liu et al. [1] demonstrated the first efficient quantum algorithm for dissipative quadratic differential equations…
We explore the utilization of higher-order discretization techniques in optimizing the gate count needed for quantum computer based solutions of partial differential equations. To accomplish this, we present an efficient approach for…
We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for…
We present an efficient, nearly optimal quantum algorithm for solving linear matrix differential equations, with applications to the simulation of open quantum systems and beyond. For unitary or dissipative dynamics, the algorithm computes…
Quantum linear system (QLS) algorithms offer the potential to solve large-scale linear systems exponentially faster than classical methods. However, applying QLS algorithms to real-world problems remains challenging due to issues such as…
Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum…
Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…
Linear equations play a pivotal role in many areas of science and engineering, making efficient solutions to linear systems highly desirable. The development of quantum algorithms for solving linear systems has been a significant…