Related papers: An efficient quantum algorithm for simulating poly…
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…
Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to…
Quantum linear system algorithms (QLSAs) have the potential to speed up algorithms that rely on solving linear systems. Interior Point Methods (IPMs) yield a fundamental family of polynomial-time algorithms for solving optimization…
We present a systematic pathway for solving differential equations within the quantum linear systems framework by combining block encoding with Quantum Singular Value Transformation (QSVT). The approach is demonstrated on a complex…
The finite element method is used to approximately solve boundary value problems for differential equations. The method discretises the parameter space and finds an approximate solution by solving a large system of linear equations. Here we…
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…
Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed…
Partial differential equations (PDEs) are central to modeling physical and engineering systems, but repeatedly solving parametric PDEs remains computationally expensive. Operator learning enables fast surrogate inference, yet typically…
This paper presents a quantum algorithm for solving the fractional Poisson equation \((-\Delta)^s u = f\) with \(s \in (0,1)\) on bounded domains. The proposed approach combines rational approximation techniques with quantum linear system…
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…
We present quantum algorithms for the estimation of n-time correlation functions, the local and non-local density of states, and dynamical linear response functions. These algorithms are all based on block-encodings - a versatile technique…
The solution for non-linear, complex partial differential Equations (PDEs) is achieved through numerical approximations, which yield a linear system of equations. This approach is prevalent in Computational Fluid Dynamics (CFD), but it…
To approximate solutions of complex nonlinear partial differential equations remains a computational challenge, especially for sets of equations relevant in industry, such as Euler or Navier-Stokes equations. Even the most sophisticated…
The Carleman embedding method is a widely used technique for linearizing a system of nonlinear differential equations, but fails to converge in regions where there are multiple fixed points. We propose and test three different versions of a…
Non-autonomous dynamical systems appear in a very wide range of interesting applications, both in classical and quantum dynamics, where in the latter case it corresponds to having a time-dependent Hamiltonian. However, the quantum…
Systems of linear equations are used to model a wide array of problems in all fields of science and engineering. Recently, it has been shown that quantum computers could solve linear systems exponentially faster than classical computers,…
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…
Many problems in linear algebra -- such as those arising from non-Hermitian physics and differential equations -- can be solved on a quantum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum…
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 introduce a quantum algorithm for simulating the dynamics of electrical circuits consisting of resistors, inductors and capacitors (aka RLC circuits) along with power sources. Given oracle access to the connectivity of the circuit and…