H-DES: a Quantum-Classical Hybrid Differential Equation Solver
- URL: http://arxiv.org/abs/2410.01130v1
- Date: Tue, 1 Oct 2024 23:47:41 GMT
- Title: H-DES: a Quantum-Classical Hybrid Differential Equation Solver
- Authors: Hamza Jaffali, Jonas Bastos de Araujo, Nadia Milazzo, Marta Reina, Henri de Boutray, Karla Baumann, Frédéric Holweck,
- Abstract summary: We introduce an original hybrid quantum-classical algorithm for solving systems of differential equations.
The algorithm relies on a spectral method, which involves encoding the solution functions in the amplitudes of the quantum states generated by different parametrized circuits.
- Score: 0.0
- License: http://creativecommons.org/licenses/by-nc-nd/4.0/
- Abstract: In this article, we introduce an original hybrid quantum-classical algorithm based on a variational quantum algorithm for solving systems of differential equations. The algorithm relies on a spectral method, which involves encoding the solution functions in the amplitudes of the quantum states generated by different parametrized circuits and transforms the task of solving the differential equations into an optimization problem. We first describe the principle of the algorithm from a theoretical point of view. We provide a detailed pseudo-code of the algorithm, on which we conduct a complexity analysis to highlight its scaling properties. We apply it to a set of examples, showcasing its applicability across diverse sets of differential equations. We discuss the advantages of our method and potential avenues for further exploration and refinement.
Related papers
- Solving Fractional Differential Equations on a Quantum Computer: A Variational Approach [0.1492582382799606]
We introduce an efficient variational hybrid quantum-classical algorithm designed for solving Caputo time-fractional partial differential equations.
Our results indicate that solution fidelity is insensitive to the fractional index and that gradient evaluation cost scales economically with the number of time steps.
arXiv Detail & Related papers (2024-06-13T02:27:16Z) - Hybrid quantum-classical and quantum-inspired classical algorithms for
solving banded circulant linear systems [0.8192907805418583]
We present an efficient algorithm based on convex optimization of combinations of quantum states to solve for banded circulant linear systems.
By decomposing banded circulant matrices into cyclic permutations, our approach produces approximate solutions to such systems with a combination of quantum states linear to $K$.
We validate our methods with classical simulations and actual IBM quantum computer implementation, showcasing their applicability for solving physical problems such as heat transfer.
arXiv Detail & Related papers (2023-09-20T16:27:16Z) - A Review on Quantum Approximate Optimization Algorithm and its Variants [47.89542334125886]
The Quantum Approximate Optimization Algorithm (QAOA) is a highly promising variational quantum algorithm that aims to solve intractable optimization problems.
This comprehensive review offers an overview of the current state of QAOA, encompassing its performance analysis in diverse scenarios.
We conduct a comparative study of selected QAOA extensions and variants, while exploring future prospects and directions for the algorithm.
arXiv Detail & Related papers (2023-06-15T15:28:12Z) - A hybrid quantum-classical algorithm for multichannel quantum scattering
of atoms and molecules [62.997667081978825]
We propose a hybrid quantum-classical algorithm for solving the Schr"odinger equation for atomic and molecular collisions.
The algorithm is based on the $S$-matrix version of the Kohn variational principle, which computes the fundamental scattering $S$-matrix.
We show how the algorithm could be scaled up to simulate collisions of large polyatomic molecules.
arXiv Detail & Related papers (2023-04-12T18:10:47Z) - Symbolic Recovery of Differential Equations: The Identifiability Problem [52.158782751264205]
Symbolic recovery of differential equations is the ambitious attempt at automating the derivation of governing equations.
We provide both necessary and sufficient conditions for a function to uniquely determine the corresponding differential equation.
We then use our results to devise numerical algorithms aiming to determine whether a function solves a differential equation uniquely.
arXiv Detail & Related papers (2022-10-15T17:32:49Z) - Alternatives to a nonhomogeneous partial differential equation quantum
algorithm [52.77024349608834]
We propose a quantum algorithm for solving nonhomogeneous linear partial differential equations of the form $Apsi(textbfr)=f(textbfr)$.
These achievements enable easier experimental implementation of the quantum algorithm based on nowadays technology.
arXiv Detail & Related papers (2022-05-11T14:29:39Z) - Twisted hybrid algorithms for combinatorial optimization [68.8204255655161]
Proposed hybrid algorithms encode a cost function into a problem Hamiltonian and optimize its energy by varying over a set of states with low circuit complexity.
We show that for levels $p=2,ldots, 6$, the level $p$ can be reduced by one while roughly maintaining the expected approximation ratio.
arXiv Detail & Related papers (2022-03-01T19:47:16Z) - QBoost for regression problems: solving partial differential equations [0.0]
The hybrid algorithm is capable of finding a solution to a partial differential equation with good precision and favorable scaling in the required number of qubits.
The classical part is composed by training several regressors, capable of solving a partial differential equation using machine learning.
The quantum part consists of adapting the QBoost algorithm to solve regression problems.
arXiv Detail & Related papers (2021-08-30T16:13:04Z) - Quadratic Unconstrained Binary Optimisation via Quantum-Inspired
Annealing [58.720142291102135]
We present a classical algorithm to find approximate solutions to instances of quadratic unconstrained binary optimisation.
We benchmark our approach for large scale problem instances with tuneable hardness and planted solutions.
arXiv Detail & Related papers (2021-08-18T09:26:17Z) - Quantum Algorithms for Solving Ordinary Differential Equations via
Classical Integration Methods [1.802439717192088]
We explore utilizing quantum computers for the purpose of solving differential equations.
We devise and simulate corresponding digital quantum circuits, and implement and run a 6$mathrmth$ order Gauss-Legendre collocation method.
As promising future scenario, the digital arithmetic method could be employed as an "oracle" within quantum search algorithms for inverse problems.
arXiv Detail & Related papers (2020-12-17T09:49:35Z) - Solving nonlinear differential equations with differentiable quantum
circuits [21.24186888129542]
We propose a quantum algorithm to solve systems of nonlinear differential equations.
We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits.
We show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space.
arXiv Detail & Related papers (2020-11-20T13:21:11Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.