Solving nonlinear differential equations on Quantum Computers: A
Fokker-Planck approach
- URL: http://arxiv.org/abs/2401.13500v1
- Date: Wed, 24 Jan 2024 14:48:55 GMT
- Title: Solving nonlinear differential equations on Quantum Computers: A
Fokker-Planck approach
- Authors: Felix Tennie and Luca Magri
- Abstract summary: We propose to transform a nonlinear dynamical system into a linear system, which we integrate with quantum algorithms.
Key to the method is the Fokker-Planck equation, which is a non-normal partial differential equation.
We emulate the integration of nonlinear systems with the proposed quantum solvers, and compare the output with the benchmark solutions of classical equations.
- Score: 5.0401589279256065
- License: http://creativecommons.org/licenses/by-nc-nd/4.0/
- Abstract: For quantum computers to become useful tools to physicists, engineers and
computational scientists, quantum algorithms for solving nonlinear differential
equations need to be developed. Despite recent advances, the quest for a solver
that can integrate nonlinear dynamical systems with a quantum advantage, whilst
being realisable on available (or near-term) quantum hardware, is an open
challenge. In this paper, we propose to transform a nonlinear dynamical system
into a linear system, which we integrate with quantum algorithms. Key to the
method is the Fokker-Planck equation, which is a non-normal partial
differential equation. Three integration strategies are proposed: (i)
Forward-Euler stepping by unitary block encoding; (ii) Schroedingerisation, and
(iii) Forward-Euler stepping by linear addition of unitaries. We emulate the
integration of prototypical nonlinear systems with the proposed quantum
solvers, and compare the output with the benchmark solutions of classical
integrators. We find that classical and quantum outputs are in good agreement.
This paper opens opportunities for solving nonlinear differential equations
with quantum algorithms.
Related papers
- Quantum Computing for nonlinear differential equations and turbulence [6.974741712647655]
We discuss progress in the development of both quantum algorithms for nonlinear equations and quantum hardware.
We propose pairings between quantum algorithms for nonlinear equations and quantum hardware concepts.
arXiv Detail & Related papers (2024-06-07T10:52:08Z) - 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) - Correspondence between open bosonic systems and stochastic differential
equations [77.34726150561087]
We show that there can also be an exact correspondence at finite $n$ when the bosonic system is generalized to include interactions with the environment.
A particular system with the form of a discrete nonlinear Schr"odinger equation is analyzed in more detail.
arXiv Detail & Related papers (2023-02-03T19:17:37Z) - Quantum Kernel Methods for Solving Differential Equations [21.24186888129542]
We propose several approaches for solving differential equations (DEs) with quantum kernel methods.
We compose quantum models as weighted sums of kernel functions, where variables are encoded using feature maps and model derivatives are represented.
arXiv Detail & Related papers (2022-03-16T18:56:35Z) - A Hybrid Quantum-Classical Algorithm for Robust Fitting [47.42391857319388]
We propose a hybrid quantum-classical algorithm for robust fitting.
Our core contribution is a novel robust fitting formulation that solves a sequence of integer programs.
We present results obtained using an actual quantum computer.
arXiv Detail & Related papers (2022-01-25T05:59:24Z) - Quantum Model-Discovery [19.90246111091863]
Quantum algorithms for solving differential equations have shown a provable advantage in the fault-tolerant quantum computing regime.
We extend the applicability of near-term quantum computers to more general scientific machine learning tasks.
Our results show a promising path to Quantum Model Discovery (QMoD) on the interface between classical and quantum machine learning approaches.
arXiv Detail & Related papers (2021-11-11T18:45:52Z) - Adiabatic Quantum Graph Matching with Permutation Matrix Constraints [75.88678895180189]
Matching problems on 3D shapes and images are frequently formulated as quadratic assignment problems (QAPs) with permutation matrix constraints, which are NP-hard.
We propose several reformulations of QAPs as unconstrained problems suitable for efficient execution on quantum hardware.
The proposed algorithm has the potential to scale to higher dimensions on future quantum computing architectures.
arXiv Detail & Related papers (2021-07-08T17:59:55Z) - Linear embedding of nonlinear dynamical systems and prospects for
efficient quantum algorithms [74.17312533172291]
We describe a method for mapping any finite nonlinear dynamical system to an infinite linear dynamical system (embedding)
We then explore an approach for approximating the resulting infinite linear system with finite linear systems (truncation)
arXiv Detail & Related papers (2020-12-12T00:01:10Z) - Generalized Quantum Assisted Simulator [0.0]
We introduce the notion of the hybrid density matrix, which allows us to disentangle the different steps of our algorithm.
Our algorithm has potential applications in solving the Navier-Stokes equation, plasma hydrodynamics, quantum Boltzmann training, quantum signal processing and linear systems.
arXiv Detail & Related papers (2020-11-30T12:40:17Z) - Electronic structure with direct diagonalization on a D-Wave quantum
annealer [62.997667081978825]
This work implements the general Quantum Annealer Eigensolver (QAE) algorithm to solve the molecular electronic Hamiltonian eigenvalue-eigenvector problem on a D-Wave 2000Q quantum annealer.
We demonstrate the use of D-Wave hardware for obtaining ground and electronically excited states across a variety of small molecular systems.
arXiv Detail & Related papers (2020-09-02T22:46:47Z) - Simulating nonnative cubic interactions on noisy quantum machines [65.38483184536494]
We show that quantum processors can be programmed to efficiently simulate dynamics that are not native to the hardware.
On noisy devices without error correction, we show that simulation results are significantly improved when the quantum program is compiled using modular gates.
arXiv Detail & Related papers (2020-04-15T05:16:24Z)
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.