Quantum Algorithm For Solving Nonlinear Algebraic Equations
- URL: http://arxiv.org/abs/2404.03810v1
- Date: Thu, 4 Apr 2024 21:20:56 GMT
- Title: Quantum Algorithm For Solving Nonlinear Algebraic Equations
- Authors: Nhat A. Nghiem, Tzu-Chieh Wei,
- Abstract summary: We build upon the classical Newton method and some recent works on quantum algorithm plus block encoding from quantum singular value transformation.
A detailed analysis are then carried out to reveal that our method polylogarithmic time in relative to the number of variables.
In particular, we show that our method can be modified with little effort to deal with various types, thus implying the generality of our approach.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Nonlinear equations are challenging to solve due to their inherently nonlinear nature. As analytical solutions typically do not exist, numerical methods have been developed to tackle their solutions. In this article, we give a quantum algorithm for solving a system of nonlinear algebraic equations, in which each equation is a multivariate polynomial of known coefficients. Building upon the classical Newton method and some recent works on quantum algorithm plus block encoding from the quantum singular value transformation, we show how to invert the Jacobian matrix to execute Newton's iterative method for solving nonlinear equations, where each contributing equation is a homogeneous polynomial of an even degree. A detailed analysis are then carried out to reveal that our method achieves polylogarithmic time in relative to the number of variables. Furthermore, the number of required qubits is logarithmic in the number of variables. In particular, we also show that our method can be modified with little effort to deal with polynomial of various types, thus implying the generality of our approach. Some examples coming from physical context, such as Gross-Pitaevski equation and Lotka-Volterra equations, involving nonlinear partial differential equations are provided to motivate the potential application, with a description on how to extend our algorithm with even less effort in such a scenario. Our work thus marks a further important step towards quantum advantage in nonlinear science, enabled by the framework of quantum singular value transformation.
Related papers
- MultiSTOP: Solving Functional Equations with Reinforcement Learning [56.073581097785016]
We develop MultiSTOP, a Reinforcement Learning framework for solving functional equations in physics.
This new methodology produces actual numerical solutions instead of bounds on them.
arXiv Detail & Related papers (2024-04-23T10:51:31Z) - Solving nonlinear differential equations on Quantum Computers: A
Fokker-Planck approach [5.0401589279256065]
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.
arXiv Detail & Related papers (2024-01-24T14:48:55Z) - Quantum Variational Solving of Nonlinear and Multi-Dimensional Partial Differential Equations [1.2670268797931266]
A variational quantum algorithm for numerically solving partial differential equations was proposed by Lubasch et al.
We generalize the method to cover a broader class of nonlinear PDEs as well as multidimensional PDEs.
We show via numerical simulations that the algorithm can solve instances of the Single-Asset Black-Scholes equation.
arXiv Detail & Related papers (2023-11-02T18:29:31Z) - Quantum algorithms for linear and non-linear fractional
reaction-diffusion equations [3.409316136755434]
We investigate efficient quantum algorithms for nonlinear fractional reaction-diffusion equations with periodic boundary conditions.
We present a novel algorithm that combines the linear combination of Hamiltonian simulation technique with the interaction picture formalism.
arXiv Detail & Related papers (2023-10-29T04:48:20Z) - 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) - Depth analysis of variational quantum algorithms for heat equation [0.0]
We consider three approaches to solve the heat equation on a quantum computer.
An exponential number of Pauli products in the Hamiltonian decomposition does not allow for the quantum speed up to be achieved.
The ansatz tree approach exploits an explicit form of the matrix what makes it possible to achieve an advantage over classical algorithms.
arXiv Detail & Related papers (2022-12-23T14:46:33Z) - Time complexity analysis of quantum algorithms via linear
representations for nonlinear ordinary and partial differential equations [31.986350313948435]
We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations.
We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations.
arXiv Detail & Related papers (2022-09-18T05:50:23Z) - Automated differential equation solver based on the parametric
approximation optimization [77.34726150561087]
The article presents a method that uses an optimization algorithm to obtain a solution using the parameterized approximation.
It allows solving the wide class of equations in an automated manner without the algorithm's parameters change.
arXiv Detail & Related papers (2022-05-11T10:06:47Z) - Variational Quantum Optimization with Multi-Basis Encodings [62.72309460291971]
We introduce a new variational quantum algorithm that benefits from two innovations: multi-basis graph complexity and nonlinear activation functions.
Our results in increased optimization performance, two increase in effective landscapes and a reduction in measurement progress.
arXiv Detail & Related papers (2021-06-24T20:16:02Z) - 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) - Quantum-Inspired Algorithms from Randomized Numerical Linear Algebra [53.46106569419296]
We create classical (non-quantum) dynamic data structures supporting queries for recommender systems and least-squares regression.
We argue that the previous quantum-inspired algorithms for these problems are doing leverage or ridge-leverage score sampling in disguise.
arXiv Detail & Related papers (2020-11-09T01:13:07Z)
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.