Quantum algorithm for time-dependent differential equations using Dyson series
- URL: http://arxiv.org/abs/2212.03544v2
- Date: Tue, 4 Jun 2024 08:30:27 GMT
- Title: Quantum algorithm for time-dependent differential equations using Dyson series
- Authors: Dominic W. Berry, Pedro C. S. Costa,
- Abstract summary: We provide a quantum algorithm for solving time-dependent linear differential equations with logarithmic dependence of the complexity on the error and derivative.
Our method is to encode the Dyson series in a system of linear equations, then solve via the optimal quantum linear equation solver.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Time-dependent linear differential equations are a common type of problem that needs to be solved in classical physics. Here we provide a quantum algorithm for solving time-dependent linear differential equations with logarithmic dependence of the complexity on the error and derivative. As usual, there is an exponential improvement over classical approaches in the scaling of the complexity with the dimension, with the caveat that the solution is encoded in the amplitudes of a quantum state. Our method is to encode the Dyson series in a system of linear equations, then solve via the optimal quantum linear equation solver. Our method also provides a simplified approach in the case of time-independent differential equations.
Related papers
- A hybrid quantum solver for the Lorenz system [0.2770822269241974]
We develop a hybrid classical-quantum method for solving the Lorenz system.
We use the forward Euler method to discretize the system in time, transforming it into a system of equations.
We present numerical results comparing the hybrid method with the classical approach for solving the Lorenz system.
arXiv Detail & Related papers (2024-10-20T15:20:28Z) - 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) - Nonlinear dynamics as a ground-state solution on quantum computers [39.58317527488534]
We present variational quantum algorithms (VQAs) that encode both space and time in qubit registers.
The spacetime encoding enables us to obtain the entire time evolution from a single ground-state computation.
arXiv Detail & Related papers (2024-03-25T14:06:18Z) - 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) - Discovering ordinary differential equations that govern time-series [65.07437364102931]
We propose a transformer-based sequence-to-sequence model that recovers scalar autonomous ordinary differential equations (ODEs) in symbolic form from time-series data of a single observed solution of the ODE.
Our method is efficiently scalable: after one-time pretraining on a large set of ODEs, we can infer the governing laws of a new observed solution in a few forward passes of the model.
arXiv Detail & Related papers (2022-11-05T07:07:58Z) - 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) - Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural
Networks on Coupled Ordinary Differential Equations [64.78260098263489]
In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs)
We show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity increases.
We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.
arXiv Detail & Related papers (2022-10-14T15:01:32Z) - 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) - Time-marching based quantum solvers for time-dependent linear
differential equations [3.1952399274829775]
The time-marching strategy is a natural strategy for solving time-dependent differential equations on classical computers.
We show that a time-marching based quantum solver can suffer from exponentially vanishing success probability with respect to the number of time steps.
This provides a path of designing quantum differential equation solvers that is alternative to those based on quantum linear systems algorithms.
arXiv Detail & Related papers (2022-08-14T23:49:19Z) - Physics Informed RNN-DCT Networks for Time-Dependent Partial
Differential Equations [62.81701992551728]
We present a physics-informed framework for solving time-dependent partial differential equations.
Our model utilizes discrete cosine transforms to encode spatial and recurrent neural networks.
We show experimental results on the Taylor-Green vortex solution to the Navier-Stokes equations.
arXiv Detail & Related papers (2022-02-24T20:46:52Z) - Quantum algorithm for nonlinear differential equations [12.386348820609626]
We present a quantum algorithm for the solution of nonlinear differential equations.
Potential applications include the Navier-Stokes equation, plasma hydrodynamics, epidemiology, and more.
arXiv Detail & Related papers (2020-11-12T18:42:02Z)
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.