Quantum algorithm for the advection-diffusion equation and the Koopman-von Neumann approach to nonlinear dynamical systems

• URL: http://arxiv.org/abs/2410.03985v1
• Date: Fri, 4 Oct 2024 23:58:12 GMT
• Title: Quantum algorithm for the advection-diffusion equation and the Koopman-von Neumann approach to nonlinear dynamical systems
• Authors: Ivan Novikau, Ilon Joseph,
• Abstract summary: We propose an explicit algorithm to simulate both the advection-diffusion equation and a nonunitary discretized version of the Koopman-von Neumann formulation of nonlinear dynamics. The proposed algorithm is universal and can be used for modeling a broad class of linear and nonlinear differential equations.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose an explicit algorithm based on the Linear Combination of Hamiltonian Simulations technique to simulate both the advection-diffusion equation and a nonunitary discretized version of the Koopman-von Neumann formulation of nonlinear dynamics. By including dissipation into the model, through an upwind discretization of the advection operator, we avoid spurious parasitic oscillations which usually accompany standard finite difference discretizations of the advection equation. In contrast to prior works on quantum simulation of nonlinear problems, we explain in detail how different components of the algorithm can be implemented by using the Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) methods. In addition, we discuss the general method for implementing the block-encoding (BE) required for QSP and QSVT circuits and provide explicit implementations of the BE oracles tailored to our specific test cases. We simulate the resulting circuit on a digital emulator of quantum fault-tolerant computers and investigate its complexity and success probability. The proposed algorithm is universal and can be used for modeling a broad class of linear and nonlinear differential equations including the KvN and Carleman embeddings of nonlinear systems, the semiclassical Koopman-van Hove (KvH) equation, as well as the advection and Liouville equations.

Related papers

• Towards Variational Quantum Algorithms for generalized linear and nonlinear transport phenomena [0.0]
  This article proposes a Variational Quantum Algorithm (VQA) to solve linear and nonlinear thermofluid dynamic transport equations. The hybrid classical-quantum framework is applied to problems governed by the heat, wave, and Burgers' equation in combination with different engineering boundary conditions.
  arXiv  Detail & Related papers  (2024-11-22T13:39:49Z)
• Quantum Algorithms for Nonlinear Dynamics: Revisiting Carleman Linearization with No Dissipative Conditions [0.7373617024876725]
  We explore the embedding of nonlinear dynamical systems into linear ordinary differential equations (ODEs) via the Carleman linearization method. Our analysis extends these findings by exploring error bounds beyond the traditional dissipative condition. We prove how this resonance condition leads to a linear convergence with respect to the truncation level $N$ in Carleman linearization.
  arXiv  Detail & Related papers  (2024-05-21T12:09:34Z)
• Improving Pseudo-Time Stepping Convergence for CFD Simulations With Neural Networks [44.99833362998488]
  Navier-Stokes equations may exhibit a highly nonlinear behavior. The system of nonlinear equations resulting from the discretization of the Navier-Stokes equations can be solved using nonlinear iteration methods, such as Newton's method. In this paper, pseudo-transient continuation is employed in order to improve nonlinear convergence.
  arXiv  Detail & Related papers  (2023-10-10T15:45:19Z)
• Solving Systems of Linear Equations: HHL from a Tensor Networks Perspective [39.58317527488534]
  We present an algorithm for solving systems of linear equations based on the HHL algorithm with a novel qudits methodology. We perform a quantum-inspired version on tensor networks, taking advantage of their ability to perform non-unitary operations such as projection.
  arXiv  Detail & Related papers  (2023-09-11T08:18:41Z)
• 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)
• Linear combination of Hamiltonian simulation for nonunitary dynamics with optimal state preparation cost [8.181184006712785]
  We propose a simple method for simulating a general class of non-unitary dynamics as a linear combination of Hamiltonian simulation problems. We also demonstrate an application for open quantum dynamics simulation using the complex absorbing potential method with near-optimal dependence on all parameters.
  arXiv  Detail & Related papers  (2023-03-02T07:37:54Z)
• 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)
• Exact solutions of interacting dissipative systems via weak symmetries [77.34726150561087]
  We analytically diagonalize the Liouvillian of a class Markovian dissipative systems with arbitrary strong interactions or nonlinearity. This enables an exact description of the full dynamics and dissipative spectrum. Our method is applicable to a variety of other systems, and could provide a powerful new tool for the study of complex driven-dissipative quantum systems.
  arXiv  Detail & Related papers  (2021-09-27T17:45:42Z)
• Hybridized Methods for Quantum Simulation in the Interaction Picture [69.02115180674885]
  We provide a framework that allows different simulation methods to be hybridized and thereby improve performance for interaction picture simulations. Physical applications of these hybridized methods yield a gate complexity scaling as $log2 Lambda$ in the electric cutoff. For the general problem of Hamiltonian simulation subject to dynamical constraints, these methods yield a query complexity independent of the penalty parameter $lambda$ used to impose an energy cost.
  arXiv  Detail & Related papers  (2021-09-07T20:01:22Z)
• Designing Kerr Interactions for Quantum Information Processing via Counterrotating Terms of Asymmetric Josephson-Junction Loops [68.8204255655161]
  static cavity nonlinearities typically limit the performance of bosonic quantum error-correcting codes. Treating the nonlinearity as a perturbation, we derive effective Hamiltonians using the Schrieffer-Wolff transformation. Results show that a cubic interaction allows to increase the effective rates of both linear and nonlinear operations.
  arXiv  Detail & Related papers  (2021-07-14T15:11:05Z)
• Fixed Depth Hamiltonian Simulation via Cartan Decomposition [59.20417091220753]
  We present a constructive algorithm for generating quantum circuits with time-independent depth. We highlight our algorithm for special classes of models, including Anderson localization in one dimensional transverse field XY model. In addition to providing exact circuits for a broad set of spin and fermionic models, our algorithm provides broad analytic and numerical insight into optimal Hamiltonian simulations.
  arXiv  Detail & Related papers  (2021-04-01T19:06:00Z)

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.