A variational quantum algorithm-based numerical method for solving
potential and Stokes flows
- URL: http://arxiv.org/abs/2303.01805v1
- Date: Fri, 3 Mar 2023 09:25:15 GMT
- Title: A variational quantum algorithm-based numerical method for solving
potential and Stokes flows
- Authors: Yangyang Liu, Zhen Chen, Chang Shu, Patrick Rebentrost, Yaguang Liu,
S. C. Chew, B. C. Khoo and Y. D. Cui
- Abstract summary: This paper presents a numerical method based on the variational quantum algorithm to solve potential and Stokes flow problems.
For the prescribed boundary conditions, the corresponding linear systems of equations can be obtained.
This work brings quantum computing to the field of computational fluid dynamics.
- Score: 5.617248827659296
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: This paper presents a numerical method based on the variational quantum
algorithm to solve potential and Stokes flow problems. In this method, the
governing equations for potential and Stokes flows can be respectively written
in the form of Laplace's equation and Stokes equations using velocity
potential, stream function and vorticity formulations. Then the finite
difference method and the generalised differential quadrature (GDQ) method are
applied to discretize the governing equations. For the prescribed boundary
conditions, the corresponding linear systems of equations can be obtained.
These linear systems are solved by using the variational quantum linear solver
(VQLS), which resolves the potential and Stokes flow problems equivalently. To
the best of authors' knowledge, this is the first study that incorporates the
GDQ method which is inherently a high-order discretization method with the VQLS
algorithm. Since the GDQ method can utilize much fewer grid points than the
finite difference method to approximate derivatives with a higher order of
accuracy, the size of the input matrix for the VQLS algorithm can be smaller.
In this way, the computational cost may be saved. The performance of the
present method is comprehensively assessed by two representative examples,
namely, the potential flow around a circular cylinder and Stokes flow in a
lid-driven cavity. Numerical results validate the applicability and accuracy of
the present VQLS-based method. Furthermore, its time complexity is evaluated by
the heuristic scaling, which demonstrates that the present method scales
efficiently in the number of qubits and the precision. This work brings quantum
computing to the field of computational fluid dynamics. By virtue of quantum
advantage over classical methods, promising advances in solving large-scale
fluid mechanics problems of engineering interest may be prompted.
Related papers
- A Catalyst Framework for the Quantum Linear System Problem via the Proximal Point Algorithm [9.804179673817574]
We propose a new quantum algorithm for the quantum linear system problem (QLSP) inspired by the classical proximal point algorithm (PPA)
Our proposed method can be viewed as a meta-algorithm that allows inverting a modified matrix via an existing texttimattQLSP_solver.
By carefully choosing the step size $eta$, the proposed algorithm can effectively precondition the linear system to mitigate the dependence on condition numbers that hindered the applicability of previous approaches.
arXiv Detail & Related papers (2024-06-19T23:15:35Z) - A two-circuit approach to reducing quantum resources for the quantum lattice Boltzmann method [41.66129197681683]
Current quantum algorithms for solving CFD problems use a single quantum circuit and, in some cases, lattice-based methods.
We introduce the a novel multiple circuits algorithm that makes use of a quantum lattice Boltzmann method (QLBM)
The problem is cast as a stream function--vorticity formulation of the 2D Navier-Stokes equations and verified and tested on a 2D lid-driven cavity flow.
arXiv Detail & Related papers (2024-01-20T15:32:01Z) - Two quantum algorithms for solving the one-dimensional
advection-diffusion equation [0.0]
Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection-diffusion equation with periodic boundary conditions.
Their accuracy and performance with increasing qubit number are compared point-by-point with each other.
arXiv Detail & Related papers (2023-12-30T21:23:15Z) - 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) - Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation [59.45669299295436]
We propose a Monte Carlo PDE solver for training unsupervised neural solvers.
We use the PDEs' probabilistic representation, which regards macroscopic phenomena as ensembles of random particles.
Our experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstrate significant improvements in accuracy and efficiency.
arXiv Detail & Related papers (2023-02-10T08:05:19Z) - 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) - Variational Quantum Solutions to the Advection-Diffusion Equation for
Applications in Fluid Dynamics [0.0]
We present one method to perform fluid dynamics calculations that takes advantage of quantum computing.
We find that reliable solutions of the equation can be obtained on even the noisy quantum computers available today.
arXiv Detail & Related papers (2022-08-24T21:29:46Z) - Application of a variational hybrid quantum-classical algorithm to heat
conduction equation [8.886131782376246]
This work applies a variational hybrid quantum-classical algorithm, namely the variational quantum linear solver (VQLS) to resolve the heat conduction equation.
Details of VQLS implementation are discussed by various test instances of linear systems.
The time complexity of the present approach is logarithmically dependent on precision epsilon and linearly dependent on the number of qubits n.
arXiv Detail & Related papers (2022-07-29T12:20:09Z) - 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) - Large-Scale Wasserstein Gradient Flows [84.73670288608025]
We introduce a scalable scheme to approximate Wasserstein gradient flows.
Our approach relies on input neural networks (ICNNs) to discretize the JKO steps.
As a result, we can sample from the measure at each step of the gradient diffusion and compute its density.
arXiv Detail & Related papers (2021-06-01T19:21:48Z) - Q-Match: Iterative Shape Matching via Quantum Annealing [64.74942589569596]
Finding shape correspondences can be formulated as an NP-hard quadratic assignment problem (QAP)
This paper proposes Q-Match, a new iterative quantum method for QAPs inspired by the alpha-expansion algorithm.
Q-Match can be applied for shape matching problems iteratively, on a subset of well-chosen correspondences, allowing us to scale to real-world problems.
arXiv Detail & Related papers (2021-05-06T17:59:38Z)
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.