Quantum Iterative Methods for Solving Differential Equations with Application to Computational Fluid Dynamics

• URL: http://arxiv.org/abs/2404.08605v1
• Date: Fri, 12 Apr 2024 17:08:27 GMT
• Title: Quantum Iterative Methods for Solving Differential Equations with Application to Computational Fluid Dynamics
• Authors: Chelsea A. Williams, Antonio A. Gentile, Vincent E. Elfving, Daniel Berger, Oleksandr Kyriienko,
• Abstract summary: We propose quantum methods for solving differential equations based on a gradual improvement of the solution via an iterative process. We benchmark the approach on paradigmatic fluid dynamics problems.
• Score: 14.379311972506791
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose quantum methods for solving differential equations that are based on a gradual improvement of the solution via an iterative process, and are targeted at applications in fluid dynamics. First, we implement the Jacobi iteration on a quantum register that utilizes a linear combination of unitaries (LCU) approach to store the trajectory information. Second, we extend quantum methods to Gauss-Seidel iterative methods. Additionally, we propose a quantum-suitable resolvent decomposition based on the Woodbury identity. From a technical perspective, we develop and utilize tools for the block encoding of specific matrices as well as their multiplication. We benchmark the approach on paradigmatic fluid dynamics problems. Our results stress that instead of inverting large matrices, one can program quantum computers to perform multigrid-type computations and leverage corresponding advances in scientific computing.

Related papers

• Double-Logarithmic Depth Block-Encodings of Simple Finite Difference Method's Matrices [0.0]
  Solving differential equations is one of the most computationally expensive problems in classical computing. Despite recent progress made in the field of quantum computing and quantum algorithms, its end-to-end application towards practical realization still remains unattainable.
  arXiv  Detail & Related papers  (2024-10-07T17:44:30Z)
• 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)
• 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)
• Improving the convergence of an iterative algorithm for solving arbitrary linear equation systems using classical or quantum binary optimization [39.58317527488534]
  We propose a novel method for solving linear systems. We transform the linear system into a binary optimization problem, drawing inspiration from the geometry of the original problem. We demonstrate that by leveraging partial knowledge of the problem's intrinsic geometry, we can decompose the original problem into smaller, independent sub-problems.
  arXiv  Detail & Related papers  (2023-09-18T16:51:03Z)
• 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)
• 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)
• Quantum algorithms for matrix operations and linear systems of equations [65.62256987706128]
  We propose quantum algorithms for matrix operations using the "Sender-Receiver" model. These quantum protocols can be used as subroutines in other quantum schemes.
  arXiv  Detail & Related papers  (2022-02-10T08:12:20Z)
• 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)
• 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)
• Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods [1.802439717192088]
  We explore utilizing quantum computers for the purpose of solving differential equations. We devise and simulate corresponding digital quantum circuits, and implement and run a 6$mathrmth$ order Gauss-Legendre collocation method. As promising future scenario, the digital arithmetic method could be employed as an "oracle" within quantum search algorithms for inverse problems.
  arXiv  Detail & Related papers  (2020-12-17T09:49:35Z)
• Fast inversion, preconditioned quantum linear system solvers, and fast evaluation of matrix functions [4.327821619134312]
  We introduce a quantum primitive called fast inversion, which can be used as a preconditioner for solving quantum linear systems. We demonstrate the application of preconditioned linear system solvers for computing single-particle Green's functions of quantum many-body systems.
  arXiv  Detail & Related papers  (2020-08-30T23:24:58Z)

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.