An Implementation of the Finite Element Method in Hybrid Classical/Quantum Computers

• URL: http://arxiv.org/abs/2411.09038v2
• Date: Mon, 03 Feb 2025 21:03:56 GMT
• Title: An Implementation of the Finite Element Method in Hybrid Classical/Quantum Computers
• Authors: Abhishek Arora, Benjamin M. Ward, Caglar Oskay,
• Abstract summary: This manuscript presents the Quantum Finite Element Method (Q-FEM) developed for use in noisy intermediate-scale quantum computers. Q-FEM keeps the structure of the finite element discretization intact allowing for the use of variable element lengths and material coefficients in FEM discretization.
• Score: 0.0
• License:
• Abstract: This manuscript presents the Quantum Finite Element Method (Q-FEM) developed for use in noisy intermediate-scale quantum (NISQ) computers and employs the variational quantum linear solver (VQLS) algorithm. The proposed method leverages the classical FEM procedure to perform the unitary decomposition of the stiffness matrix and employs generator functions to design explicit quantum circuits corresponding to the unitaries. Q-FEM keeps the structure of the finite element discretization intact allowing for the use of variable element lengths and material coefficients in FEM discretization. The proposed method is tested on a steady-state heat equation discretized using linear and quadratic shape functions. Numerical verification studies are performed on the IBM QISKIT simulator and it is demonstrated that Q-FEM is effective in converging to the correct solution for a variety of problems and model discretizations, including with different element lengths, variable coefficients, and different boundary conditions. The formalism developed herein is general and can be extended to problems with higher dimensions. However, numerical examples also demonstrate that the number of parameters for the variational ansatz scale exponentially with the number of qubits to increase the odds of convergence. Moreover, the deterioration of system conditioning with problem size results in barren plateaus and convergence difficulties.

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)
• Demonstration of Scalability and Accuracy of Variational Quantum Linear Solver for Computational Fluid Dynamics [0.0]
  This paper presents an exploration of quantum methodologies aimed at achieving high accuracy in solving such a large system of equations. We consider the 2D, transient, incompressible, viscous, non-linear coupled Burgers equation as a test problem. Our findings demonstrate that our quantum methods yield results comparable in accuracy to traditional approaches.
  arXiv  Detail & Related papers  (2024-09-05T04:42:24Z)
• Quantum Circuits for the heat equation with physical boundary conditions via Schrodingerisation [33.76659022113328]
  This paper explores the explicit design of quantum circuits for quantum simulation of partial differential equations (PDEs) with physical boundary conditions. We present two methods for handling the inhomogeneous terms arising from time-dependent physical boundary conditions. We then apply the quantum simulation technique from [CJL23] to transform the resulting non-autonomous system to an autonomous system in one higher dimension.
  arXiv  Detail & Related papers  (2024-07-22T03:52:14Z)
• Evaluation of phase shifts for non-relativistic elastic scattering using quantum computers [39.58317527488534]
  This work reports the development of an algorithm that makes it possible to obtain phase shifts for generic non-relativistic elastic scattering processes on a quantum computer.
  arXiv  Detail & Related papers  (2024-07-04T21:11:05Z)
• Characterizing randomness in parameterized quantum circuits through expressibility and average entanglement [39.58317527488534]
  Quantum Circuits (PQCs) are still not fully understood outside the scope of their principal application. We analyse the generation of random states in PQCs under restrictions on the qubits connectivities. We place a connection between how steep is the increase on the uniformity of the distribution of the generated states and the generation of entanglement.
  arXiv  Detail & Related papers  (2024-05-03T17:32:55Z)
• Boundary Treatment for Variational Quantum Simulations of Partial Differential Equations on Quantum Computers [1.6318838452579472]
  The paper presents a variational quantum algorithm to solve initial-boundary value problems described by partial differential equations. The approach uses classical/quantum hardware that is well suited for quantum computers of the current noisy intermediate-scale quantum era.
  arXiv  Detail & Related papers  (2024-02-28T18:19:33Z)
• Efficient estimation of trainability for variational quantum circuits [43.028111013960206]
  We find an efficient method to compute the cost function and its variance for a wide class of variational quantum circuits. This method can be used to certify trainability for variational quantum circuits and explore design strategies that can overcome the barren plateau problem.
  arXiv  Detail & Related papers  (2023-02-09T14:05:18Z)
• Analyzing Prospects for Quantum Advantage in Topological Data Analysis [35.423446067065576]
  We analyze and optimize an improved quantum algorithm for topological data analysis. We show that super-quadratic quantum speedups are only possible when targeting a multiplicative error approximation. We argue that quantum circuits with tens of billions of Toffoli can solve seemingly classically intractable instances.
  arXiv  Detail & Related papers  (2022-09-27T17:56:15Z)
• 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)
• Say NO to Optimization: A Non-Orthogonal Quantum Eigensolver [0.0]
  A balanced description of both static and dynamic correlations in electronic systems with nearly degenerate low-lying states presents a challenge for multi-configurational methods on classical computers. We present here a quantum algorithm utilizing the action of correlating cluster operators to provide high-quality wavefunction ans"atze.
  arXiv  Detail & Related papers  (2022-05-18T16:20:36Z)
• Q-FW: A Hybrid Classical-Quantum Frank-Wolfe for Quadratic Binary Optimization [44.96576908957141]
  We present a hybrid classical-quantum framework based on the Frank-Wolfe algorithm, Q-FW, for solving quadratic, linear iterations problems on quantum computers.
  arXiv  Detail & Related papers  (2022-03-23T18:00:03Z)

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.