Abstract This paper introduces a quantum-enhanced finite element method (FEM) designed for noisy intermediate-scale quantum (NISQ) devices, leveraging variational quantum algorithms (VQAs) to solve engineering partial differential equations. We demonstrate the framework by solving the Euler–Bernoulli beam and the NAFEMS T4 heat transfer problems, which involve Dirichlet, Neumann, and Robin boundary conditions. A key innovation is a ‘set-to-zero’ strategy that incorporates boundary conditions through a correction matrix, K bc , allowing for flexible imposition at any node without domain decomposition. The global stiffness matrix is decomposed into a constant number of Pauli terms, O (1), using the method by Sato et al while boundary terms are handled with a sublinearly scaling partial Pauli measurement technique. The algorithm achieves logarithmic qubit scaling ( n = ⌈ log 2 N ⌉ qubits for N degrees of freedom(DOF)) and employs shallow, hardware-efficient circuits with empirically trainable depth for small-scale systems. Validation on the Qiskit statevector simulator shows high accuracy. For the Euler–Bernoulli beam problem with 4 to 64 DOF, the algorithm achieves relative errors of 0.5%–1.5% and fidelities of 0.998–0.999. For the NAFEMS T4 heat transfer benchmark, a 5.4% relative error is observed. The VQA converges robustly within 77–350 iterations, though barren plateaus are a known challenge for scaling to larger systems. This work establishes a scalable framework for quantum FEM, offering a significant memory advantage over classical methods and advancing the potential for quantum-enhanced engineering simulations.
Abstract This paper introduces a quantum-enhanced finite element method (FEM) designed for noisy intermediate-scale quantum (NISQ) devices, leveraging variational quantum algorithms (VQAs) to solve engineering partial differential equations. We demonstrate the framework by solving the Euler–Bernoulli beam and the NAFEMS T4 heat transfer problems, which involve Dirichlet, Neumann, and Robin boundary conditions. A key innovation is a ‘set-to-zero’ strategy that incorporates boundary conditions through a correction matrix, K bc , allowing for flexible imposition at any node without domain decomposition. The global stiffness matrix is decomposed into a constant number of Pauli terms, O (1), using the method by Sato et al while boundary terms are handled with a sublinearly scaling partial Pauli measurement technique. The algorithm achieves logarithmic qubit scaling ( n = ⌈ log 2 N ⌉ qubits for N degrees of freedom(DOF)) and employs shallow, hardware-efficient circuits with empirically trainable depth for small-scale systems. Validation on the Qiskit statevector simulator shows high accuracy. For the Euler–Bernoulli beam problem with 4 to 64 DOF, the algorithm achieves relative errors of 0.5%–1.5% and fidelities of 0.998–0.999. For the NAFEMS T4 heat transfer benchmark, a 5.4% relative error is observed. The VQA converges robustly within 77–350 iterations, though barren plateaus are a known challenge for scaling to larger systems. This work establishes a scalable framework for quantum FEM, offering a significant memory advantage over classical methods and advancing the potential for quantum-enhanced engineering simulations.
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, and increases the odds of convergence. Moreover, the deterioration of system conditioning with problem size results in barren plateaus and convergence difficulties.
We proposed a general quantum-computing-based algorithm that harnesses the exponential power of noisy intermediate-scale quantum (NISQ) devices in solving partial differential equations (PDE). This variational quantum eigensolver (VQE)-inspired approach transcends previous idealized model demonstrations constrained by strict and simplistic boundary conditions. It enables the imposition of arbitrary boundary conditions, significantly expanding its potential and adaptability for real-world applications, achieving this "from ad-hoc to systematic" concept. We have implemented this method using the fourth-order PDE (the Euler-Bernoulli beam) as example and showcased its effectiveness with four different boundary conditions. This framework enables expectation evaluations independent of problem size, harnessing the exponentially growing state space inherent in quantum computing, resulting in exceptional scalability. This method paves the way for applying quantum computing to practical engineering applications.
The paper presents a variational quantum algorithm to solve initial–boundary value problems described by second-order partial differential equations. The approach uses hybrid classical/quantum framework that is well suited for quantum computers of the current noisy intermediate-scale quantum era. The partial differential equation is initially translated into an optimal control problem with a modular control-to-state operator (ansatz). The objective function and its derivatives required by the optimizer can efficiently be evaluated on a quantum computer by measuring an ancilla qubit, while the optimization procedure employs classical hardware. The focal aspect of the study is the treatment of boundary conditions, which is tailored to the properties of the quantum hardware using a correction technique. For this purpose, the boundary conditions and the discretized terms of the partial differential equation are decomposed into a sequence of unitary operations and subsequently compiled into quantum gates. The accuracy and gate complexity of the approach are assessed for second-order partial differential equations by classically emulating the quantum hardware. The examples include steady and unsteady diffusive transport equations for a scalar property in combination with various Dirichlet, Neumann, or Robin conditions. The results of this flexible approach display a robust behavior and a strong predictive accuracy in combination with a remarkable polylog complexity scaling in the number of qubits of the involved quantum circuits. Remaining challenges refer to adaptive ansatz strategies that speed up the optimization procedure.
The finite element method (FEM) is a cornerstone numerical technique for solving partial differential equations (PDEs). Here, we present Qu-FEM, a fault-tolerant era quantum algorithm for the finite element method. In contrast to other quantum PDE solvers, Qu-FEM preserves the geometric flexibility of FEM by introducing two new primitives, the unit of interaction and the local-to-global indicator matrix, which enable the assembly of global finite element arrays with a constant-size linear combination of unitaries. We study the modified Poisson equation as an elliptic problem of interest, and provide explicit circuits for Qu-FEM in Cartesian domains. For problems with constant coefficients, our algorithm admits block-encodings of global arrays that require only O~(d2p2n) Clifford+T gates for d-dimensional, order-p tensor product elements on grids with 2n degrees of freedom in each dimension, where n is the number of qubits representing the N=2n discrete grid points. For problems with spatially varying coefficients, we perform numerical integration directly on the quantum computer to assemble global arrays and force vectors. Dirichlet boundary conditions are enforced via the method of Lagrange multipliers, eliminating the need to modify the block-encodings that emerge from the assembly procedure. This work presents a framework for extending the geometric flexibility of quantum PDE solvers while preserving the possibility of a quantum advantage.
Partial differential equations (PDEs) play a central role in scientific and engineering analysis, with applications spanning fluid dynamics, heat and mass transfer, electromagnetism, quantum mechanics, and financial modeling, where they are used to describe diffusion processes, wave propagation, and the evolution of complex systems over space and time. Solving multidimensional partial differential equations (PDEs) is a computationally challenging problem, even for the most advanced classical systems. Over the past decade, quantum computing has attracted significant interest as a potential approach for solving complex computational problems, including multidimensional PDEs. Although a variety of approaches have been proposed for solving PDEs, most of the existing techniques are based on variational quantum algorithms (VQAs). Despite being promising, these VQA-based approaches suffer from low accuracy, long execution times, and limited scalability. In this work, we propose a scalable and efficient quantum algorithm for solving multidimensional PDEs. Our algorithm has two variants. One variant is based on the finite difference method (FDM), classical-to-quantum (C2Q) encoding, and numerical instantiation, whereas the other is based on FDM, C2Q, and column-by-column decomposition (CCD). We have also evaluated our algorithm using several practical case studies; namely, Poisson, heat, Black–Scholes, and Navier–Stokes equations. The results show that our proposed approach achieves higher accuracy, greater scalability, and faster execution time than the VQA-based approaches. We validated these findings on both noise-free and noisy simulators, as well as on a hardware emulator and real IBM quantum hardware.
Finite-element methods are industry standards for finding numerical solutions to partial differential equations. However, the application scale remains pivotal to the practical use of these methods, even for modern-day supercomputers. Large, multi-scale applications, for example, can be limited by their requirement of prohibitively large linear system solutions. It is therefore worthwhile to investigate whether near-term quantum algorithms have the potential for offering any kind of advantage over classical linear solvers. In this study, we investigate the recently proposed variational quantum linear solver (VQLS) for discrete solutions to partial differential equations. This method was found to scale polylogarithmically with the linear system size, and the method can be implemented using shallow quantum circuits on noisy intermediate-scale quantum (NISQ) computers. Herein, we utilize the hybrid VQLS to solve both the steady Poisson equation and the time-dependent heat and wave equations.