Quantum finite element algorithm for solving Euler–Bernoulli and heat transfer PDEs with Dirichlet, Neumann, and Robin boundary conditions

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.