Design parallel and sequential quantum algorithms via spectral element method on distributed quantum architectures.

In this work, we present a solution to the critical limitation of qubit capacity in near-term quantum hardware by giving a hybrid framework that integrates the spectral element method (SEM) with distributed quantum computing. Using domain decomposition techniques, the additive and multiplicative Schwarz methods, the global problem is divided into smaller subproblems that each require significantly fewer qubits, making them suitable for current quantum devices. The HHL algorithm is applied locally to these subproblems, enabling quantum speedups without exceeding hardware constraints. Numerical results validate the approach, demonstrating a scalable and practical solution for high-accuracy simulations despite qubit limitations.