VQA_POISSON: A quantum library for solving two-dimensional poisson equations with mixed boundary conditions

Variational quantum algorithms (VQAs) have emerged as promising approaches for solving partial differential equations on near-term quantum hardware, offering hybrid optimization schemes that are robust to noise. A key challenge lies in balancing decomposition complexity and circuit depth: reducing the number of unitary terms lowers measurement cost but increases circuit depth, thereby amplifying hardware-induced noise. This work benchmarks three VQA-based solvers for the Poisson equation-Permutation Operator (PO), Sparse Decomposition (SD), and Fourier Diagonalization (FD) methods-all built on a common cost function. The FD method, in particular, leverages the Quantum Fourier Transform to diagonalize circulant Laplacians, enabling a decomposition with only one term per spatial dimension. The trade-off between gate count and noise resilience is systematically analyzed under realistic hardware constraints, including limited qubit connectivity and native gate sets. Empirical results on IBM quantum hardware indicate that while the FD method achieves superior cost scaling and requires fewer measurements, its deeper circuits are more sensitive to hardware noise. In contrast, the SD method provides better noise resilience with shallower circuits, but at the expense of a larger number of measurements. To facilitate reproducibility and further research, an open-source Qiskit-based library is provided for solving Poisson equations with mixed boundary conditions, supporting both IonQ simulators and IBM devices.