On the Efficacy of Solving the Poisson Equation through a QPINN
By Vyaas
Physics-informed neural networks (PINNs) solve differential equations by training a neural net- work while penalizing violations of the governing equation and boundary conditions in the loss function [1]. Quantum physics-informed neural networks (QPINNs) extend this idea by embedding a parameterized quantum circuit within part of the classical network [2, 3]. This project will first test how three approaches solve the same one-dimensional Poisson prob- lem on $x∈[−1,1]$: a classical numerical method, a classical “vanilla” PINN, and a hybrid QPINN containing a quantum layer implemented with the Python library PennyLane. A fourth imple- mentation will discretize the equation and apply the Harrow–Hassidim–Lloyd (HHL) quantum linear-systems algorithm as a distinct quantum baseline [4]. The prescribed profile $u_0(x) = sin(πx)$ will be treated as a manufactured exact solution, since the time-independent Poisson equation has boundary conditions rather than an initial condition. After the 1D implementations are validated, the numerical, PINN, and QPINN pipelines will be extended to a 2D Poisson equation on $[−1,1]^2$; HHL will also be tested in 2D if the required circuit is worth it. We will measure relative L2 solution error against the exact solution, physics residual, convergence, training time, circuit resources, and variation across repeated runs. The study will alow the quantum methods to improve, match, or underperform the classical methods.
The project will isolate the effect of a quantum layer through controlled benchmarking rather than evaluating whether a hybrid model can produce a reasonable solution. Comparing the QPINN with a matched classical PINN, an exact solution, a conventional discretization, and HHL may clarify whether observed differences arise from representation, parameter count, optimization dynamics, discretization, quantum noise, or computational overhead.
Clear negative or neutral results are useful in quantum machine learning because they help identify where quantum components should be used and whether their use is justified. The benchmark, code, and analysis workflow developed here could also be reused by future QLab projects evaluating hybrid scientific-machine-learning models or quantum algorithms for computational fluid dynamics (CFD) and partial differential equations more generally.