This arXiv preprint (not yet peer-reviewed) from May 2026 presents end-to-end circuit implementations for solving the advection, wave, and Poisson equations in the frequency domain using the quantum Fourier transform. Researchers compiled circuits with two approaches: a first-order approximation yielding shallow but error-uncontrolled gates, and quantum signal processing (QSP) polynomials that allow tunable accuracy at the cost of deeper circuits. Numerical simulations validated both, while IBMQ hardware runs demonstrated that QSP versions maintained solution fidelity under realistic noise for homogeneous cases; the team further extended the method to non-homogeneous Dirichlet boundaries and tested it on a plasma-physics-derived Poisson source term. Methodology relied on standard gate decompositions without error correction, limiting scale to small grids. The work signals a concrete pivot from high-level quantum algorithms toward executable near-term scientific computing, yet understates calibration overheads and does not benchmark against optimized classical spectral methods on equivalent problem sizes. Related analyses appear in Low et al. (2017) on QSP foundations and in recent IBM studies of variational PDE solvers, which together highlight that hardware noise still dominates before any asymptotic quantum advantage emerges. The original coverage misses how QSP's tunable error directly trades against decoherence time, a practical constraint that may delay utility until logical qubits arrive.