arXiv:2604.02064v4 Announce Type: replace-cross Abstract: We prove a quantitative universal approximation theorem for noisy quantum neural networks (QNNs), with explicit non-asymptotic error bounds in terms of the number of accuracy blocks, the qubit count, the number of two-qubit gates, and hardware noise parameters. Motivated by quantitative finance, where target functions are expectations of payoffs, we first extend existing noiseless bounds to expectation functions and derive explicit constants for exponential Levy models, covering the Black-Scholes price of (truncated) European Put options. We then formulate the noisy circuit via quantum channels, obtain a fidelity-based bound for general noise models, and specialise to depolarising noise with readout error, calibrated to real hardware. Finally, we provide a detailed numerical analysis for option pricing in the Black-Scholes model, testing our results on actual noisy quantum hardware.