Transition from Statistical to Hardware-Limited Scaling in Photonic Quantum State Reconstruction

The theoretical efficiency of classical shadow tomography is predicated on a perfect Haar-random unitary ensemble, yet this mathematical ideal remains physically unattainable in near-term hardware.

Here, we report the experimental discovery of a fundamental accuracy bound on integrated photonic processors: a ``Hardware Horizon'' where the reconstruction error undergoes a sharp phase transition.

While the error initially obeys the predicted statistical scaling $\mathcal{O}(M^{-1/2})$, it abruptly saturates at a floor determined by the spectral distortions of the realized unitary group.

By deriving a phenomenological error model, we decouple the competing mechanisms of static coherent spectral distortion and dynamic decoherence, demonstrating that this intrinsic noise floor imposes a hard bound that statistical accumulation cannot overcome.

These findings establish that the utility of shadow tomography on NISQ (noisy intermediate-scale quantum) hardware is defined by a specific scaling law involving hardware parameters, necessitating active compensation strategies to bridge the gap between theoretical purity and the noisy reality of integrated photonics.