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

Abstract

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.

Figures (7)

• Figure 1: Schematic of the experimental arrangement. The LASER source acts as the carrier, while the Fiber Switch (FS) prepares the initial state, denoted by the density matrix $\rho$. This state propagates through the Photonic Processor (PP), which implements the specific unitary transformation $U$. Finally, the Photonic Diode Array (PDA) detects the output intensity distribution, generating the input for the reconstruction of the rotated density matrix.
• Figure 2: Sketch of the unit cell. Phase sifters $\phi$ and $\theta$ are denoted by red lines and the 50:50 beam splitter by blue lines.
• Figure 3: Scaling of reconstruction error. The plot shows the dependence of the Frobenius norm on the number of unitary measurements ($M$) for both 4-dimensional and 8-dimensional systems. The simulated data (points) adhere strictly to the analytical prediction $\sqrt{(d-1)/M}$ (lines). The dash-dotted green lines mark the theoretical accuracy limit for the experimentally chosen sample size of $M=5000$.
• Figure 4: Numerical validation of density matrix reconstruction. Left: Reconstruction of an 8-dimensional trivial state. Right: Reconstruction of a 4-dimensional random state. The heatmaps display the real and imaginary components of the density matrices for the original random state ($4 \times 4$ case) and after $M=10$ and $M=5000$ (both cases) unitary transformations. The numerical values within the cells indicate the reconstructed amplitudes, demonstrating convergence toward the target state.
• Figure 5: Observation of the Hardware Horizon. The scaling of reconstruction error (Frobenius norm) versus measurement count ($M$) for 4- and 8-dimensional states. The data initially follows the predicted statistical scaling law $\mathcal{O}(M^{-1/2})$ (solid black lines). However, a distinct phase transition occurs at a critical sample size ($M_{crit}$), where the error diverges from the theoretical prediction and saturates at a hardware-imposed floor. This plateau defines the "Hardware Horizon," governed by the intrinsic spectral distortion of the photonic processor, which prevents convergence to the theoretical accuracy limit (green dash-dotted lines).