Light-Cone Scaling of In-Circuit Noise in Randomized Measurements

Randomized measurements, such as classical shadows, are powerful tools for characterizing quantum states. However, hardware noise—including gate and readout errors—biases the resulting estimators. This paper provides a microscopic framework to understand how local noise accumulates in shallow, locally scrambled circuits, showing that the noise bias is directly linked to the space-time history of operator evolution.

In the Heisenberg picture, a local noise channel only damps the signal if it overlaps with the evolving support of the measured Pauli operator. The authors define this as an activated noise event. For a contiguous Pauli string of size k, the light cone of the operator encompasses a bulk volume proportional to k times the circuit depth d, plus boundary contributions. By averaging over the paths of operator evolution, the authors derive an exponential damping ratio for the measurement coefficients.

The study demonstrates that the damping ratio follows a linear scaling in the log-domain: log(η) = -αk - β. This relationship holds even under spatially fluctuating noise or deterministic temporal drift, as these effects self-average over the light-cone bulk. Because this scaling is robust, it enables a practical calibration protocol: by measuring small, known string observables (like Z-strings on a product state), researchers can fit the parameters α and β to predict and correct the bias for much larger, complex observables without needing to learn the full noisy measurement channel.

This work bridges the gap between abstract noise models and the physical dynamics of quantum circuits. By relating measurement bias to the geometry of operator spreading, it provides a scalable path to high-fidelity quantum state characterization on near-term hardware, bypassing the need for full quantum tomography or exhaustive channel reconstruction.