Fig. 4: Since the number of samples required to achieve a certain statistical error is proportional to the variance (square of the standard deviation) of the estimator, we use the standard deviation to indicate sample complexity.
From: Demonstration of robust and efficient quantum property learning with shallow shadows
a The standard deviation of fidelity predictions decreases with increasing circuit depth, indicating reduced sample complexity. b The standard deviation for estimating Pauli operator expectations is plotted as a function of the Pauli weight k. We observe excellent agreement between experimental data (solid dots with error bars) and theoretical predictions (solid lines). Notably, shallow circuits (d = 2, 4) exhibit favorable sample complexity scaling for higher-weight Pauli operators, outperforming the d = 0 scaling (proportional to 3k), as well as the theoretical upper bound of 2.28k.
