On the Classical Shadow Nonparametric Bootstrap
Eric Ghysels, Jack Morgan
TL;DR
The paper addresses the limitations of Gaussian-based error characterizations for classical shadow estimators by applying nonparametric bootstrap to the shadow data, treating the shadow samples as a dataset and generating bootstrap replicates $T_N^{*,b}$. This approach reveals non-Gaussian, heavy-tailed tail behavior in general observables and shows that standard MoM error bounds are not tight, highlighting the value of bootstrap-based uncertainty quantification. By introducing tail risk measures such as Value-at-Risk and Expected Shortfall for quantum circuit predictions, the work demonstrates meaningful tail-risk differences compared to Gaussian approximations, underscoring the need for tail-focused analysis. The findings broaden the applicability of classical shadows, enabling risk-aware assessments for a range of tasks including entropies, correlators, and error mitigation, and pave the way for more robust, tail-conscious quantum state property estimation.
Abstract
Classical shadows are an efficient method for constructing an approximate classical description of a quantum state using very few measurements. In the paper we propose to enhance classical shadow methods using bootstrap resampling methods. We apply nonparametric bootstrapping to assess the variability and accuracy of estimators by repeatedly sampling with replacement from the observed data, i.e. in our case the classical shadow measurements. We show that the bootstrap distributions are very different from the Gaussian approximations. Likewise, the theoretical error bounds are not tight compared to the bootstrap percentiles. Finally, we suggest using resampling tools to make risk assessments.