Batched and Complete U-Statistics for Trace-Polynomial Estimation from Classical Shadows

Abstract: We study estimation of the trace polynomial $\operatorname{tr} p(PρP)$ from global classical shadows, where $ρ$ is an unknown quantum state and $P$ is a fixed projector. Disjoint batching and complete U-statistics yield unbiased estimators of the same trace moments, but assign different sample-size factors to the degenerate terms in their Hoeffding decompositions. Under the global Clifford protocol, exact degree-two variance formulas show that, on a null projected block of rank $s$, the quadratic degenerate term has order $s<sup>2/N$ under batching and $s<sup>2/N<sup>2$ under complete symmetrization. For a logarithmic-degree polynomial used in entropy approximation, the quadratic coefficient raises the batched variance to at least order $s<sup>2N\log<sup>2N$ at the classical entropy cutoff. For complete U-statistics, we derive a cross-degree covariance identity and an exact variance decomposition for polynomial estimators. We also bound every Hoeffding order at a fixed degree and obtain a growing-dimensional risk bound for a small-spectrum entropy functional. The higher-order bounds retain a polynomial dependence on the ambient dimension and therefore do not cover logarithmically increasing degrees. Monte Carlo experiments confirm the degree-two formulas, and exact calculations illustrate the entropy risks.