Xiao Shi

Estimating nonlinear properties of quantum states, particularly observable-weighted moments $\mathrm{Tr}(\mathcal{O}\rho^k)$, is a central task in quantum information science. In this work, we present a unified framework establishing the optimal sample complexity for this task and a resource-efficient protocol for its implementation. Theoretically, we prove that $\tilde{\Theta}(k)$ samples of an $m$-qubit state $\rho$ are sufficient and necessary to \textit{simultaneously} estimate the full hierarchy of moments $\mathrm{Tr}(\mathcal{O}\rho), \dots, \mathrm{Tr}(\mathcal{O}\rho^k)$. This reveals that estimating the entire hierarchy is asymptotically as efficient as estimating the single highest-order term. To realize this, we introduce a circuit architecture leveraging qubit reuse that requires only $2m+1$ physical qubits and $\mathcal{O}(k)$ depth. This approach achieves the near-optimal sample complexity of $\mathcal{O}(k \log k / \varepsilon^2)$ with significantly reduced hardware overhead. We demonstrate the framework's utility by bounding maximum eigenvalues, performing virtual cooling on the Heisenberg model, and experimentally measuring higher-order Rényi entropies on a superconducting processor.