Jiyu Jiang

Random thermal states of many-body Hamiltonians underpin studies of thermalization, chaos, and quantum phase transitions, yet their generation remains costly when each Hamiltonian must be prepared individually. We introduce the thermal-drift channel, a measurement-based operation that implements a tunable nonunitary drift along a chosen Pauli term. Based on this channel, we present a measurement-controlled sampling algorithm that generates thermal states together with their Hamiltonian ``labels'' for general physical models. We prove that the total gate count of our algorithm scales cubically with system size, quadratically with inverse temperature, and as the inverse error tolerance to the two-thirds power, with logarithmic dependence on the allowed failure probability. We also show that the induced label distribution approaches a normal distribution reweighted by the thermal partition function, which makes an explicit trade-off between accuracy and effective range. Numerical simulations for a 2D Heisenberg model validate the predicted scaling and distribution. As an application, we compute unfolding-free level-spacing ratio statistics from sampled thermal states of a 2D transverse-field Ising model and observe a crossover toward the Wigner-Dyson prediction, demonstrating a practical and scalable route to chaos diagnostics and random matrix universality studies on near-term quantum hardware.

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.