We study the problem of learning a Hamiltonian $H$ to precision $\varepsilon$, supposing we are given copies of its Gibbs state $\rho=\exp(-\beta H)/\operatorname{Tr}(\exp(-\beta H))$ at a known inverse temperature $\beta$. Anshu, Arunachalam, Kuwahara, and Soleimanifar (Nature Physics, 2021, arXiv:2004.07266) recently studied the sample complexity (number of copies of $\rho$ needed) of this problem for geometrically local $N$-qubit Hamiltonians. In the high-temperature (low $\beta$) regime, their algorithm has sample complexity poly$(N, 1/\beta,1/\varepsilon)$ and can be implemented with polynomial, but suboptimal, time complexity. In this paper, we study the same question for a more general class of Hamiltonians. We show how to learn the coefficients of a Hamiltonian to error $\varepsilon$ with sample complexity $S = O(\log N/(\beta\varepsilon)^{2})$ and time complexity linear in the sample size, $O(S N)$. Furthermore, we prove a matching lower bound showing that our algorithm's sample complexity is optimal, and hence our time complexity is also optimal. In the appendix, we show that virtually the same algorithm can be used to learn $H$ from a real-time evolution unitary $e^{-it H}$ in a small $t$ regime with similar sample and time complexity.

翻译：我们研究在已知逆温度 $\beta$ 下，给定吉布斯态 $\rho=\exp(-\beta H)/\operatorname{Tr}(\exp(-\beta H))$ 的副本，学习哈密顿量 $H$ 至精度 $\varepsilon$ 的问题。Anshu、Arunachalam、Kuwahara 和 Soleimanifar（Nature Physics, 2021, arXiv:2004.07266）近期研究了针对几何局域 $N$ 量子比特哈密顿量的此问题的样本复杂度（所需 $\rho$ 的副本数）。在高温（低 $\beta$）区域，他们的算法具有样本复杂度 poly$(N, 1/\beta,1/\varepsilon)$，且可用多项式但非最优的时间复杂度实现。本文针对更一般的哈密顿量类型研究相同问题。我们展示了如何以样本复杂度 $S = O(\log N/(\beta\varepsilon)^{2})$ 和时间复杂度与样本量线性相关 $O(S N)$ 来学习哈密顿量系数至误差 $\varepsilon$。此外，我们证明了一个匹配的下界，表明我们算法的样本复杂度是最优的，因此时间复杂度也是最优的。在附录中，我们展示了几乎相同的算法可用于从小 $t$ 区域的实时演化幺正 $e^{-it H}$ 学习 $H$，并具有相似的样本和时间复杂度。