We study the problem of Hamiltonian structure learning from real-time evolution: given the ability to apply $e^{-\mathrm{i} Ht}$ for an unknown local Hamiltonian $H = \sum_{a = 1}^m \lambda_a E_a$ on $n$ qubits, the goal is to recover $H$. This problem is already well-understood under the assumption that the interaction terms, $E_a$, are given, and only the interaction strengths, $\lambda_a$, are unknown. But how efficiently can we learn a local Hamiltonian without prior knowledge of its interaction structure? We present a new, general approach to Hamiltonian learning that not only solves the challenging structure learning variant, but also resolves other open questions in the area, all while achieving the gold standard of Heisenberg-limited scaling. In particular, our algorithm recovers the Hamiltonian to $\varepsilon$ error with total evolution time $O(\log (n)/\varepsilon)$, and has the following appealing properties: (1) it does not need to know the Hamiltonian terms; (2) it works beyond the short-range setting, extending to any Hamiltonian $H$ where the sum of terms interacting with a qubit has bounded norm; (3) it evolves according to $H$ in constant time $t$ increments, thus achieving constant time resolution. As an application, we can also learn Hamiltonians exhibiting power-law decay up to accuracy $\varepsilon$ with total evolution time beating the standard limit of $1/\varepsilon^2$.

翻译：我们研究了从实时演化中学习哈密顿量结构的问题：假设能够对未知的局部哈密顿量 $H = \sum_{a = 1}^m \lambda_a E_a$（作用于 $n$ 个量子比特）应用 $e^{-\mathrm{i} Ht}$，目标是重构 $H$。在已知相互作用项 $E_a$、仅相互作用强度 $\lambda_a$ 未知的前提下，该问题已被充分理解。然而，若事先不知道其相互作用结构，我们能够以多高效率学习一个局部哈密顿量？本文提出了一种新颖、通用的哈密顿量学习方法，该方法不仅解决了具有挑战性的结构学习变体，还解决了该领域的其他开放性问题，同时达到了海森堡极限标度的黄金标准。具体而言，我们的算法以总演化时间 $O(\log (n)/\varepsilon)$ 将哈密顿量恢复至 $\varepsilon$ 误差，并具有以下吸引人的特性：（1）无需事先知道哈密顿量项；（2）其适用范围超越短程情形，可扩展至任何与单个量子比特相互作用的项之和具有有界范数的哈密顿量 $H$；（3）它以恒定时间增量 $t$ 根据 $H$ 进行演化，从而实现了恒定时间分辨率。作为应用，我们还能以总演化时间学习表现出幂律衰减的哈密顿量，其精度 $\varepsilon$ 突破了标准的 $1/\varepsilon^2$ 限制。