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 $H = \sum_{a = 1}^m λ_a E_a$ 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, $λ_a$, 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：我们研究从实时演化中学习哈密顿量结构的问题：给定对未知局域哈密顿量 $H = \sum_{a = 1}^m λ_a E_a$（作用于 $n$ 个量子比特）施加 $e^{-\mathrm{i} Ht}$ 的能力，目标是恢复 $H$。在假设相互作用项 $E_a$ 已知而仅相互作用强度 $λ_a$ 未知的情况下，该问题已得到充分理解。然而，在没有先验了解相互作用结构的情况下，我们能够以多高的效率学习局域哈密顿量？我们提出了一种新的通用哈密顿量学习方法，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问题，同时达到了海森堡极限标尺的黄金标准。特别地，我们的算法以总演化时间 $O(\log (n)/\varepsilon)$ 将哈密顿量恢复到 $\varepsilon$ 误差，并具有以下吸引人的特性：(1) 无需知道哈密顿量项；(2) 适用于短程设定之外，扩展到任何与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) 按恒定时间 $t$ 增量根据 $H$ 演化，从而实现恒定时间分辨率。作为一个应用，我们还可以以总演化时间突破标准 $1/\varepsilon^2$ 极限的方式，学习具有幂律衰减的哈密顿量，精度达到 $\varepsilon$。