We initiate the study 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-studied under the assumption that the interaction terms, $E_a$, are given, and only the interaction strengths, $\lambda_a$, are unknown. But is it possible to 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 an evolution time scaling with $1/\varepsilon$, 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. To our knowledge, no prior algorithm with Heisenberg-limited scaling existed with even one of these properties. 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：我们开启了从实时演化中学习哈密顿量结构的研究：给定对未知局域哈密顿量 $H = \sum_{a = 1}^m \lambda_a E_a$（作用于 $n$ 个量子比特）应用 $e^{-\mathrm{i} Ht}$ 的能力，目标是恢复 $H$。这一问题在假设相互作用项 $E_a$ 已知而仅相互作用强度 $\lambda_a$ 未知的情况下已被充分研究。然而，能否在不事先知道相互作用结构的情况下学习一个局域哈密顿量？我们提出了一种新的、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问题，同时达到了海森堡极限标度的黄金标准。特别地，我们的算法将哈密顿量恢复至 $\varepsilon$ 误差，所需的演化时间标度为 $1/\varepsilon$，并具有以下吸引人的性质：(1) 无需知道哈密顿量项；(2) 适用于超出短程范围的情形，可推广至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) 以恒定时间增量 $t$ 按 $H$ 演化，因此实现恒定时间分辨率。据我们所知，在此之前没有任何具有海森堡极限标度的算法具备这些性质中的任意一个。作为应用，我们还能学习具有幂律衰减的哈密顿量，在总演化时间突破标准极限 $1/\varepsilon^2$ 的情况下达到 $\varepsilon$ 精度。