We study the problem of learning an unknown quantum many-body Hamiltonian $H$ from black-box queries to its time evolution $e^{-\mathrm{i} H t}$. Prior proposals for solving this task either impose some assumptions on $H$, such as its interaction structure or locality, or otherwise use an exponential amount of computational postprocessing. In this paper, we present efficient algorithms to learn any $n$-qubit Hamiltonian, assuming only a bound on the number of Hamiltonian terms, $m \leq \mathrm{poly}(n)$. Our algorithms do not need to know the terms in advance, nor are they restricted to local interactions. We consider two models of control over the time evolution: the first has access to time reversal ($t < 0$), enabling an algorithm that outputs an $\epsilon$-accurate classical description of $H$ after querying its dynamics for a total of $\widetilde{O}(m/\epsilon)$ evolution time. The second access model is more conventional, allowing only forward-time evolutions; our algorithm requires $\widetilde{O}(\|H\|^3/\epsilon^4)$ evolution time in this setting. Central to our results is the recently introduced concept of a pseudo-Choi state of $H$. We extend the utility of this learning resource by showing how to use it to learn the Fourier spectrum of $H$, how to achieve nearly Heisenberg-limited scaling with it, and how to prepare it even under our more restricted access models.

翻译：我们研究从黑盒查询其时间演化 $e^{-\mathrm{i} H t}$ 来学习未知量子多体哈密顿量 $H$ 的问题。先前解决此任务的方案要么对 $H$ 施加某些假设（如其相互作用结构或局域性），要么使用指数级计算量的后处理。本文中，我们提出高效算法来学习任意 $n$-量子比特哈密顿量，仅假设其哈密顿量项数存在上界 $m \leq \mathrm{poly}(n)$。我们的算法无需预先知晓这些项，也不局限于局域相互作用。我们考虑两种对时间演化的控制模型：第一种可访问时间反演（$t < 0$），使得算法在总共查询 $\widetilde{O}(m/\epsilon)$ 演化时间后，能输出 $H$ 的 $\epsilon$-精度经典描述。第二种访问模型更为传统，仅允许正向时间演化；在此设置下，我们的算法需要 $\widetilde{O}(\|H\|^3/\epsilon^4)$ 的演化时间。我们结果的核心是最近引入的 $H$ 的伪 Choi 态概念。我们通过展示如何利用它来学习 $H$ 的傅里叶谱、如何用它实现近海森堡极限标度，以及如何在更受限的访问模型下制备它，从而扩展了这一学习资源的实用性。