  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
$1/\varepsilon^2$.
