Structure learning of Hamiltonians from real-time evolution
This solves a challenging structure learning variant in quantum Hamiltonian learning, with potential applications in quantum simulation and error correction, though it is incremental in advancing known methods for Hamiltonian learning.
The paper tackles the problem of learning the structure and parameters of an unknown local Hamiltonian from its real-time evolution, presenting an algorithm that achieves Heisenberg-limited scaling with total evolution time O(log(n)/ε) and works without prior knowledge of the interaction terms, extending to Hamiltonians with bounded norm interactions and power-law decay.
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 $1/\varepsilon^2$.