We study the problem of certifying local Hamiltonians from real-time access to their dynamics. Given oracle access to $e^{-itH}$ for an unknown $k$-local Hamiltonian $H$ and a fully specified target Hamiltonian $H_0$, the goal is to decide whether $H$ is exactly equal to $H_0$ or differs from $H_0$ by at least $\varepsilon$ in normalized Frobenius norm, while minimizing the total evolution time. We introduce the first intolerant Hamiltonian certification protocol that achieves optimal performance for all constant-locality Hamiltonians. For general $n$-qubit, $k$-local, traceless Hamiltonians, our procedure uses $O(c^k/\varepsilon)$ total evolution time for a universal constant $c$, and succeeds with high probability. In particular, for $O(1)$-local Hamiltonians, the total evolution time becomes $Θ(1/\varepsilon)$, matching the known $Ω(1/\varepsilon)$ lower bounds and achieving the gold-standard Heisenberg-limit scaling. Prior certification methods either relied on implementing inverse evolution of $H$, required controlled access to $e^{-itH}$, or achieved near-optimal guarantees only in restricted settings such as the Ising case ($k=2$). In contrast, our algorithm requires neither inverse evolution nor controlled operations: it uses only forward real-time dynamics and achieves optimal intolerant certification for all constant-locality Hamiltonians.

翻译：我们研究了基于对其动力学实时访问的局域哈密顿量认证问题。给定对未知k-局域哈密顿量H的酉演化算子$e^{-itH}$的预言机访问，以及完全确定的目标哈密顿量$H_0$，目标是在最小化总演化时间的前提下，判定H是否精确等于$H_0$，或在归一化Frobenius范数下与$H_0$至少相差$\varepsilon$。我们提出了首个针对所有常数局域哈密顿量达到最优性能的严格容错哈密顿量认证协议。对于一般的n-量子比特、k-局域、无迹哈密顿量，我们的算法使用$O(c^k/\varepsilon)$的总演化时间（其中c为普适常数），并以高概率成功。特别地，对于$O(1)$-局域哈密顿量，总演化时间达到$Θ(1/\varepsilon)$，与已知的$Ω(1/\varepsilon)$下界匹配，实现了黄金标准的海森堡极限标度。先前的认证方法要么依赖于实现H的逆演化，要么需要$e^{-itH}$的受控访问，或仅在受限场景（如伊辛模型$k=2$）中达到近最优保证。相比之下，我们的算法既不需要逆演化也不需要受控操作：仅利用前向实时动力学，即可对所有常数局域哈密顿量实现最优的严格容错认证。