提出最优常数局部哈密顿量认证方法，仅用正向演化即达理论极限。
Optimal certification of constant-local Hamiltonians
- 仅依赖正向时间演化，无需逆演化或受控操作
- 总演化时间仅需 $O(1/\varepsilon)$，达到海森堡极限
- 适用于任意常数局部哈密顿量，通用性强
我们研究从真实时间访问动力学中认证局部哈密顿量的问题。给定对未知 $k$-局部哈密顿量 $H$ 的 $e^{-itH}$ 量子查询访问，以及一个完全指定的目标哈密顿量 $H_0$，目标是判断 $H$ 是否等于 $H_0$，或在归一化 Frobenius 范数下至少相差 $\varepsilon$，同时最小化总演化时间。本文提出首个对容忍度不敏感的哈密顿量认证协议，对所有常数局部哈密顿量实现最优性能。对于一般 $n$-量子比特、$k$-局部、迹为零的哈密顿量，该方法总演化时间为 $O(c^k/\varepsilon)$（$c$ 为通用常数），高概率成功。特别地，对 $O(1)$-局部哈密顿量，总演化时间降至 $Θ(1/\varepsilon)$，与已知的 $Ω(1/\varepsilon)$ 下界匹配，达到黄金标准的海森堡极限。此前方法或依赖 $H$ 的逆演化、需受控访问 $e^{-itH}$，或仅在特定情形（如 $k=2$ 的伊辛模型）中近似最优。本算法无需逆演化或受控操作，仅使用正向实时演化，实现了所有常数局部哈密顿量的最优不容忍认证。
原文摘要 · Abstract (English)
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.
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