In this work, we study the problems of certifying and learning quantum $k$-local Hamiltonians, for a constant $k$. Our main contributions are as follows: - Certification of Hamiltonians. We show that certifying a local Hamiltonian in normalized Frobenius norm via access to its time-evolution operator can be achieved with only $O(1/\varepsilon)$ evolution time. This is optimal, as it matches the Heisenberg-scaling lower bound of $Ω(1/\varepsilon)$. To our knowledge, this is the first optimal algorithm for testing a Hamiltonian property. A key ingredient in our analysis is the Bonami Hypercontractivity Lemma from Fourier analysis. - Learning Gibbs states. We design an algorithm for learning Gibbs states of local Hamiltonians in trace norm that is sample-efficient in all relevant parameters. In contrast, previous approaches learned the underlying Hamiltonian (which implies learning the Gibbs state), and thus inevitably suffered from exponential sample complexity scaling in the inverse temperature. - Certification of Gibbs states. We give an algorithm for certifying Gibbs states of local Hamiltonians in trace norm that is both sample and time-efficient in all relevant parameters, thereby solving a question posed by Anshu (Harvard Data Science Review, 2022).

翻译：本文研究常数$k$下量子$k$-局域哈密顿量的认证与学习问题。主要贡献如下： - **哈密顿量认证**。我们证明：通过访问局域哈密顿量的时间演化算子，可在仅需$O(1/\varepsilon)$演化时间的条件下，对其归一化Frobenius范数进行认证。该结果达到海森堡标度下限$\Omega(1/\varepsilon)$的最优性。据我们所知，这是首个针对哈密顿量性质测试的最优算法，其分析关键引入傅里叶分析中的Bonami超收缩引理。 - **吉布斯态学习**。我们设计了一种算法，可在所有相关参数下实现局域哈密顿量吉布斯态的迹范数高效采样学习。相比之下，传统方法通过学习底层哈密顿量（隐含吉布斯态学习）必然产生随逆温度指数增长的样本复杂度。 - **吉布斯态认证**。我们提出一种算法，可在所有相关参数下实现局域哈密顿量吉布斯态迹范数的样本与时间高效认证，从而解决了Anshu（《哈佛数据科学评论》，2022）提出的问题。