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

翻译：在本工作中，我们研究了量子伊辛哈密顿量的验证与学习问题。我们的主要贡献如下：伊辛哈密顿量的验证。我们证明，通过访问其时间演化算子，以归一化Frobenius范数验证伊辛哈密顿量仅需$\widetilde O(1/\varepsilon)$次时间演化。这在对数因子内匹配了$\Omega(1/\varepsilon)$的海森堡尺度下界。据我们所知，这是首个测试哈密顿量性质的近乎最优算法。我们分析中的一个关键要素是傅里叶分析中的博纳米引理。伊辛吉布斯态的学习。我们设计了一种在迹范数下学习伊辛吉布斯态的算法，该算法在所有参数上均具有样本效率。相比之下，先前的方法通过学习底层哈密顿量（这隐含着对吉布斯态的学习）来实现，但受限于逆温度参数的指数级样本复杂度。伊辛吉布斯态的验证。我们提出了一种在迹范数下验证伊辛吉布斯态的算法，该算法同时具有样本和时间效率，从而解决了Anshu（《哈佛数据科学评论》，2022年）提出的一个问题。最后，我们将吉布斯态的学习和验证结果推广到任意常数$k$的一般$k$-局域哈密顿量。