We consider the problems of testing and learning an $n$-qubit $k$-local Hamiltonian from queries to its evolution operator with respect the 2-norm of the Pauli spectrum, or equivalently, the normalized Frobenius norm. For testing whether a Hamiltonian is $\epsilon_1$-close to $k$-local or $\epsilon_2$-far from $k$-local, we show that $O(1/(\epsilon_2-\epsilon_1)^{8})$ queries suffice. This solves two questions posed in a recent work by Bluhm, Caro and Oufkir. For learning up to error $\epsilon$, we show that $\exp(O(k^2+k\log(1/\epsilon)))$ queries suffice. Our proofs are simple, concise and based on Pauli-analytic techniques.

翻译：我们考虑从演化算符的查询中测试和学习$n$比特$k$-局域哈密顿量的问题，其中误差度量采用泡利谱对应的2范数（等价于归一化弗罗贝尼乌斯范数）。针对判断哈密顿量是$\epsilon_1$-接近$k$-局域还是$\epsilon_2$-远离$k$-局域的问题，我们证明$O(1/(\epsilon_2-\epsilon_1)^{8})$次查询即可满足要求。这解决了Bluhm、Caro和Oufkir近期工作中提出的两个问题。对于学习误差$\epsilon$，我们证明$\exp(O(k^2+k\log(1/\epsilon)))$次查询即可。我们的证明过程简洁明了，基于泡利分析技术。