We consider the problems of testing and learning an unknown $n$-qubit Hamiltonian $H$ from queries to its evolution operator $e^{-iHt}$ under the normalized Frobenius norm. We prove: 1. Local Hamiltonians: We give a tolerant testing protocol to decide if $H$ is $\epsilon_1$-close to $k$-local or $\epsilon_2$-far from $k$-local, with $O(1/(\epsilon_2-\epsilon_1)^{4})$ queries, solving open questions posed in a recent work by Bluhm et al. For learning a $k$-local $H$ up to error $\epsilon$, we give a protocol with query complexity $\exp(O(k^2+k\log(1/\epsilon)))$ independent of $n$, by leveraging the non-commutative Bohnenblust-Hille inequality. 2. Sparse Hamiltonians: We give a protocol to test if $H$ is $\epsilon_1$-close to being $s$-sparse (in the Pauli basis) or $\epsilon_2$-far from being $s$-sparse, with $O(s^{6}/(\epsilon_2^2-\epsilon_1^2)^{6})$ queries. For learning up to error $\epsilon$, we show that $O(s^{4}/\epsilon^{8})$ queries suffice. 3. Learning without memory: The learning results stated above have no dependence on $n$, but require $n$-qubit quantum memory. We give subroutines that allow us to learn without memory; increasing the query complexity by a $(\log n)$-factor in the local case and an $n$-factor in the sparse case. 4. Testing without memory: We give a new subroutine called Pauli hashing, which allows one to tolerantly test $s$-sparse Hamiltonians with $O(s^{14}/(\epsilon_2^2-\epsilon_1^2)^{18})$ queries. A key ingredient is showing that $s$-sparse Pauli channels can be tolerantly tested under the diamond norm with $O(s^2/(\epsilon_2-\epsilon_1)^6)$ queries. Along the way, we prove new structural theorems for local and sparse Hamiltonians. We complement our learning results with polynomially weaker lower bounds. Furthermore, our algorithms use short time evolutions and do not assume prior knowledge of the terms in the support of the Pauli spectrum of $H$.

翻译：我们考虑在归一化Frobenius范数下，通过对其演化算子$e^{-iHt}$的查询来测试和学习未知$n$-量子比特哈密顿量$H$的问题。我们证明：1. 局域哈密顿量：我们给出一个容忍性测试协议，能以$O(1/(\epsilon_2-\epsilon_1)^{4})$次查询判断$H$是$\epsilon_1$-接近$k$-局域还是$\epsilon_2$-远离$k$-局域，解决了Bluhm等人近期工作中提出的开放性问题。对于以误差$\epsilon$学习$k$-局域$H$，我们通过利用非交换Bohnenblust-Hille不等式，给出了查询复杂度为$\exp(O(k^2+k\log(1/\epsilon)))$且与$n$无关的协议。2. 稀疏哈密顿量：我们给出一个协议，能以$O(s^{6}/(\epsilon_2^2-\epsilon_1^2)^{6})$次查询测试$H$是$\epsilon_1$-接近$s$-稀疏（在Pauli基下）还是$\epsilon_2$-远离$s$-稀疏。对于以误差$\epsilon$学习，我们证明$O(s^{4}/\epsilon^{8})$次查询足够。3. 无记忆学习：上述学习结果均不依赖于$n$，但需要$n$-量子比特量子存储器。我们给出了允许无记忆学习的子程序；在局域情形中将查询复杂度增加$(\log n)$因子，在稀疏情形中增加$n$因子。4. 无记忆测试：我们提出了一个称为Pauli哈希的新子程序，允许以$O(s^{14}/(\epsilon_2^2-\epsilon_1^2)^{18})$次查询容忍性测试$s$-稀疏哈密顿量。一个关键要素是证明$s$-稀疏Pauli通道可以在菱形范数下以$O(s^2/(\epsilon_2-\epsilon_1)^6)$次查询进行容忍性测试。在此过程中，我们证明了关于局域和稀疏哈密顿量的新结构定理。我们通过多项式级较弱的下界补充了学习结果。此外，我们的算法使用短时间演化，且不假设事先知道$H$的Pauli谱支撑项。