Simple algorithms to test and learn local Hamiltonians

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 ε1-close to k-local or ε2-far from k-local, we show that O(1/(ε2-ε1)8) queries suffice. This solves two questions posed in a recent work by Bluhm, Caro and Oufkir. For learning up to error ε, we show that exp(O(k2+klog(1/ε))) queries suffice. Our proofs are simple, concise and based on Pauli-analytic techniques.