Learning $k$-body Hamiltonians via compressed sensing

We study the problem of learning a $k$-body Hamiltonian with $M$ unknown Pauli terms that are not necessarily geometrically local. We propose a protocol that learns the Hamiltonian to precision $\epsilon$ with total evolution time ${\mathcal{O}}(M^{1/2+1/p}/\epsilon)$ up to logarithmic factors, where the error is quantified by the $\ell^p$-distance between Pauli coefficients. Our learning protocol uses only single-qubit control operations and a GHZ state initial state, is non-adaptive, is robust against SPAM errors, and performs well even if $M$ and $k$ are not precisely known in advance or if the Hamiltonian is not exactly $M$-sparse. Methods from the classical theory of compressed sensing are used for efficiently identifying the $M$ terms in the Hamiltonian from among all possible $k$-body Pauli operators. We also provide a lower bound on the total evolution time needed in this learning task, and we discuss the operational interpretations of the $\ell^1$ and $\ell^2$ error metrics. In contrast to most previous works, our learning protocol requires neither geometric locality nor any other relaxed locality conditions.