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Simple algorithms to test and learn local Hamiltonians
Published 9 Apr 2024 in quant-ph, cs.CC, cs.DS, cs.IT, cs.LG, and math.IT | (2404.06282v1)
Abstract: 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 $\exp(O(k2+k\log(1/\epsilon)))$ queries suffice. Our proofs are simple, concise and based on Pauli-analytic techniques.
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