Robust Structure Learning of $k$-local Lindbladians

Abstract: We present an efficient protocol for learning an unknown $k$-local Lindblad generator on $n$ qubits using only product-state preparations, short-time evolution, and single-qubit Pauli measurements, without prior knowledge of the interaction structure. For fixed $k$ and bounded weighted interaction strength, the protocol estimates all Hamiltonian and dissipative Pauli--GKSL coefficients to entrywise accuracy $\varepsilon$ with probability at least $1-δ$ using $$\widetilde{\mathcal O}_k(\varepsilon<sup>{-2}n<sup>{2k}\log(1/δ))$$ samples and polylogarithmically many evolution times. A semidefinite projection converts these estimates into a valid $k$-local Lindblad generator with diamond-norm error at most $\varepsilon$ using $$\widetilde{\mathcal O}_k(\varepsilon<sup>{-2}n<sup>{4k}\log(1/δ))$$ samples and polynomial-time classical postprocessing. If a suitable set of influential coefficients is supplied and satisfies a stable sparsity condition, the dependence on $n$ can improve from polynomial to logarithmic; in particular, exact supports of bounded intersection degree require only $$\widetilde{\mathcal O}_k(\varepsilon<sup>{-2}\log(n/δ))$$ samples, with analogous reductions in system-size dependence for sufficiently decaying long-range interactions. We also provide a robust structure-learning procedure, extend the guarantees to model misspecification, and prove complementary sample-complexity lower bounds. To our knowledge, these are the first efficient learning guarantees for general $k$-local dissipative quantum dynamics under such limited experimental control.