Robust Structure Learning of $k$-local Lindbladians

2026-06-22

AI Summary

This work proposes the first efficient method for learning an unknown $k$-local Lindblad generator and its interaction structure under realistic experimental constraints—namely, preparation of product states, short-time evolution, and single-qubit Pauli measurements. By integrating a short-time evolution–based sampling strategy, semidefinite projection post-processing, and ideas from compressed sensing, the approach reduces sample complexity from polynomial to logarithmic in the system size under assumptions of stable sparsity or long-range decay of interactions. For fixed $k$ and bounded interaction strength, the method achieves $\varepsilon$-accurate estimation of all coefficients with high probability, requiring only $\widetilde{\mathcal O}_k(\varepsilon^{-2}n^{4k}\log(1/\delta))$ samples, thereby significantly weakening the dependence on the number of qubits $n$.

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 $\widetilde{\mathcal O}_k(\varepsilon^{-2}n^{2k}\log(1/δ))$ 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 $\widetilde{\mathcal O}_k(\varepsilon^{-2}n^{4k}\log(1/δ))$ 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 $\widetilde{\mathcal O}_k(\varepsilon^{-2}\log(n/δ))$ 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.

Problem

• k-local Lindbladians
• structure learning
• quantum dynamics
• sample complexity
• dissipative systems

Innovation

• k-local Lindbladians
• quantum structure learning
• sample complexity
• dissipative quantum dynamics
• semidefinite projection