Learning Arbitrary Lindbladians from Time Evolution
Zhili Chen, Zhan Yu
We study the problem of learning arbitrary $n$-qubit Lindbladians, the generators of Markovian open-system dynamics, from their time evolution. We give the first known efficient algorithm that learns arbitrary Lindbladians without any structural assumptions. For an $n$-qubit Lindbladian of dynamical strength at most $Λ$, the algorithm estimates every coefficient to error $\varepsilon$ using $\widetilde O(Λ^2/\varepsilon^2)$ experiments and $\widetilde O(Λ/\varepsilon^2)$ total evolution time, with classical running time polynomial in $n$ and $Λ/\varepsilon$. Both experimental costs match known lower bounds up to polylogarithmic factors. The algorithm identifies candidate labels through product Pauli experiments and estimates their coefficients using random stabilizer preparations and Clifford measurements. It requires no ancillas or control during the unknown evolution. For constant $k$, our algorithm learns generic $k$-local Lindbladians of local dynamical strength at most $Λ_{\mathrm{loc}}$ using $\widetilde O(n^2Λ_{\mathrm{loc}}^2/\varepsilon^2)$ experiments and $\widetilde O(nΛ_{\mathrm{loc}}/\varepsilon^2)$ total evolution time, improving the upper bounds of prior works in terms of the dependence on $n$. We further improve the lower bounds of Arad, Chen, Guo, Rebentrost, and Yu (SODA 2027) using a family of $3$-local Lindbladians. They already hold when the support is known, and only one Hamiltonian coefficient is requested, even for adaptive protocols with arbitrary ancillas and quantum memory. These lower bounds establish that our algorithm is optimal up to polylogarithmic factors in both experiment count and total evolution time for every fixed $k \geq 3$.
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