We study the problem of learning local Lindbladians from black-box access to the physical evolution, where the goal is to estimate all Hamiltonian and dissipative coefficients. For local Lindbladians with low-intersection dissipators and local dynamical strength at most $Λ$, we design an algorithm that learns the coefficients to target accuracy $\varepsilon$, whose number of channel uses and total evolution time scale as $\widetilde{O}(Λ^2/\varepsilon^2)$ and $\widetilde{O}(Λ/\varepsilon^2)$, respectively, with only logarithmic dependence on the number of qubits. The algorithm is built directly from finite-time channel probes. It runs the unknown evolution for short times, estimates the corresponding Pauli transfer matrices from classical shadows, and converts these estimates into Lindbladian coefficients by stable local Fourier inversions.
The algorithm is non-adaptive, uses no ancillas, and requires only random product states as inputs followed by random Pauli measurements. The method does not require knowing the structure of the Lindbladian in advance. We prove matching lower bounds that establish the near-optimality of the algorithm in both resources. By constructing a family of single-qubit dephasing Lindbladians, we show that any algorithm, even an adaptive one with arbitrary ancillas and measurements, requires $Ω(Λ^2/\varepsilon^2)$ channel uses and $Ω(Λ/\varepsilon^2)$ total evolution time. In particular, the lower bounds imply that the Heisenberg-limited scaling achievable for Hamiltonian learning is information-theoretically impossible once dissipative coefficients must be estimated.
We study the problem of learning local Lindbladians from black-box access to the physical evolution, where the goal is to estimate all Hamiltonian and dissipative coefficients. For local Lindbladians with low-intersection dissipators and local dynamical strength at most Λ, we design an algorithm that learns the coefficients to target accuracy ε, whose number of channel uses and total evolution time scale as O(Λ2/ε2) and O(Λ/ε2), respectively, with only logarithmic dependence on the number of qubits. The algorithm is built directly from finite-time channel probes. It runs the unknown evolution for short times, estimates the corresponding Pauli transfer matrices from classical shadows, and converts these estimates into Lindbladian coefficients by stable local Fourier inversions.
The algorithm is non-adaptive, uses no ancillas, and requires only random product states as inputs followed by random Pauli measurements. The method does not require knowing the structure of the Lindbladian in advance. We prove matching lower bounds that establish the near-optimality of the algorithm in both resources. By constructing a family of single-qubit dephasing Lindbladians, we show that any algorithm, even an adaptive one with arbitrary ancillas and measurements, requires Ω(Λ2/ε2) channel uses and Ω(Λ/ε2) total evolution time. In particular, the lower bounds imply that the Heisenberg-limited scaling achievable for Hamiltonian learning is information-theoretically impossible once dissipative coefficients must be estimated.
We study the problem of learning an unknown Markovian open-system generator from access to its physical time evolution. This generator, called a Lindbladian, contains Hamiltonian and dissipative coefficients indexed by an exponentially large family of possible Pauli terms. We propose an efficient algorithm that learns arbitrary Lindbladians from time evolution under minimal assumptions. For a Lindbladian of dynamical strength at most Λ, the algorithm estimates every coefficient to error ε using O(Λ2/ε2) experiments and O(Λ/ε2) total evolution time, together with polynomial classical running time. The algorithm consists of two nonadaptive, ancilla-free, and control-free stages: 1. The support-learning stage outputs a candidate support of size poly(Λ/η) that contains every Hamiltonian and dissipative coordinate of magnitude at least η, using O(Λ2/η2) experiments with preparations of product Pauli eigenstates and single-qubit Pauli measurements. 2.The coefficient-learning stage estimates all coefficients in any candidate support of size M to error ε, using O(Λ2logM/ε2) experiments with preparations of random stabilizer states and measurements in random Clifford bases. Composing the two stages identifies and estimates every coefficient of an arbitrary Lindbladian in polynomial time. The experiment-count and total-evolution-time scalings match the lower bounds up to logarithmic factors, so the algorithm is nearly optimal for learning arbitrary Lindbladians.
We design an algorithm for learning the coefficients of an n-qubit constant-local Lindbladian to ε error with O(gd2log(n)/ε2) total evolution time, where g is the single-site energy and d is the (approximate) degree of the interaction graph. Though Lindbladians present new challenges not present in the special case of Hamiltonians, our algorithm achieves the suite of desiderata attained by state-of-the-art Hamiltonian learning algorithms: (1) it uses non-adaptive, ancilla-free randomized Pauli measurement circuits with a time resolution of only Θ(1/g); (2) it works without knowledge of the structure of the unknown Lindbladian; (3) it depends on a smooth form of degree, thereby supporting the learning of quasi-local and power-law Lindbladians. Moreover, we prove a lower bound showing that our algorithm is optimal in each parameter up to logarithmic factors.
Our algorithm is a simple iterative method, where the objective function consists of Fourier coefficients of the Lindbladian restricted to few-site regions. Its analysis identifies the difficulty unique to open systems, which we call "confusing" terms. For settings where the "confusion" is limited, the performance of the algorithm improves. We demonstrate this for the case of structure learning of Hamiltonians from access to real-time evolution, where we obtain a new algorithm that is significantly simpler than previous work. In addition, using the same iterative method, we design the first efficient algorithm for structure learning Hamiltonians from high-temperature Gibbs states.
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 a supplied weighted interaction-strength bound αˉ, the protocol estimates all Hamiltonian and dissipative Pauli--GKSL coefficients with total coefficient error at each site at most ε with probability at least 1−δ using Ok(αˉ2n2k−2ε−2log(n/δ)) samples and polylogarithmically many distinct evolution times. This guarantee requires no degree, sparsity, or tail promise. A convex optimization converts these estimates into a valid k-local Lindblad generator with diamond-norm error at most ε using Ok(αˉ2n2kε−2log(n/δ)) samples and polynomial-time classical postprocessing. With a supplied local effective-sparsity bound r and a sufficiently small sitewise subthreshold tail Ok(ε), coefficient recovery requires only Ok(αˉ2r2ε−2log(n/δ)) samples, without a supplied threshold or coefficient locations and without a coefficient-gap assumption. In particular, exactly sparse bounded-degree models have logarithmic sample dependence on n. We also provide guard-band support recovery and complementary supplied-candidate guarantees, extend the guarantees to model misspecification, and prove complementary sample-complexity lower bounds.
Learning the generator of an open many-body system is more challenging than Hamiltonian learning: local responses, which can directly reveal coherent interaction terms in closed-system dynamics, may also contain dissipative contributions in open-system dynamics. In this paper, we address this challenge by developing an efficient Lindbladian learning framework for a known local candidate generator dictionary with bounded dissipative support and either bounded dual-interaction-graph degree or bounded unweighted local strength. The framework resolves the coherent-dissipative ambiguity by treating local Pauli responses as a linear system over both types of generator terms. Inverting this response system separates their contributions and makes the individual Lindbladian coefficients accessible from local response data in a fixed short-time window. Within this framework, we develop two efficient learning algorithms: Chebyshev--Lobatto response interpolation, which uses logarithmically many short evolution times and has a post-mean cost linear in M, with the stated dependence on ε, and Single-time projected response contraction, which uses a single fixed evolution time and globally inverts a truncated response function. Both procedures estimate M candidate coefficients to entrywise accuracy ε using O(M/ε2) sample and classical post-processing complexity. Our theoretical results establish local response inversion as a scalable paradigm for learning, calibrating, and diagnosing complex quantum systems from experimentally accessible short-time data.
Characterizing the features of a Hamiltonian that governs a quantum system serves as a fundamental subroutine of quantum device calibration, signal sensing, and error correction. Recent works have proposed protocols achieving the optimal Heisenberg-limited scaling learning ansatz-free Hamiltonians from their real-time evolutions without fully specifying interaction structures. However, these protocols rely on both deep circuits with interleaving probes and control, and extremely short time resolution, making them difficult to implement on near- and intermediate-term in situ quantum experiments. In this work, we propose a computationally efficient, control-free, and ancilla-free algorithm that uses only Pauli product state preparation and measurement, and learns an ansatz-free Hamiltonian H with ∣∣H∣∣≤Λ in total evolution time of Θ(ε2Λlog(εΛ)). The evolution time cost of our algorithm is optimal for any control-free protocols as we further prove a lower bound of Ω(ε2Λlog(εΛ)). Technically, our method introduces a randomized-sampling framework that combines band-limited kernel-based time sampling with a displacement sieve for Hamiltonian structure learning. The characteristic probe time resolution depends only on Λ instead of ε, which makes our protocol especially appealing in the high-precision regime for sensing and calibration applications. We also show that the algorithm maintains the same asymptotic total evolution time in the presence of state-preparation-and-measurement (SPAM) noise when the Hamiltonian is local after calibration. Our results demonstrate the fundamental cost of experimentally friendly Hamiltonian learning and provide a practical route to rigorous in situ characterization of near-term quantum platforms.
Experimental implementations of Hamiltonian dynamics are often affected by dissipative noise arising from interactions with the environment. This raises the question of whether one can detect the presence or absence of such dissipation using only access to the observed time evolution of the system. We consider the following decision problem: given black-box access to the time-evolution channels etL generated by an unknown time-independent Lindbladian L, determine whether the dynamics are purely Hamiltonian or contain dissipation of magnitude at least ε in normalized Frobenius norm. We give a randomized procedure that solves this task using total evolution time O(ε−1), which is information-theoretically optimal. This guarantee holds under the assumptions that the Lindblad generator has bounded strength and its dissipative part is of constant locality with bounded degree. Our work provides a practical method for detecting dissipative noise in experimentally implemented quantum dynamics.