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arXiv:2607.23044

Characterizing Arbitrary Lindbladian Dynamics with a Few Pauli Measurements

Taiqi Zhou, Weiyuan Gong

quant-phcs.ITcs.LGSubmitted Jul 25, 2026• Announced Jul 28, 2026• 0 citations
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Abstract

Quantum devices are open systems whose dynamics interleave coherent evolution with dissipation, and benchmarking, error mitigation, and error correction all rest on a faithful model of both. Existing characterization protocols either assume prior knowledge of the interaction and noise structure, or demand ancillas, entangled probes, or mid-circuit control, or capture only the Pauli-diagonal part of the noise. Here, we present a protocol that reconstructs an arbitrary sparse Markovian generator, including every Hamiltonian together with the jump operator coefficients, using only product Pauli state preparation, single uninterrupted forward evolutions, and product Pauli measurements. Given a sparsity budget $M_0$ and a strength bound $Γ$ of the Lindbladian, every coefficient is learned to precision $ε$ from $\widetilde{O}(Γ^2M_0^2/ε^4)$ experiments and $\widetilde{O}(ΓM_0^2/ε^2)$ total evolution time, with both supports identified from data without locality assumptions. The protocol runs at a logarithmic number of positive evolution times on a hardware clock lattice and is provably robust to calibrated state-preparation and measurement errors.

Comments
24 pages, 1 figure

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Quantum devices are open systems whose dynamics interleave coherent evolution with dissipation, and benchmarking, error mitigation, and error correction all rest on a faithful model of both. Existing characterization protocols either assume prior knowledge of the interaction and noise structure, or demand ancillas, entangled probes, or mid-circuit control, or capture only the Pauli-diagonal part of the noise. Here, we present a protocol that reconstructs an arbitrary sparse Markovian generator, including every Hamiltonian together with the jump operator coefficients, using only product Pauli state preparation, single uninterrupted forward evolutions, and product Pauli measurements. Given a sparsity budget M0M_0M0​ and a strength bound ΓΓΓ of the Lindbladian, every coefficient is learned to precision εεε from O~(Γ2M02/ε4)\widetilde{O}(Γ^2M_0^2/ε^4)O(Γ2M02​/ε4) experiments and O~(ΓM02/ε2)\widetilde{O}(ΓM_0^2/ε^2)O(ΓM02​/ε2) total evolution time, with both supports identified from data without locality assumptions. The protocol runs at a logarithmic number of positive evolution times on a hardware clock lattice and is provably robust to calibrated state-preparation and measurement errors.
Characterizing the dynamics of open quantum systems at the level of microscopic interactions and error mechanisms is essential for calibrating quantum hardware, designing robust simulation protocols, and developing tailored error-correction methods. Under Markovian noise/dissipation, a natural characterization approach is to identify the full Lindbladian generator that gives rise to both coherent (Hamiltonian) and dissipative dynamics. Prior protocols for learning Lindbladians from dynamical data assumed pre-specified interaction structure, which can be restrictive when the relevant noise channels or control imperfections are not known in advance. In this paper, we present a sample-efficient protocol for learning sparse Lindbladians without assuming any a priori structure. Our protocol is ancilla-free, uses only product-state preparations and Pauli-basis measurements, admits provably stable coefficient reconstruction, and achieves near-optimal time resolution, making it compatible with near-term experimental capabilities. Together, this provides a systematic route to scalable characterization of open-system quantum dynamics, especially in settings where the error mechanisms of interest are unknown.
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 ΛΛΛ, the algorithm estimates every coefficient to error εεε using O~(Λ2/ε2)\widetilde O(Λ^2/ε^2)O(Λ2/ε2) experiments and O~(Λ/ε2)\widetilde O(Λ/ε^2)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)\mathrm{poly}(Λ/η)poly(Λ/η) that contains every Hamiltonian and dissipative coordinate of magnitude at least ηηη, using O~(Λ2/η2)\widetilde O(Λ^2/η^2)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 MMM to error εεε, using O~(Λ2log⁡M/ε2)\widetilde O(Λ^2\log M/ε^{2})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 present an efficient protocol for learning an unknown kkk-local Lindblad generator on nnn qubits using only product-state preparations, short-time evolution, and single-qubit Pauli measurements, without prior knowledge of the interaction structure. For fixed kkk and a supplied weighted interaction-strength bound αˉ\barααˉ, the protocol estimates all Hamiltonian and dissipative Pauli--GKSL coefficients with total coefficient error at each site at most ε\varepsilonε with probability at least 1−δ1-δ1−δ using O~k(αˉ2n2k−2ε−2log⁡(n/δ))\widetilde{\mathcal O}_k(\barα^2n^{2k-2}\varepsilon^{-2}\log(n/δ))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 kkk-local Lindblad generator with diamond-norm error at most ε\varepsilonε using O~k(αˉ2n2kε−2log⁡(n/δ))\widetilde{\mathcal O}_k(\barα^2n^{2k}\varepsilon^{-2}\log(n/δ))Ok​(αˉ2n2kε−2log(n/δ)) samples and polynomial-time classical postprocessing. With a supplied local effective-sparsity bound rrr and a sufficiently small sitewise subthreshold tail Ok(ε)\mathcal O_k(\varepsilon)Ok​(ε), coefficient recovery requires only O~k(αˉ2r2ε−2log⁡(n/δ))\widetilde{\mathcal O}_k(\barα^2r^2\varepsilon^{-2}\log(n/δ))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 nnn. 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.
Describing the dynamics of open systems is essential for fault-tolerant quantum computation. Under Markovian assumptions, we can characterize dissipative dynamics via the Lindbladian. Using Bell sampling, we provide an efficient, ansatz-free Lindbladian learning algorithm with polynomial-time classical postprocessing. Motivated by the prevalence of state preparation and measurement (SPAM) noise on near-term devices, we also introduce the first efficient SPAM-robust protocol capable of learning the gauge-independent components of sparse Lindbladians to arbitrary precision in the presence of constant-order SPAM error. In doing so, we provide the first rigorous characterization of the gauge degrees of freedom in noisy Lindbladian learning, precisely identifying which components remain learnable under SPAM noise.
We study ansatz-free Lindbladian learning, the problem of reconstructing the generator of an open quantum system without prior knowledge of its Hamiltonian or dissipator structures. This problem exhibits two fundamental precision limits. Hamiltonian components not obscured by dissipation are Heisenberg-limited, while full Lindbladian reconstruction is subject to the quadratically worse standard quantum limit. This creates an apparent algorithmic impasse: achieving the Heisenberg-limited Hamiltonian learning requires suppressing unknown dissipation, which seemingly demands prior reconstruction of the noise and thereby incurs the standard-quantum-limit cost. In this work, we resolve this obstruction by giving an algorithm that learns the Hamiltonian disjoint from dissipator (HDD) at the Heisenberg limit without prior knowledge of either the Hamiltonian or dissipator supports. Our main technical ingredient is a randomized recursive stabilizer-code construction that progressively identifies and suppresses the dominant dissipative terms without reconstructing the full dissipator. Building on this framework, we also introduce an efficient end-to-end algorithm that learns the entire sparse Lindbladian at the standard quantum limit. Finally, we prove that in the ansatz-free setting, the HDD terms constitute the maximal set of Hamiltonian coefficients uniformly learnable at the Heisenberg limit. We show that every term outside HDD is fundamentally standard-quantum limited, extending prior no-go results to the ansatz-free setting. Together, our protocols provide a scalable framework for characterizing open quantum systems, with quantum error correction serving as a key learning primitive.
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 ΛΛΛ, we design an algorithm that learns the coefficients to target accuracy ε\varepsilonε, whose number of channel uses and total evolution time scale as O~(Λ2/ε2)\widetilde{O}(Λ^2/\varepsilon^2)O(Λ2/ε2) and O~(Λ/ε2)\widetilde{O}(Λ/\varepsilon^2)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)Ω(Λ^2/\varepsilon^2)Ω(Λ2/ε2) channel uses and Ω(Λ/ε2)Ω(Λ/\varepsilon^2)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.