Ansatz-Free Learning of Lindbladian Dynamics In Situ

Abstract

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 the first sample-efficient protocol for learning sparse Lindbladians without assuming any a priori structure or locality. Our protocol is ancilla-free, uses only product-state preparations and Pauli-basis measurements, and achieves near-optimal time resolution, making it compatible with near-term experimental capabilities. The final sample complexity depends on linear-system conditioning, which we find empirically to be moderate for a broad class of physically motivated models. 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.

Figures (6)

• Figure 1: Structure learning from Pauli error rates at short times. The protocol outputs sets $\widehat{\mathcal{S}}_D$ and $\widehat{\mathcal{S}}_H$ containing the true Hamiltonian and dissipator structures $\mathcal{S}_D$ and $\mathcal{S}_H$, respectively. (a) We fix a set of short evolution times $\{t_m\}$ and, for each $t_m$, query the channel $e^{\mathcal{L}t_m}$ using the population-recovery protocol flammia2021pauli to obtain a distribution of Pauli error rates $\{\widehat{\chi}_{ii}(t_m)\}$. Each query consists of preparing a random Pauli product state, evolving for time $t_m$, and measuring in a random Pauli basis. (b) Collecting these estimated distributions over $\{t_m\}$ yields time traces of Pauli error rates. For very short times, only the identity component is appreciable. As $t$ increases, the rates associated with Hamiltonian terms (red), dissipator terms (blue), and terms in the overlap $\mathcal{S}_H\cap\mathcal{S}_D$ (purple) begin to grow, spreading the rate distribution. (c) For each observed trace, we fit a low-degree Chebyshev interpolant and use the first and second derivatives of the fit at $t=0$ to decide whether the Pauli enters through dissipation (first order) or through the Hamiltonian (second order): Pauli rates corresponding to dissipator terms $P_D \in \mathcal{S}_D$ exhibit linear short-time growth, while purely Hamiltonian rates $P_H \in \mathcal{S}_H \setminus \mathcal{S}_D$ enter only at second order and exhibit quadratic short-time growth. (d) Finally, we threshold these derivative estimates to obtain candidate supports $\widehat{\mathcal{S}}_D$ and $\widehat{\mathcal{S}}_H$ that satisfy $\widehat{\mathcal{S}}_D = \mathcal{S}_D$ and $\widehat{\mathcal{S}}_H \supseteq \mathcal{S}_H$.
• Figure 2: Coefficient learning from Pauli observables at short-times. (a) The (unknown) Lindbladian is a sparse sum of Pauli terms: coherent Hamiltonian terms (red) and dissipative terms (blue), each acting nontrivially on some subset of qubits, represented by white circles. (b) In the dual picture, we define an interaction graph whose vertices are the nonzero Lindbladian Pauli terms and whose edges connect terms with overlapping qubit support. The maximum degree $\mathfrak d$ of the dual graph upper bounds how rapidly Heisenberg-evolved Pauli observables can spread. If $\mathfrak d$ is not known in advance, it can be upper bounded from the terms induced by the candidate structures $\widehat{\mathcal{S}}_H, \widehat{\mathcal{S}}_D$. (c) To learn the coefficients, we use the candidate supports $\widehat{\mathcal{S}}_H$ and $\widehat{\mathcal{S}}_D$ returned by structure learning to select a sufficient set of Pauli probes $(\rho_{Q},O)$ so that the resulting design matrix $C$ is full rank. For each chosen probe, we sample $f(t)=\tr \{\rho\,e^{t\mathcal{L}^\dagger}(O)\}$ at short Gauss--Chebyshev times $\{t_m\}$, fit a low-degree interpolant, and extract $\left.\frac{d}{dt}f(t)\right|_{t=0}$. Stacking these derivative estimates yields a linear system $\vb d=C\vb x$ for the Lindbladian coefficients, which we solve classically.
• Figure 3: Conditioning factor distribution across system sizes. The conditioning factor $\nu=\|C^{-1}\|_{\infty\to\infty}$ quantifies error amplification when solving for Lindbladian coefficients with design matrix $C$. We construct $C$ via Pauli patchwise tomography using candidate structures $\widehat{\mathcal{S}}_H$ and $\widehat{\mathcal{S}}_D$ that include (in a lattice model) all single-qubit Paulis and all two-qubit Paulis on nearest-, next-nearest-, and next-next-nearest-neighbour pairs, plus $\Theta(n)$ randomly sampled nonlocal three- and four-qubit Paulis in $\widehat{\mathcal{S}}_H$ to model unexpected couplings. This family covers common lattice Hamiltonians, arbitrary local noise, and collective jump operators $J_{\alpha}=\sum_{i=1}^{n}(\alpha_{i,x}X_i+\alpha_{i,y}Y_i+\alpha_{i,z}Z_i)$ arising from correlated processes such as collective decay or global drive fluctuations kraft2025bounded. We use $16$ seeds for each $n$, randomizing the nonlocal terms in $\widehat{\mathcal{S}}_H$ and the probe-selection order.
• Figure 4: Block structure of the linear system $\vb*{d} = C \vb*{x}$.
• Figure 5: Simulated candidate interaction patterns. For visualization, we depict all candidate Lindbladian patches that contain a fixed reference qubit: thin line segments depict two-qubit patches, while filled shapes depict single-, three-, and four-qubit patches corresponding to candidate Lindbladian terms. (a) The candidate Hamiltonian structure $\widehat{\mathcal{S}}_H$ contains all single-qubit Paulis and all two-qubit Paulis supported on nearest-neighbour, next-nearest-neighbour, and next-next-nearest-neighbour pairs, and in addition a small number ($\propto n$) of randomly sampled nonlocal three- and four-qubit Paulis to model unexpected couplings. (b) The candidate dissipator structure $\widehat{\mathcal{S}}_D$ contains all single-qubit Paulis and all two-qubit Paulis supported on nearest-neighbour, next-nearest-neighbour, and next-next-nearest-neighbour pairs. Moreover, by including all dissipator terms $P_k\rho P_m$ formed from pairs of single-qubit Paulis $P_k,P_m\in\widehat{\mathcal{S}}_D$, we cover the couplings induced by arbitrary collective jump operators of the form $J_{\alpha}=\sum_{i=1}^{n}(\alpha_{i,x}X_i+\alpha_{i,y}Y_i+\alpha_{i,z}Z_i)$, which appear as all-to-all two-qubit patches on the figure.

Theorems & Definitions (75)

• Theorem B.1: Noisy derivative estimators
• proof
• Theorem B.2: Derivative error bound for Lagrange interpolants mason2002chebyshev
• Corollary 1: First derivative: sufficient $\tau_{\max}$, $r$, and $\varepsilon_s$
• proof
• Corollary 2: Second derivative: sufficient $\tau_{\max}$, $r$, and $\varepsilon_s$
• proof
• Corollary 3: First derivative: sufficient $\tau_{\max}$, $r$, and $\varepsilon_s$
• proof
• Lemma 1