Lindbladian Learning with Neural Differential Equations

Timothy Heightman; Roman Aseguinolaza Gallo; Edward Jiang; JRM Saavedra; Antonio Acín; Marcin Płodzień

quant-phMar 8, 2026

Lindbladian Learning with Neural Differential Equations

Authors: Timothy Heightman, Roman Aseguinolaza Gallo, Edward Jiang, JRM Saavedra, Antonio Acín, Marcin Płodzień

Organizations: ICFO – Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain · Quside Technologies SL, Carrer d’Esteve Terradas 1, 08860 Castelldefels, Barcelona, Spain · CIC nanoGUNE Consolider, Tolosa Hiribidea 76, E-20018 Donostia-San Sebastian, Spain · Quantum Motion, 9 Sterling Way, London, N7 9HJ, United Kingdom ·

Abstract

Inferring the dynamical generator of a many-body quantum system from measurement data is essential for the verification, calibration, and control of quantum processors. When the system is open, this task becomes considerably harder than in the purely unitary case, because coherent and dissipative mechanisms can produce similar measurement statistics and long-time data can be insensitive to coherent couplings. Here we tackle this so-called Lindbladian learning problem of open-system characterisation with maximum-likelihood on Pauli measurements at multiple experimentally friendly transient times, exploiting the richer information content of transient dynamics. To navigate the resulting non-convex likelihood loss-landscape, we augment the physical model neural differential-equation term, which is progressively removed during training to distil an interpretable Lindbladian solution. Our method reliably learns open-system dynamics across neutral-atom (with 2D connectivity) and superconducting Hamiltonians, as well as the Heisenberg XYZ, and PXP models on a spin-1/2 chain. For the dissipative part, we show robustness over phase noise, thermal noise, and their combination. Our algorithm can robustly infer these dissipative systems over noise-to-signal ratios spanning four orders of magnitude, and system sizes up to $N=6$ qubits with fewer than $5 \times 10^5$ shots.

Figures & tables

Panel of Figure 1 — Figure 1: The white-box Lindbladian Learning (LL) setup. The structure of the true Hamiltonian $H_{T}=\sum_{j}c_{j}P_{j}$ and open-system terms $\sum_{\alpha}\gamma_{\alpha}\left(L_{\alpha}\rho(t)L_{\alpha}^{\dagger}-\tfrac{1}{2}\{L_{\alpha}^{\dagger}L_{\alpha},\rho(t)\}\right)$ structure the ansatz, and the coefficients $c_{j},\gamma_{\alpha}\in\mathbb{R}$ are unknown.

Panel of Figure 2 — Figure 2: Computational graphs for automatic differentiation of an ordinary differential equation solver ODEint for a vanilla model (a), and a NDE model, (b). Here by ODE-int we refer to using a numerical solver compatible with automatic differentiation. The subscripts $\theta$ and $\varphi$ indicate the variational parameters assigned to the Lindbladian and Neural component respectively in both (a) and (b). As mentioned in Sec. 2 , we enforce physical output states by renormalising via the trace and averaging over the conjugate transpose once on the output of the ODEint numerical solver.

Figure 3: Success rates for a 1D Transverse-Field Ising Hamiltonian model with phase (left), thermal (centre) and combined phase and thermal (right) noise terms over four orders of magnitude of noise-to-unitary ratios,

R

. The top row shows the robustness metric over the Hamiltonian parameters and the shows the robustness metric for Lindbladian parameters. Shaded regions correspond to when the NDE term increases (red) or decreases (green) the fitting error.

Panel of Figure 4 — Figure 4: Initial loss landscape of two random orthogonal directions in the Hamiltonian variational subspace of $V$ defined in Eq. ( 19 ) for the vanilla (left) architecture and the NDE architecture (right), for all four noise-to-unitary ratios $R=0.01,0.1,1,10$ . In all four cases, notice the heatmap of the contour lines indicating a barren landscape on the vanilla architecture, against a much richer loss-landscape for the NDE. This explains the performance gains of the NDE’s robustness from Fig. 3 , corresponding to the experiments in the top-left of Fig. 3

Panel of Figure 5 — Figure 5: Initial loss landscape of two random orthogonal directions in the Lindbladian variational subspace of $V$ defined in Eq. ( 19 ), for the vanilla (left) architecture and the NDE architecture (right), over all four noise-to-unitary ratios $R=0.01,0.1,1,10$ . In all four cases, notice the heatmap of the contour lines indicating a barren landscape on the vanilla architecture, against a much richer loss-landscape for the NDE. The legend for each heatmap shows the NDE having a landscape with heights that vary $\sim 14\times$ that of the vanilla model. This explains the performance gains of the NDE’s robustness from Fig. 3 , corresponding to the experiments in the bottom-left of Fig. 3

Figure 6: Infidelities against renormalised time for the vanilla (top row) and NDE (bottom row) architectures for the neutral atom TFIM with phase (left column), thermal (central) and combined (right) noise. The shaded region corresponds to the training window, i.e. times over which snapshots were taken, and each system size

N=3,4,5,6

is plotted with increasing darker lines for

R=0.1

(green),

R=1

(red) and

R=10

(blue). Here, we renormalise by the noise-to-unitary ratio,

R

. We see NDE infidelities are shifted lower than the vanilla ones for phase (left) noise, with systems undergoing strong noise performing best as explained in the main text. Notice the vanilla and NDE models do not differ as much for the thermal (central) case, despite the increase in robustness shown in Fig. 3 . This is due the unique steady-state effect explained in the main text.

Figure 7: Success rates for a 1D superconducting Hamiltonian model with thermal, phase, and combined noise terms as a function of noise-to-unitary ratio

R

over four orders of magnitude. The top row contains the superconducting model’s Hamiltonian errors and the bottom row contains its Lindbladian errors. These are shown from left to right for (a) phase damping, (b) thermal damping and (c) combined damping. Shaded regions correspond to when the NDE term increases (red) or decreases (green) the fitting error.

Figure 8: Infidelities against renormalised time for the vanilla (top row) and NDE (bottom row) architectures for the superconducting system with phase (left column), thermal (central) and combined (right) noise. The shaded region corresponds to the training window, i.e. times over which snapshots were taken, and each system size

N=3,4,5,6

is plotted with increasing darker lines for

R=0.1

(green),

R=1

(red) and

R=10

(blue). Here, we renormalise by the noise-to-unitary ratio,

R

Figure 9: Success rates for the Heisenberg XYZ model with thermal, phase, and combined noise terms as a function of noise ratio

R

over four orders of magnitude. The top row contains the XYZ model errors and the bottom row contains the noise model errors. These are shown from left to right for (a) phase damping, (b) thermal damping and (c) combined damping. Shaded regions correspond to when the NDE term increases (red) or decreases (green) the fitting error.

Figure 10: The trajectory of two random orthogonal directions in the Hamiltonian variational subspace of

V

defined in Eq. ( 19 ) (left) and the combined Hamiltonian and Lindbladian variational subspace of

V

(right) for the XYZ Heisenberg model with

R=0.01

noise-to-unitary ratio on

N=3

bodies. Comparing the vanilla (blue) and NDE architecture (red), we see the NDE architecture converging to the true values (set at the origin) in both cases. Whereas the vanilla architecture fails to converge in both cases, and its trajectory appears to get stuck, indicating the presence of local minima on those epochs of the vanilla model’s training. Integer values on the plots refer to the start (0) and final (20) epochs.

Figure 11: The loss-landscape of two random orthogonal directions in the Hamiltonian variational subspace of

V

defined in Eq. ( 19 ), for the XYZ Heisenberg Model at the start of epochs

0,3,6,9,12,15,18

and

20

. The current parameter value shown by a red marker

\boldsymbol{\times}

. We see an initially rugged loss-landscape smoothing out as training proceeds until we arrive at a smooth landscape that is quasi-quadratic around the true value. The trajectory is shown by a grey line with the red

\times

marking the current parameter values along the 20-epoch path.

Figure 12: Infidelities against renormalised time for the vanilla (top row) and NDE (bottom row) architectures for the Heisenberg XYZ model with phase (left column), thermal (central) and combined (right) noise. The shaded region corresponds to the training window, i.e. times over which snapshots were taken, and each system size

N=3,4,5,6

is plotted with increasing darker lines for

R=0.1

(green),

R=1

(red) and

R=10

(blue). Here, we renormalise by the noise-to-unitary ratio,

R

. There is little change in infidelity between these architectures, despite the robustness for phase noise improving in Fig. 9 . This is due again to the unique steady-state arising from this combined Hamiltonian and phase-noise model as explained in the main text - see also Sec. 4.1 .

Figure 13: Success rates for Hamiltonian (left) and Lindbladian (right) parameters on the PXP model with thermal noise, as a function of noise ratio

R

over four orders of magnitude. Solid lines represent vanilla architecture success rates and dashed lines represent NODE architecture success rates, with green (red) regions indicated the NODE architecture improving (diminishing) performance. Here we show

N=3,4,5,6

Figure 14: Initial loss landscape for two random orthogonal directions in the Lindbladian variational subspace of

V

defined in Eq. ( 19 ) for the PXP hamiltonian with thermal noise, with ratio

R=0.01

. Here we plot the two random orthogonal variational directions on the dissipative component for the vanilla (left) architecture and the NDE architecture (right). We see the NDE’s rugged landscape, contains many walls, is hindering convergence. This explains the drop in performance we see in the success rate figure for the PXP Hamiltonian with thermal noise, as explained in the main text.

Figure 15: Infidelities against renormalised time for the vanilla (left) and NDE (right) architectures for the PXP system undergoing thermal noise. Here, we renormalise by the noise-to-unitary ratio,

R

. The shaded region corresponds to the training window, i.e. times over which snapshots were taken, and each system size

N=3,4,5,6

is plotted with increasing darker lines for

R=0.1

(green),

R=1

(red) and

R=10

(blue). There is little change in infidelity between these architectures, despite the robustness for phase noise improving in Fig. 9 . This is due again to the unique steady-state arising from this combined Hamiltonian and phase-noise model as explained in the main text - see also Sec. 4.1

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Quantum system identification infers a quantum device’s dynamics, parameters, and environmental noise from experimental measurements.

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Jun 29, 2026J/K move · Enter open · S save

Learning the structure of open quantum systems

Laura Lewis, Ewin Tang, John Wright

UC Berkeley

Quantum System Identification Quantum Machine Learning System Identification Quantum Many-Body Physics

Open paper AskPDF arXiv

We design an algorithm for learning the coefficients of an

n

-qubit constant-local Lindbladian to

\varepsilon

error with

O(g d^2 \log(n) / \varepsilon^2)

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. 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.

Abstract

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