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arXiv:2505.06928 · Physical Review A

Unraveling quantum environments: Transformer-assisted learning in Lindblad dynamics

Chi‐Sheng Chen, En-Jui Kuo

Published Sep 16, 2025• 1 citation
arXivHTMLPDF

Abstract

Understanding dissipation in open quantum systems is crucial for the development of robust quantum technologies. In this work, we introduce a transformer-based machine learning framework to infer time-dependent dissipation rates in systems governed by the Lindblad master equation. Our method requires only time series of observable quantities, such as 〈 σ x ( t ) 〉 , 〈 σ y ( t ) 〉 , and 〈 σ z ( t ) 〉 , and does not require prior knowledge of the initial quantum state or the system Hamiltonian during inference. We demonstrate the effectiveness of our approach across a hierarchy of open quantum models, including single-qubit systems with time-independent and time-dependent jump rates, two-qubit interacting systems (e.g., Heisenberg and transverse Ising models), and the Jaynes-Cummings model with cavity loss and time-dependent decay. Our method accurately reconstructs both fixed and time-varying dissipation profiles from observable trajectories. We further show that under reasonable assumptions, the jump rates in these models are uniquely determined by a small set of measurable observables. The results suggest that transformer-based learning provides a scalable and data-driven route for identifying environmental effects in open quantum systems.

DOI
10.1103/gsxk-45mk

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Understanding dissipation in open quantum systems is crucial for the development of robust quantum technologies. In this work, we introduce a transformer-based machine learning framework to infer time-dependent dissipation rates in systems governed by the Lindblad master equation. Our method requires only time series of observable quantities, such as 〈 σ x ( t ) 〉 , 〈 σ y ( t ) 〉 , and 〈 σ z ( t ) 〉 , and does not require prior knowledge of the initial quantum state or the system Hamiltonian during inference. We demonstrate the effectiveness of our approach across a hierarchy of open quantum models, including single-qubit systems with time-independent and time-dependent jump rates, two-qubit interacting systems (e.g., Heisenberg and transverse Ising models), and the Jaynes-Cummings model with cavity loss and time-dependent decay. Our method accurately reconstructs both fixed and time-varying dissipation profiles from observable trajectories. We further show that under reasonable assumptions, the jump rates in these models are uniquely determined by a small set of measurable observables. The results suggest that transformer-based learning provides a scalable and data-driven route for identifying environmental effects in open quantum systems.
The Lindblad master equation is the standard framework for describing the non-unitary evolution of open quantum systems, where environmental interactions induce dissipation and decoherence. When both the system Hamiltonian and the dissipation rates are partially unknown or explicitly time-dependent, traditional analytical inversion and system-identification techniques become intractable. Recent works have demonstrated that Transformer-based models can infer unknown dissipation rates from observable time series, yet these approaches typically rely on hand-crafted statistical features under idealized and highly restricted conditions. Here we advance the paradigm by introducing a raw time-series Transformer that directly ingests the full trajectories of Pauli expectation values ⟨σx(t)⟩\langleσ_x(t)\rangle⟨σx​(t)⟩, ⟨σy(t)⟩\langleσ_y(t)\rangle⟨σy​(t)⟩, and ⟨σz(t)⟩\langleσ_z(t)\rangle⟨σz​(t)⟩, thereby fully exploiting the self-attention mechanism for temporal modeling. The architecture is further extended to jointly learn unknown Hamiltonian parameters, handle multiple dissipation channels, and operate robustly under realistic measurement noise. Across all tested scenarios the model achieves consistently high reconstruction accuracy while eliminating manual feature engineering. This provides a scalable, robust, and versatile framework for quantum environment sensing in realistic open quantum systems.
Learning dissipation rates in large-scale open quantum systems is a major obstacle for near-term quantum technologies, as existing Lindblad estimation methods are typically limited to small system sizes due to the computational complexity of repeatedly solving the Lindblad equation during optimization. Here, we propose a scalable noise-learning framework for Lindblad dissipation rates that combines a stochastic simulation method, the Tensor Jump Method (TJM), with gradient-free optimization of a least-squares cost-function defined on time series of local-observable expectation values. We demonstrate the approach on two noise models in the Ising model: a site-resolved (local) model, in which independent dissipation rates are learned for each site up to Nsite=16N_{\mathrm{site}}=16Nsite​=16, and a spatially homogeneous (global) model with only seven parameters, scaled to Nsite=160N_{\mathrm{site}}=160Nsite​=160 sites.We complement these numerical results with a series of exact, provable guarantees: the Frobenius variance of the TJM density-matrix estimator is shown to equal (1−Tr[ρ2])/Ntraj(1-\mathrm{Tr}[ρ^2])/N_{\mathrm{traj}}(1−Tr[ρ2])/Ntraj​, an exact purity-based characterization of the stochastic estimation error; the corresponding purity evolution is proven to be monotonically non-increasing for Hermitian jump operators; and, under a finite covariance distance assumption, the standard deviation of the cost-function is shown to decrease with system size, so that fewer trajectories are needed to reach a fixed target accuracy as the system grows. Together, this combination of scalable numerics and rigorous theoretical guarantees positions TJM-based noise learning as a practical foundation for characterizing dissipation in large quantum devices and for guiding future work on error mitigation and quantum error correction.
The dynamics of open quantum systems play a crucial role in quantum information science. However, obtaining numerically exact solutions for the Lindblad master equation is often computationally expensive. Recently, machine learning techniques have gained considerable attention for simulating open quantum system dynamics. In this paper, we propose a deep learning model based on time series prediction (TSP) to forecast the dynamical evolution of open quantum systems. We employ the positive operator-valued measure (POVM) approach to convert the density matrix of the system into a probability distribution and construct a TSP model based on Transformer neural networks. This model effectively captures the historical evolution patterns of the system and accurately predicts its future behavior. Our results show that the model achieves high-fidelity predictions of the system's evolution trajectory in both short- and long-term scenarios, and exhibits robust generalization under varying initial states and coupling strengths. Moreover, we successfully predicted the steady-state behavior of the system, further proving the practicality and scalability of the method.
A precise understanding of the influence of a quantum system's environment on its dynamics, which is at the heart of the theory of open quantum systems, is crucial for further progress in the development of controllable large-scale quantum systems. However, existing approaches to account for complex system-environment interaction in the presence of memory effects are either based on heuristic and oversimplified principles or give rise to computational difficulties. In practice, one can leverage on available experimental data and replace first-principles simulations with a data-driven analysis that is often much simpler. Inspired by recent advances in data analysis and machine learning, we propose a data-driven approach to the analysis of the non-Markovian dynamics of open quantum systems. Our method allows, on the one hand, capturing the most important characteristics of open quantum systems such as the effective dimension of the environment and the spectrum of the joint system-environment quantum dynamics, and, on the other hand, reconstructing a predictive model of non-Markovian quantum dynamics, and denoising the measured quantum trajectories. We demonstrate the performance of the proposed approach with various models of open quantum systems, including a qubit coupled with a finite environment, a spin-boson model, and the damped Jaynes-Cummings model.
Learned open-quantum-system models have broad applications in device characterization, device calibration, realistic noise model construction, and bounding observable estimates in quantum simulation. To this end, we develop a highly scalable Lindblad learning protocol assuming an effective time-independent Lindblad generator with local interactions. For a given operation that can be applied repeatedly on a quantum computer, the protocol learns the Lindbladian by forming a linear system of equations for the model parameters in terms of a set of observable values and their gradients. The required gradient information is obtained by fitting time-series data with sums of exponentially damped sinusoids and differentiating those curves. We develop a robust curve-fitting procedure that finds a parsimonious representation of the data up to shot noise. We demonstrate the approach by learning the Lindbladian for a full layer of gates on a 156-qubit superconducting quantum processor, providing the first learning experiment of this kind. We study the effects of state-preparation and measurement errors and limitations on the operations that can be learned. For improved performance under readout errors, we propose an optional fine-tuning strategy that improves the fit between the time-evolved model and the measured data.
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 \emph{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=6N=6N=6 qubits with fewer than 5×1055 \times 10^55×105 shots.