Unraveling Quantum Environments: Transformer-Assisted Learning in Lindblad Dynamics

TL;DR

The paper addresses inferring time-dependent dissipation rates in open quantum systems governed by the Lindblad master equation from observable time-series data. It introduces a Transformer-based regression framework that parameterizes dissipation rates $\gamma(t)$ with a Bernstein polynomial basis, enabling learning of the nonnegative profiles from trajectories of Pauli and photon observables without requiring the initial state or full Hamiltonian knowledge. The authors prove identifiability under mild assumptions and demonstrate accurate recovery of $\gamma(t)$ across single-qubit, two-qubit, and Jaynes–Cummings models, highlighting scalability and robustness. This data-driven approach provides a scalable tool for characterizing environments in quantum technologies and suggests extensions to broader Markovian descriptions and classical stochastic dynamics.

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 quantum systems governed by the Lindblad master equation. Our approach uses time series of observable quantities, such as expectation values of single Pauli operators, as input to learn dissipation profiles without requiring knowledge of the initial quantum state or even the system Hamiltonian. We demonstrate the effectiveness of our approach on a hierarchy of open quantum models of increasing complexity, including single-qubit systems with time-independent or time-dependent jump rates, two-qubit interacting systems (e.g., Heisenberg and transverse Ising models), and the Jaynes--Cummings model involving light--matter interaction and cavity loss with time-dependent decay rates. Our method accurately reconstructs both fixed and time-dependent decay rates from observable time series. To support this, we prove that under reasonable assumptions, the jump rates in all these models are uniquely determined by a finite set of observables, such as qubit and photon measurements. In practice, we combine Transformer-based architectures with lightweight feature extraction techniques to efficiently learn these dynamics. Our results suggest that modern machine learning tools can serve as scalable and data-driven alternatives for identifying unknown environments in open quantum systems.

Figures (5)

• Figure 1: Schematic illustration of the machine learning pipeline. Time-series data of local observables $\langle \sigma_{x,y,z}(t) \rangle$ are used as input features to a Transformer model trained to regress the time-dependent dissipation rate $\gamma(t)$. The dissipation profile is parameterized using a Bernstein polynomial expansion. This approach is applied to single- and two-qubit systems, as well as light–matter interaction models.
• Figure 2: Predictions of dissipation rates for single- and two-qubit systems. (a) shows results from the first three single-qubit examples in Table \ref{['main:table:ex']}. The top row illustrates prediction of time-dependent dissipation rates. The first panel corresponds to Example 1, with a single channel $\gamma_-(t)$ and jump operator $\sigma_-$. The second and third panels correspond to Example 2, which includes both $\gamma_+(t)$ and $\gamma_-(t)$ associated with jump operators $\sigma_+$ and $\sigma_-$. The bottom row shows the predicted Bernstein coefficients $a_0$, $a_1$, and $a_2$ for Example 3, where $\gamma_-(t) = \sum_{k=0}^{2} a_k b_{k,2}(t)$ is expanded in the Bernstein basis. (b) shows the predicted dissipation rates for the Heisenberg two-qubit model, involving four constant rates $\gamma_{\pm,j}$ acting on each qubit $j = 1,2$.
• Figure 3: Prediction of time-dependent dissipation rates in the open Jaynes–Cummings model. The decay rates $\gamma(t)$ and $\kappa(t)$ are each parameterized by a second-order Bernstein polynomial with three unknown coefficients. The model accurately reconstructs all six coefficients from time series of observables $\langle \sigma_z(t) \rangle$, $\langle a^\dagger a(t) \rangle$, and $\langle X \otimes \sigma_z(t) \rangle$. See Table \ref{['main:table:ex']}, column 7 for full input/output specification.
• Figure 4: Prediction results for Example 4 (Main Text) involving a single-qubit system with two quadratic time-dependent loss channels. The dissipation rates are represented by second-order Bernstein polynomials: $\gamma_{\pm}(t) = \sum_{j=0}^{2} a_{j,\pm} b_{j,2}(t)$, and six coefficients are predicted from local observables.
• Figure 5: Prediction results for the transverse-field Ising model with four constant dissipation rates $\gamma_{\pm,j}$ ($j = 1,2$). Each dissipation rate is inferred from local observables $\langle \sigma_{x,y,z}^{(j)}(t) \rangle$ using a Transformer-based regression model. This confirms the method's generalizability beyond the Heisenberg model shown in the main text.

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof