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Learning spectral density functions in open quantum systems

Felipe Peleteiro, João Victor Shiguetsugo Kawanami Lima, Pedro Marcelo Prado, Felipe Fernandes Fanchini, Ariel Norambuena

Abstract

Spectral density functions quantify how environmental modes couple to quantum systems and govern their open dynamics. Inferring such frequency-dependent functions from time-domain measurements is an ill-conditioned inverse problem. Here, we use exactly solvable spin-boson models with pure-dephasing and amplitude-damping channels to reconstruct spectral density functions from noisy simulated data. First, we introduce a parameter estimation approach based on machine learning regressors to infer Lorentzian and Ohmic-like spectral density parameters, quantifying robustness to noise. Second, we show that a cosine transform inversion yields a physics-consistent spectral prior estimation, which is refined by a constrained neural network enforcing positivity and correct asymptotic behaviour. Our neural network framework robustly reconstructs structured spectral densities by filtering simulated noisy signals and learning general functional dependencies.

Learning spectral density functions in open quantum systems

Abstract

Spectral density functions quantify how environmental modes couple to quantum systems and govern their open dynamics. Inferring such frequency-dependent functions from time-domain measurements is an ill-conditioned inverse problem. Here, we use exactly solvable spin-boson models with pure-dephasing and amplitude-damping channels to reconstruct spectral density functions from noisy simulated data. First, we introduce a parameter estimation approach based on machine learning regressors to infer Lorentzian and Ohmic-like spectral density parameters, quantifying robustness to noise. Second, we show that a cosine transform inversion yields a physics-consistent spectral prior estimation, which is refined by a constrained neural network enforcing positivity and correct asymptotic behaviour. Our neural network framework robustly reconstructs structured spectral densities by filtering simulated noisy signals and learning general functional dependencies.
Paper Structure (15 sections, 21 equations, 4 figures)

This paper contains 15 sections, 21 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Models
  3. Phenomenological models and parameter estimation
  4. Simulated data and machine learning regressors
  5. Neural network learning with prior cosine transform
  6. Results of SDF reconstruction for the PD model using neural networks refinement
  7. Conclusions
  8. GKSL derivation and dependence on the SDF
  9. Algorithmic protocol for reconstructing spectral density functions using cosine transform
  10. Configuration.
  11. Preconditioning.
  12. Cosine-based prior.
  13. Neural-network refinement.
  14. Diagnostics and outputs.
  15. Robustness checks.

Figures (4)

  • Figure 1: Conceptual overview of the spectral density reconstruction problem. A quantum system coupled to a structured environment is characterized by a spectral density function $J(\omega)$, which determines measurable dynamical signals $f_m(t)$ through a known map $J(\omega) \mapsto f_m(t)$. The inverse problem, inferring $J(\omega)$ from noisy time-domain data, is ill-conditioned under finite time sampling and measurement noise. We address this challenge via two complementary AI-based strategies: (i) a parametric regression scheme for estimating SDF parameters, and (ii) a nonparametric approach combining discrete cosine transform (DCT) inversion with a constrained neural network refinement to obtain physically consistent spectral reconstructions.
  • Figure 2: Machine learning accuracy for parametric SDF inference in the amplitude-damping (AD) channel. Each row corresponds to a target parameter $(\lambda,\gamma_0,\omega_b)$ and each column to a regression model. Colors show $\log_{10}(\mathrm{MSE})$ and $\log_{10}(\mathrm{MAE})$ for noisy (top) and noiseless (bottom) signals, highlighting robustness to measurement noise and model-dependent performance.
  • Figure 3: Noise-free reconstruction using the discrete cosine transform. (a) Original coherence signal generated from the phenomenological SDF (Eqs. \ref{['Jbulk']}-\ref{['Jloc2']}). (b) Signal $H(t) = G"(t)$ used in the cosine transform \ref{['J_from_cosine_add']}. (c) True and estimated SDFs $J(\omega)$ using the discrete cosine transform, without machine learning processing.
  • Figure 4: Reconstruction of the SDF using a signal with $5\%$ of noise in the coherence. (a) Original coherence for the phenomenological model including noise. (b) True and discrete cosine transform (DCT) SDFs. (c) Neural network reconstruction of the SDF using $J_{\rm DCT}(\omega)$ in the preprocessing stage. (d) Loss function convergence for the two states: preprocessing (phase A) and postprocessing (phase B). Phase B corresponds to the learning of the SDF via a neural network.