Machine learning non-Markovian two-level quantum noise spectroscopy

TL;DR

The paper introduces a data-driven framework to automatically characterize quantum noise spectra in non-Hermitian two-level systems by mapping time-resolved TLS dynamics to environmental parameters. Using pure dephasing and spin-boson models, it constructs numerically exact datasets and trains FFNN, Random Forest, and SVR models to classify Ohmicity, regress coupling strengths, and quantify non-Markovianity via a time-averaged trace-distance metric. The results show near-perfect accuracy for Ohmicity classification and state-of-the-art regression performance for SD parameters and non-Markovian labels, with FFNN and RFR generally outperforming SVR. This work enables automated extraction of environmental properties from TLS dynamics and has potential applications in quantum dissipation studies and noise spectroscopy of complex baths. The approach, validated on 1–2 TLS models and datasets generated via HEOM, provides a path toward applying ML-based spectroscopy to experimental data where the bath details are unknown.

Abstract

We develop machine learning models for the automated characterization of quantum noise spectroscopy for non-Hermitian two-level systems. We use the Random Forest, Support Vector and Feed-Forward Neural Network regression algorithms to perform a highly accurate regression of the two-level system-bath coupling strength. High accuracy Ohmicity classification was implemented to provide a complete characterization of the spectral density function. We define a time-averaged trace-distance metric to feed the machine learning algorithms which, together with numerically exact populations as inputs, produce a highly accurate non-Markovian regression spanning the transition from fast to slow baths and from weak to strong coupling regimes of the interaction. The dynamics database of the non-Hermitian systems has been built up within the independent spin-boson and pure dephasing model.

Figures (9)

• Figure 1: Sketch of a generic TLS $\mathbf{S}$ of transition frequency $\omega_0$ and tunneling energy $\Delta$, coupled to a bosonic bath $\mathbf{B}$ of infinitely many oscillators. The coupling constants $g_k$ define the S-B interaction strength of each mode $\omega_k$.
• Figure 2: Comparison of different spectral density functions. The Lorentz-Drude SDF with $\gamma = 0.1$ and $\omega_c = 0.5$ is shown in blue. The orange, green and red curves represent Ohmic SDFs $J_{1}(\omega)$ with $\eta=0.5$, $\eta=0.25$, and $\eta=0.1$, respectively, and cut-off $\omega_c=0.5$. The inset shows the SDFs from Eq. \ref{['Eq:Js(w)']} for fixed $\omega_c=0.5$, $\eta=0.25$ and different $s$ values.
• Figure 3: Case $\Delta= 0$: Dynamics of the real part of the coherence $\text{Re }\rho_{01}$ as a function of Ohmicity $s$ and coupling strength $\eta$; (a) sub-Ohmic ($s=0.1$), Ohmic ($s=1$) and super-Ohmic ($s=3.5$) spectral densities; $\omega_c=0.5$, $\eta=0.25$ and $\omega_0=1$; (b) Ohmic dynamics for $\eta=0.1, 0.4, 0.9$, $\omega_c=0.5$ and $\omega_0=1$. In both cases, Eq. \ref{['Eq:rhoS_analytical']} was used with initial condition $\hat{\rho}_S(0)= |+\rangle \langle+|$ at $T=0$. (c) Case $\Delta\neq 0$: TLS population difference $P(t)$ as a function of time and the ratio of the bath bandwidth to the tunnelling energy, $\omega_c/\Delta$ ($\hbar=1$). The numerically exact calculation allows the full transition from fast (Markovian) to slow (non-Markovian) baths; $\gamma = 0.25$, $\omega_0=1$, $k_\text{B}T = 0.5 \Delta$ and TLS initial state $\hat{\rho}_S(0)= |1\rangle \langle1|$.
• Figure 4: Normalized confusion matrix showing the results of Ohmicity classification using a FFNN on the test set with non-separated data.
• Figure 5: (a) Results of the $\eta$ regression using RFR showing the comparison between predicted and true values. The inset shows the regression for separated data and the colorbar shows the absolute error. (b) The contrast between actual values and model predictions is shown over the parameter space of the test set.