Compact representation and long-time extrapolation of real-time data for quantum systems using the ESPRIT algorithm

Abstract

Representing real-time data as a sum of complex exponentials provides a compact form that enables both denoising and extrapolation. As a fully data-driven method, the Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm is agnostic to the underlying physical equations, making it broadly applicable to various observables and experimental or numerical setups. In this work, we consider applications of the ESPRIT algorithm primarily to extend real-time dynamical data from simulations of quantum systems. We evaluate ESPRIT's performance in the presence of noise and compare it to other extrapolation methods. We demonstrate its ability to extract information from short-time dynamics to reliably predict long-time behavior and determine the minimum time interval required for accurate results. We discuss how this insight can be leveraged in numerical methods that propagate quantum systems in time, and show how ESPRIT can predict infinite-time values of dynamical observables, offering a purely data-driven approach to characterizing quantum phases.

Figures (11)

• Figure 1: Extrapolation of the function $f(t)$ (Eq. (\ref{['eq:test_func']})) using different methods (from left to right: linear prediction, higher-order dynamic mode decomposition (HO-DMD), recurrent neural network (RNN), and ESPRIT). The function $f(t)$ is shown as solid and dashed black lines for the real and imaginary parts, respectively. Colored lines represent predictions from various methods: different orders $p$ for linear prediction and $n_g$ for HO-DMD, as well as different strategies for incorporating the final value in ESPRIT. The threshold used for the SVD decomposition in both HO-DMD and ESPRIT is $10^{-6}$. The details of the RNN used to predict the dynamics are specified in App. \ref{['sec:append_ML_dyn']}. Shaded regions correspond to extrapolations with rapid oscillations, causing the area to appear filled. Panels (a): Without noise. For ESPRIT and HO-DMD, the predictions are so accurate that they coincide with the black lines representing the analytical test function $f(t)$. Panels (b): With Gaussian noise with $\sigma=10^{-2}$. In both panels, the sampling time $t_{\text{samp}}$ (red vertical dashed line), that is the time up to which the data is available to the algorithms, increases from top to bottom with $t_{\text{samp}}=2.5,\ 5.0,\ 7.5,$ and $10.0$, respectively.
• Figure 2: Influence of noise on the fidelity of long-term predictions by ESPRIT. Panels (a): Predicted values of $f_{\infty}^{\text{predict}}$ as a function of sampling time $t_{\text{samp}}$ for different noise strengths $\sigma$ and numbers of exponents $M$ (including the zero exponent). The gray dashed line indicates the true value of $f_\infty=0.2$. Panel (b): Sampling time $t_{\text{samp}}$ required to achieve an accuracy of $|f_\infty-f_\infty^{\text{predict}}|<0.02$, shown for various values of $M$ (which includes the zero exponent). The red dashed lines in panels (a) and (b) highlight the values $t_{\text{samp}}=5.0$ and $t_{\text{samp}}=10.0$ and serve as a guide for the eye and indicate a regime of $t_{\text{samp}}$ for which an accurate extrapolation is desirable. Panels (c): Influence of the final value $f_\infty$ on the predictability of the ESPRIT algorithm at different noise levels. Sampling time required to reach a fixed accuracy of $0.05$ (left) and a relative accuracy of $10\%$ of $f_\infty$ (right), whereby colors indicate different noise levels $\sigma$. All three panels show results from ten independent runs with distinct random noise seeds.
• Figure 3: Application of the ESPRIT algorithm to QMC data across increasing inverse temperatures $\beta$ (left to right). Panels (a) show results for the standard ESPRIT algorithm with all exponents included and no additional postprocessing, while panels (b) show results where exponentially increasing components have been removed (see Sec. \ref{['sec:post_processing']} for details). In both panels, the top two rows show the particle-hole symmetric restricted propagators $\varphi$, with black solid and dashed lines representing the real and imaginary parts obtained from QMC, respectively. The red dashed vertical line marks the sampling time $t_{\text{samp}}$, where ESPRIT achieves a prediction accuracy of $10^{-3}$ with respect to the QMC data. Bright green lines show ESPRIT extrapolations from this $t_{\text{samp}}$. Faint red lines indicate ESPRIT predictions using shorter sampling times $0.2/\Gamma$, $0.5/\Gamma$, $1.0/\Gamma$, and $2.0/\Gamma$, where applicable. The bottom row in both panels displays the extracted exponents $\xi_p$, which are shared by all restricted propagators, as a function of time provided to the algorithm, with scatter point opacity determined by the corresponding maximum absolute value of the prefactor $C_p$, making more significant exponents visually more prominent. The tolerance for the SVD decomposition in the ESPRIT algorithm was set to $10^{-9}$.
• Figure 4: Spin-polarization in the spin–boson model in the deep sub-Ohmic regime ($s=0.2$) at low temperature ($\beta=100/\Delta$). Panels (a), top: Time evolution of the spin-polarization for various coupling strengths $\alpha$; black lines are inchworm QMC data from Ref. goulko_transient_2025, orange lines are ESPRIT extrapolations. Panels (a), bottom: Infinite-time spin-polarization. Red dashed lines are values from Ref. goulko_transient_2025, black dots show ESPRIT predictions from data up to time $t$, and the light green dashed line fits these to an exponential decay. Panel (b): Onset of localization as a function of $\alpha$. Red dashed lines are from Ref. goulko_transient_2025, gray crosses show ESPRIT predictions based on the full time series, successively truncated by removing up to the last five time steps, where many of these crosses are so close that they overlap in the plot and appear as black crosses. Blue dots show the final value obtained by fitting an exponentially decaying function to the ESPRIT predictions over time (corresponding to light green dashed lines in panels (a), bottom). Green arrows indicate the data sets shown in panel (a).
• Figure 5: Illustration of the function $f(t)$, which consists of a finite number of exponential poles (Eq. (\ref{['eq:test_FT_1']})), and its Fourier transform under different noise conditions. Panles (a) Real and imaginary parts of the function $f(t)$ defined in Eq. (\ref{['eq:test_FT_1']}) as a function of time. Noiseless data is shown in the left panel, and data with Gaussian noise with $\sigma = 10^{-2}$ is shown in the right panel. Fourier transform of the same function computed with ESPRIT (panel b) and FFT (panel c), shown for noiseless (left panels) and noisy (right panels) cases. For both algorithms, the sampling time $t_{\mathrm{samp}}$ indicates the time up to which data is available to the algorithms.