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Data-Driven Discovery of Dynamical Signatures of Quantum Phase Transitions via Symbolic Regression

igor merlini

Published Apr 2, 2026• 0 citations
Publisher

Abstract

We present a data-driven framework for discovering dynamical signatures of quantum phase transitions directly from simulated observables. By combining sparse identification of nonlinear dynamics (SINDy), symbolic regression (PySR), and spectral analysis, we extract interpretable features from time series generated by QuTiP simulations of the transverse-field Ising chain. We demonstrate that the dominant oscillation frequency ω of the mean magnetization M(t) scales with system size as ω ~ N^z, with the exponent z taking distinct values in the three phases: z ≈ -1.35 (ferromagnetic), z ≈ -0.91 (critical), z ≈ -0.35 (paramagnetic). At the quantum critical point J = h, the exponent z ≈ -1 is consistent with the known finite-size scaling of the energy gap Δ ~ N⁻¹ in the transverse-field Ising chain, establishing a direct connection between the dynamical frequency and the spectral gap. The frequency signal is robust to measurement noise up to 5% and stable with simulation time, making it a reliable probe of phase structure accessible without prior knowledge of the order parameter or the Hamiltonian. Our results suggest that spectral analysis combined with symbolic regression can serve as an automated discovery tool for identifying dynamical phase structure in quantum many-body systems.

DOI
10.5281/zenodo.19391158

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We present a data-driven framework for discovering dynamical signatures of quantum phase transitions directly from simulated observables. By combining sparse identification of nonlinear dynamics (SINDy), symbolic regression (PySR), and spectral analysis, we extract interpretable features from time series generated by QuTiP simulations of the transverse-field Ising chain. We demonstrate that the dominant oscillation frequency ω of the mean magnetization M(t) scales with system size as ω ~ N^z, with the exponent z taking distinct values in the three phases: z ≈ -1.35 (ferromagnetic), z ≈ -0.91 (critical), z ≈ -0.35 (paramagnetic). At the quantum critical point J = h, the exponent z ≈ -1 is consistent with the known finite-size scaling of the energy gap Δ ~ N⁻¹ in the transverse-field Ising chain, establishing a direct connection between the dynamical frequency and the spectral gap. The frequency signal is robust to measurement noise up to 5% and stable with simulation time, making it a reliable probe of phase structure accessible without prior knowledge of the order parameter or the Hamiltonian. Our results suggest that spectral analysis combined with symbolic regression can serve as an automated discovery tool for identifying dynamical phase structure in quantum many-body systems.
Sparse identification of nonlinear dynamics (SINDy) is a data-driven framework for estimating classical nonlinear dynamical systems from time-series data. In this approach, system dynamics is represented as a linear combination of a predefined set of basis functions, and the corresponding coefficients are sparsely estimated from observed time-series data. In this study, we propose sparse identification of quantum Hamiltonian dynamics (SIQHDy), a SINDy-inspired quantum circuit learning framework for estimating quantum Hamiltonian dynamics from time-series data of quantum measurement outcomes. In SIQHDy, the unitary evolution of a quantum Hamiltonian system is expressed as a product of basis quantum circuits, and the corresponding circuit parameters are estimated through sparsity-promoting optimization. We numerically demonstrate that SIQHDy accurately reconstructs the dynamics of single-, three-, and five-spin systems, and exhibits robustness to measurement noise in the three-spin case. Furthermore, we propose an extension of SIQHDy for scenarios with limited accessible observables and evaluate its performance in identifying two-spin systems and in network-structure identification for five-spin systems.
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We provide a general machine learning methodology that integrates classical shadow representations with unsupervised principal component analysis (PCA) to explore various quantum phase transitions. By sampling spin configurations from random Pauli measurements, our approach can effectively analyze hidden statistical patterns in the data, thereby capturing the distinct signatures of quantum criticality through their fluctuations. We benchmark this approach across various spin-1/2 systems, including the 1D XZX cluster-Ising model, the 1D bond-alternating XXZ model, the 2D transverse-field Ising model, and the 2D Kitaev honeycomb model. We show that PCA not only reliably detects and distinguishes both symmetry-breaking and topological transitions, but also enables their qualitative classification based on characteristic fluctuation patterns. Our data-driven approach does not require any knowledge of the Hamiltonian or explicit order parameters, and can therefore be a general and applicable tool for probing new quantum phases.