Equation-Free Discovery of Open Quantum Systems via Paraconsistent Neural Networks

TL;DR

This work presents the ParaQNN (ParaQuantum neural network) architecture, an equation-free framework for physical discovery that succeeds in maintaining oscillatory and damping dynamics with high accuracy even in extrapolation regions where training data are unavailable, by discovering the underlying structural invariants from noisy measurements.

Abstract

Modeling the dynamics of open quantum systems on noisy intermediate-scale quantum (NISQ) devices constitutes a major challenge, as high noise levels and environmental degradations lead to the decay of pure quantum states (decoherence) and energy losses. This situation represents one of the most important problems in the field of quantum information technologies. While existing data-driven methods struggle to generalize beyond the training data (extrapolation), physics-informed neural networks (PINNs) require predefined governing equations, which limit their discovery capability when the underlying physics is incomplete or unknown. In this work, we present the ParaQNN (ParaQuantum neural network) architecture, an equation-free framework for physical discovery. ParaQNN disentangles multi-scale dynamics without relying on a priori laws by employing a dialetheist logic layer that models coherent signal and decoherent noise as independent yet interacting channels. Through extensive benchmark tests performed on Rabi oscillations, Lindblad dynamics, and particularly complex ``mixed regimes'' where relaxation and dephasing processes compete, we show that ParaQNN exhibits a consistent performance advantage compared to Random Forest, XGBoost, and PINN models with incomplete physical information. Unlike its competitors, ParaQNN succeeds in maintaining oscillatory and damping dynamics with high accuracy even in extrapolation regions where training data are unavailable, by ``discovering'' the underlying structural invariants from noisy measurements. These results demonstrate that paraconsistent logic provides a structurally more stable epistemic foundation than classical methods for learning quantum behavior in situations where mathematical equations prove insufficient.

Figures (6)

• Figure 1: Equation-free reconstruction of damped Rabi oscillations under mixed noise.(a) Synthetic measurements of the excited-state population $P(|1\rangle)$ ($N=10,000$, $t \in [0, 8.0] \, \mu\text{s}$), corrupted by mixed noise; the solid curve indicates the hidden noise-free trajectory. (b) Convergence of the paraconsistent loss $\mathcal{L}_{\mathrm{ParaQNN}}$ (train/validation). (c) Adaptive evolution of the contradiction interaction parameter $\alpha$ during training. (d) Reconstruction result: ParaQNN output (red) overlaid on noisy measurements (grey) and the ideal trajectory (dashed), demonstrating robust recovery of oscillations and damping in an equation-free manner.
• Figure 2: Quantitative fidelity benchmarking across architectural paradigms (Rabi regime). Values are reported as mean $\pm$ standard deviation over 5 independent training seeds on the fixed dataset (seed 42). Test-set MSE (log scale) comparing ParaQNN against Random Forest (RF), XGBoost (XGB), GAN, and two PINN variants (Incomplete vs. Known physics). The reported values represent the mean performance computed over 5 independent random seeds. Numerical annotations indicate the exact MSE for each model; ParaQNN achieves the lowest error ($1.9 \times 10^{-4}$), outperforming the closest baselines by approximately two orders of magnitude.
• Figure 3: Equation-free discovery of open-system quantum dynamics (Lindblad regime).(a) Noisy measurements of $P(|1\rangle)$ ($N=25,000$, $t\in[0,5]$) generated from Lindblad dynamics; the solid curve indicates the hidden noise-free trajectory. (b) Convergence of the paraconsistent loss $\mathcal{L}_{\mathrm{ParaQNN}}$. (c) Evolution of the contradiction interaction parameter $\alpha$. (d) Reconstruction result: ParaQNN output (red) overlaid on noisy measurements (grey) and the ideal trajectory (dashed), demonstrating high-fidelity recovery in a dissipative regime.
• Figure 4: Quantitative benchmarking of open-system reconstruction (Lindblad regime). Values are reported as mean $\pm$ standard deviation over 5 independent training seeds on the fixed dataset (seed 42). Test-set MSE (log scale) comparing ParaQNN against RF, XGBoost, GAN, and PINN variants (Incomplete vs. Known physics). ParaQNN attains the lowest error ($4.9 \times 10^{-7}$), demonstrating robust equation-free recovery even when standard physics priors are incomplete or mismatched.
• Figure 5: Equation-free discovery of time-dependent quantum dynamics (mixed regime).(a) Synthetic measurements of $P(|1\rangle)$ ($N=50{,}000$ over $\texttt{time\_span}=10.0$) with regime transitions encoded in regime_switches; the solid curve indicates the hidden noise-free Hamiltonian dynamics. (b) Convergence of the paraconsistent loss $\mathcal{L}_{\mathrm{ParaQNN}}$. (c) Adaptive evolution of $\alpha$, reflecting regime-dependent calibration of contradiction coupling. (d) Reconstruction result: ParaQNN output (red) overlaid on noisy measurements (grey) and the ideal trajectory (dashed), capturing transitions and steady-state behavior without explicit boundary information.

Theorems & Definitions (2)

• Lemma 1: Convergence of the Contradiction Regulator $\alpha$
• proof : Proof Sketch