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Physics-informed neural networks and beyond: enforcing physical constraints in quantum dissipative dynamics

Arif Ullah, Yu Huang, Ming Yang, Pavlo O. Dral

Published Jan 1, 2024• 13 citations
Publisher

Abstract

This study shows that NNs do not conserve trace in quantum dissipative dynamics and proposes an uncertainty-aware approach that enforces perfect trace conservation by design.
DOI
10.1039/d4dd00153b

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Abstract Reconstructing the unobservable dynamics of quantum many-body systems from sequential weak measurements presents fundamental information-theoretic challenges due to measurement back-action. In this study, we introduce a Physics-Informed Neural Network (PINN) designed for continuous quantum state filtering on a 14-qubit Transverse Field Ising Model (TFIM). We demonstrate that while unconstrained models can exploit thermodynamic memorization, enforcing causal masking and Hamiltonian differential constraints forces the network to synthesize true unitary evolution. Under these rigorous physical constraints, we observe that the predictive accuracy for non-commuting transverse magnetization invariably reaches a strict empirical asymptote at $R^2 \approx 0.39$. We establish that this ceiling is not an architectural bottleneck, but the direct empirical manifestation of the Quantum Cramér-Rao Bound (QCRB), proving that deep learning models remain fundamentally bound by the Heisenberg uncertainty principle. Key Findings The Heisenberg Wall: We identify a fundamental predictive ceiling ($R^2 \approx 0.39$) for non-commuting observables in continuous quantum state filtering. Physics-Informed Architecture: We demonstrate that enforcing Hamiltonian consistency and causal masking eliminates "identity mapping" artifacts, allowing for true phenomenological simulation. Information Erasure: Our results map the neural network's performance bound to the residual information capacity dictated by the Quantum Cramér-Rao Bound. Keywords Physics-Informed Neural Networks; Quantum Many-Body Systems; Transverse Field Ising Model; Sequential Weak Measurements; Quantum Cramér-Rao Bound; Quantum State Filtering.

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Neural networks (NNs) accelerate simulations of quantum dissipative dynamics. Ensuring that these simulations adhere to fundamental physical laws is crucial, but has been largely ignored in the state-of-the-art NN approaches. We show that this may lead to implausible results measured by violation of the trace conservation. To recover the correct physical behavior, we develop physics-informed NNs (PINNs) that mitigate the violations to a good extend. Beyond that, we propose a novel uncertainty-aware approach that enforces perfect trace conservation by design, surpassing PINNs.
Machine learning has emerged as a promising tool for simulating quantum dissipative dynamics. However, existing methods often struggle to enforce key physical constraints, such as trace conservation, when modeling reduced density matrices (RDMs). While physics-informed neural networks (PINN) aim to address these challenges, they frequently fail to achieve full physical consistency. In this work, we introduce a novel approach that leverages the su(n) Lie algebra to represent RDMs as a combination of an identity matrix and n2 - 1 Hermitian, traceless, and orthogonal basis operators, where n is the system's dimension. By learning only the coefficients associated with the operators, our framework inherently ensures exact trace conservation, as the traceless nature of the operators restricts the trace contribution solely to the identity matrix. This eliminates the need for explicit trace-preserving penalty terms in the loss function, simplifying optimization and improving learning efficiency. We validate our approach on two benchmark quantum systems: the spin-boson model and the Fenna-Matthews-Olson complex. By comparing the performance of four neural network architectures-purely data-driven physics-uninformed neural networks (PUNN), su(n) Lie algebra-based PUNN (su(n)-PUNN), traditional PINN, and su(n) Lie algebra-based PINN (su(n)-PINN)-we highlight the limitations of conventional methods and demonstrate the superior accuracy, robustness, and efficiency of our approach in learning quantum dissipative dynamics.
We propose a physics-informed neural network (PINN) scheme for designing shortcuts to adiabaticity in a three-level non-Hermitian quantum system. The PINN is used to solve an inverse control problem in which the state amplitudes and the auxiliary driving field are learned simultaneously from the Schrödinger residual, the initial and target population constraints and a probability conservation constraint on the control pulse. The learned compensation field counteracts the loss of the intermediate state and enables high-fidelity population inversion in an open-system setting. Importantly, the imposed probability conservation is treated as an auxiliary constraint along the learned trajectory rather than as an intrinsic property of the non-Hermitian Hamiltonian. Under this constrained evolution, the Hamiltonian expectation value evaluated on the obtained state remains real within numerical accuracy. Numerical simulations and independent propagation with the fitted control field verify the population inversion and exhibit strong generalization capability over a range of coupling strengths and dissipation rates, as verified by retraining the network independently for each parameter set. The results demonstrate that PINNs provide a flexible inverse design tool for non-Hermitian shortcut to adiabaticity protocols when the governing dynamics and physical constraints are explicitly incorporated into the loss function.
Abstract Reconstructing the unobservable dynamics of quantum many-body systems from sequential weak measurements presents fundamental information-theoretic challenges due to measurement back-action. In this study, we introduce a Physics-Informed Neural Network (PINN) designed for continuous quantum state filtering on a 14-qubit Transverse Field Ising Model (TFIM). We demonstrate that while unconstrained models can exploit thermodynamic memorization, enforcing causal masking and Hamiltonian differential constraints forces the network to synthesize true unitary evolution. Under these rigorous physical constraints, we observe that the predictive accuracy for non-commuting transverse magnetization invariably reaches a strict empirical asymptote at R2≈0.39R^2 \approx 0.39R2≈0.39. We establish that this ceiling is not an architectural bottleneck, but the direct empirical manifestation of the Quantum Cramér-Rao Bound (QCRB), proving that deep learning models remain fundamentally bound by the Heisenberg uncertainty principle. Key Findings The Heisenberg Wall: We identify a fundamental predictive ceiling (R2≈0.39R^2 \approx 0.39R2≈0.39) for non-commuting observables in continuous quantum state filtering. Physics-Informed Architecture: We demonstrate that enforcing Hamiltonian consistency and causal masking eliminates "identity mapping" artifacts, allowing for true phenomenological simulation. Information Erasure: Our results map the neural network's performance bound to the residual information capacity dictated by the Quantum Cramér-Rao Bound. Keywords Physics-Informed Neural Networks; Quantum Many-Body Systems; Transverse Field Ising Model; Sequential Weak Measurements; Quantum Cramér-Rao Bound; Quantum State Filtering.