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arXiv:2502.15141 · The Journal of Chemical Physics

Machine learning meets su(n) Lie algebra: Enhancing quantum dynamics learning with exact trace conservation

Arif Ullah, Jeremy O. Richardson

Published Jun 24, 2025• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/5.0266604

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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.
Accurately modeling quantum dissipative dynamics remains challenging due to environmental complexity and non-Markovian memory effects. Although machine learning provides a promising alternative to conventional simulation techniques, most existing models employ real-valued neural networks (RVNNs) that inherently mismatch the complex-valued nature of quantum mechanics. By decoupling the real and imaginary parts of the density matrix, RVNNs can obscure essential amplitude-phase correlations, compromising physical consistency. Here, we introduce complex-valued neural networks (CVNNs) as a physics-consistent framework for learning quantum dissipative dynamics. CVNNs operate directly on complex-valued inputs, preserve the algebraic structure of quantum states, and naturally encode quantum coherences. Through numerical benchmarks on the spin-boson model and few variants of the Fenna-Matthews-Olson complex, we demonstrate that CVNNs outperform RVNNs in convergence speed, training stability, and physical fidelity-including significantly improved trace conservation and Hermiticity. These advantages increase with system size and coherence complexity, establishing CVNNs as a robust, scalable, quantum-aware classical approach for simulating open quantum systems in the pre-fault-tolerant quantum era.
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.
Physics-informed neural networks (PINNs) have emerged as a versatile approach for solving nonlinear partial differential equations (PDEs), yet achieving high accuracy efficiently using these techniques remains challenging for high-dimensional or multiscale systems. Here, we present a hybrid quantum-classical framework that enhances Quantum PINNs (QPINNs) through adaptive collocation point sampling and loss-aware attention mechanisms. By dynamically prioritizing points in regions with large PDE residuals or steep solution gradients, our method mitigates the spectral bias inherent in conventional PINNs. Current Quantum Physics-Informed Neural Networks are commonly assumed to be limited by the expressive power of quantum circuits. In our work, we observed that, across diverse differential equations, optimization - not only expressivity - can be an important bottleneck. Furthermore, a trainable loss-weighting scheme balances contributions from physics residuals, boundary conditions, and data fidelity during training. Integrating these strategies with quantum computing techniques (including variational quantum circuits and quantum gradient estimation) can yield at least a 60% improvement in solution accuracy under specific regimes for benchmark fluid flows and reaction-diffusion systems. Finally, we argue that merely increasing model expressivity is insufficient for resolving complex PDEs via QPINNs, as they remain constrained by the structural optimization limitations of classical PINNs. This framework provides a scalable pathway for quantum-enhanced scientific machine learning, bridging physics-based modeling with emerging quantum computational capabilities.
Quantum many-body systems are of great interest for many research areas, including physics, biology, and chemistry. However, their simulation is extremely challenging, due to the exponential growth of the Hilbert space with system size, making it exceedingly difficult to parameterize the wave functions of large systems by using exact methods. Neural networks and machine learning, in general, are a way to face this challenge. For instance, methods like tensor networks and neural quantum states are being investigated as promising tools to obtain the wave function of a quantum mechanical system. In this tutorial, we focus on a particularly promising class of deep learning algorithms. We explain how to construct a Physics-Informed Neural Network (PINN) able to solve the Schrödinger equation for a given potential, by finding its eigenvalues and eigenfunctions. This technique is unsupervised, and utilizes a novel computational method in a manner that is barely explored. PINNs are a deep learning method that exploit automatic differentiation to solve integro-differential equations in a mesh-free way. We show how to find both the ground and the excited states. The method discovers the states progressively by starting from the ground state. We explain how to introduce inductive biases in the loss to exploit further knowledge of the physical system. Such additional constraints allow for a faster and more accurate convergence. This technique can then be enhanced by a smart choice of collocation points in order to take advantage of the mesh-free nature of the PINN. The methods are made explicit by applying them to the infinite potential well and the particle in a ring, a challenging problem to be learned by an artificial intelligence agent due to the presence of complex-valued eigenfunctions and degenerate states
In this paper, authors present a physics-informed neural networks (PINN) approach for solving the optimal control problem (OCP) of the stochastic Schrödinger equation (SSE), with an emphasis on applications in quantum optics. The SSE models quantum systems under uncertainty, a scenario often encountered in open quantum systems where environmental interactions introduce randomness into the system dynamics. These stochastic dynamics are particularly relevant in quantum optics, where precise control over quantum states is crucial for applications such as quantum computing, secure communication, and measurement protocols. Traditional numerical methods for solving such problems face limitations due to high computational costs and the complexity associated with stochastic partial differential equations. The proposed approach utilises PINN, a type of deep learning model that integrates physical laws directly into the learning process by incorporating loss functions based on governing equations, initial and boundary conditions. The fidelity-based objective function is constructed to achieve the desired quantum state so as to develop an OCP solving algorithm. The proposed work is distinguished between deterministic drift-based training and post-training stochastic validation. Extensive numerical experiments are demonstrated near-unity deterministic fidelity. Further, a systematic architecture study validated that a moderately deep network yielded optimal performance with reduced computational cost. Fidelity dissipation analysis confirmed that the method maintained accuracy even under strong damping. These findings established the PINN as an efficient and physically consistent framework for quantum optimal control in open systems.
Implementing quantum gates on quantum computers can require the application of carefully shaped pulses for high-fidelity operations. We explore the use of physics-informed neural networks (PINNs) for quantum optimal control to assess their usefulness in predicting such pulses. Our PINN is a feedforward neural network that utilizes an unsupervised learning approach, whose loss function includes terms that enforce the equations that govern the evolution of a quantum system, measure how close the learned unitary is to the target unitary operation, and ensure state normalization. We use a sinusoidal activation function and adopt variance-type weight initialization, tailored to our activation function. By analyzing the model's performance with important machine learning metrics, we demonstrate that the choice of our architecture is well-suited for this type of problem. We ensure that our network avoids the vanishing and exploding gradients with our relevant choices. We build two different PINNs, one based on the Schrödinger equation and another one based on the Lindblad equation. Our PINNs are able to discover high-fidelity two-qubit gate pulses for a variety of quantum operations, demonstrating its flexibility and robustness. We build two different PINNs, one based on the Schrödinger equation and another one based on the Lindblad equation. Our PINNs are able to discover high-fidelity two-qubit gate pulses for a variety of quantum operations, demonstrating its flexibility and robustness.