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.
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.
Abstract To efficiently complete quantum information processing tasks, quantum neural networks (QNNs) should be introduced rather than the common classical neural networks, but the QNNs in the current noisy intermediate-scale quantum era cannot perform better than classical neural networks because of scale and the efficiency limits. So if the quantum properties can be introduced into classical neural networks, more efficient classical neural networks may be constructed for tasks in the field of quantum information. Complex numbers play an indispensable role in the standard quantum theory, and constitute an important feature in quantum theory. So if complex numbers are introduced in classical neural networks, they may outperform the common classical neural networks in dealing with the tasks in the quantum information field. In this paper, we verify this conjecture by studying quantum state classification via complex-valued neural networks (CVNNs). The numerical results show that the performance of CVNNs is much better than the real-valued neural network in classifying the entangled states. Our results not only provide a new way to improve the performance of artificial neural networks in quantum state classifiers, but also might shed light on the study of CVNNs in the field of other quantum information processing tasks before the appearance of the universal quantum computer.
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.
Simulating non-Markovian quantum dissipative dynamics remains a major challenge in theoretical and computational chemistry. While traditional numerical methods such as hierarchical equations of motion are numerically exact, they suffer from prohibitive computational costs when modeling systems with complex environmental couplings or strong non-Markovian effects. To address this limitation, we propose a deep learning framework based on two-dimensional convolutional neural networks (2D-CNN) for efficiently predicting long-time quantum dissipative dynamics using only short-time trajectory data. Our approach incorporates a 1D-to-2D feature reconstruction strategy, which transforms 1D time-series data into 2D images, and a multi-timescale fusion network to resolve complex dynamical features. We validate the framework on two paradigmatic cases — dissipative relaxation in a two-level system and Rabi oscillations in a dissipative spin system — achieving prediction mean absolute errors of 10−3 and 10−2, respectively. The results highlight the effectiveness of our 2D-CNN approach in capturing long-time temporal correlations, providing a computationally efficient pathway for simulating quantum dynamics in realistic open systems.
Abstract Despite the complexity of quantum systems in the real world, models with just a few effective many-body states often suffice to describe their quantum dynamics, provided decoherence is accounted for. We show that a machine learning algorithm is able to construct such models, given a straightforward set of quantum dynamics measurements. The effective Hilbert space can be a black box, with variations of the coupling to just one accessible output state being sufficient to generate the required training data. We demonstrate through simulations of a Markovian open quantum system that a neural network can automatically detect the number N of effective states and the most relevant Hamiltonian terms and state-dephasing processes and rates. For systems with N ⩽ 5 we find typical mean relative errors of predictions in the 10 % range. With more advanced networks and larger training sets, it is conceivable that a future single software can provide the automated first stop solution to model building for an unknown device or system, complementing and validating the conventional approach based on physical insight into the system.
Quantum machine learning (QML) is rapidly transitioning from theoretical promise to practical relevance across data-intensive scientific domains. In this Review, we provide a structured overview of recent advances that bridge foundational quantum learning principles with real-world applications. We survey foundational QML paradigms, including variational quantum algorithms, quantum kernel methods, and neural-network quantum states, with emphasis on their applicability to complex quantum systems. We examine neural-network quantum states as expressive variational models for correlated matter, non-equilibrium dynamics, and open quantum systems, and discuss fundamental challenges associated with training and sampling. Recent advances in quantum-enhanced sampling and diagnostics of learning dynamics, including information-theoretic tools, are reviewed as mechanisms for improving scalability and trainability. The Review further highlights application-driven QML frameworks in drug discovery, cancer biology, and agro-climate modeling, where data complexity and constraints motivate hybrid quantum-classical approaches. We conclude with a discussion of federated quantum machine learning as a route to distributed, privacy-preserving quantum intelligence. Overall, this Review presents a unified perspective on the opportunities and limitations of QML for complex systems.
Quantum machine learning (QML) is rapidly transitioning from theoretical promise to practical relevance across data-intensive scientific domains. In this review, we provide a structured overview of recent advances that bridge foundational quantum learning principles with real-world applications. We survey foundational QML paradigms, including variational quantum algorithms, quantum kernel methods, and neural-network quantum states, with emphasis on their applicability to complex quantum systems. We examine neural-network quantum states as expressive variational models for correlated matter, non-equilibrium dynamics, and open quantum systems, and discuss fundamental challenges associated with training and sampling. Recent advances in quantum-enhanced sampling and diagnostics of learning dynamics, including information-theoretic tools, are reviewed as mechanisms for improving scalability and trainability. The review further highlights application-driven QML frameworks in drug discovery, cancer biology, and agro-climate modeling, where data complexity and constraints motivate hybrid quantum–classical approaches. We conclude with a discussion of federated quantum machine learning as a route to distributed, privacy-preserving quantum learning. Overall, this review presents a unified perspective on the opportunities and limitations of QML for complex systems.