We present an approach to simulate the Schrödinger equation through continuous time quantum walks. The CTQW-based simulation applies unitary evolution driven by a quantum walk to generate probability amplitude distributions at various time steps. Additionally, we implemented a supervised neural network model to evaluate the effectiveness of data-driven techniques. The model learns to predict the squared modulus of the wavefunction given spatial and temporal coordinates. A comparative analysis demonstrates that the ML model can reproduce the qualitative structure and temporal progression of the quantum system with high accuracy. This study provides the synergy between quantum walk-based simulation and machine learning for solving quantum dynamical equations.
We present an approach to simulate the Schrödinger equation through continuous time quantum walks. The CTQW-based simulation applies unitary evolution driven by a quantum walk to generate probability amplitude distributions at various time steps. Additionally, we implemented a supervised neural network model to evaluate the effectiveness of data-driven techniques. The model learns to predict the squared modulus of the wavefunction given spatial and temporal coordinates. A comparative analysis demonstrates that the ML model can reproduce the qualitative structure and temporal progression of the quantum system with high accuracy. This study provides the synergy between quantum walk-based simulation and machine learning for solving quantum dynamical equations.
The machine learning approaches are applied in the dynamical simulation of open quantum systems. The long short-term memory recurrent neural network (LSTM-RNN) models are used to simulate the long-time quantum dynamics, which are built based on the key information of the short-time evolution. We employ various hyperparameter optimization methods, including simulated annealing, Bayesian optimization with tree-structured parzen estimator, and random search, to achieve the automatic construction and adjustment of the LSTM-RNN models. The implementation details of three hyperparameter optimization methods are examined, and among them, the simulated annealing approach is strongly recommended due to its excellent performance. The uncertainties of the machine learning models are comprehensively analyzed by the combination of bootstrap sampling and Monte Carlo dropout approaches, which give the prediction confidence of the LSTM-RNN models in the simulation of the open quantum dynamics. This work builds an effective machine learning approach to simulate the dynamics evolution of open quantum systems. In addition, the current study provides an efficient protocol to build optimal neural networks and estimate the trustiness of the machine learning models.
Describing the dynamics of strong-laser driven open quantum systems is a very challenging task that requires the solution of highly involved equations of motion. While machine learning techniques are being applied with some success to simulate the time evolution of individual quantum states, their use to approximate time-dependent operators (that can evolve various states) remains largely unexplored. In this work, we develop driven (NQP), a universal neural network framework that solves driven-dissipative quantum dynamics by approximating propagators rather than wave functions or density matrices. NQP can handle arbitrary initial quantum states, adapt to various external fields, and simulate long-time dynamics, even when trained on far shorter time windows. Furthermore, by appropriately configuring the external fields, our trained NQP can be transferred to systems governed by different Hamiltonians. We demonstrate the effectiveness of our approach by studying the spin-boson and the three-state transition Gamma models.
Capturing the dynamics of quantum many-body systems under time-dependent driving protocols is a central challenge for numerical simulations. Existing methods such as tensor networks and time-dependent neural quantum states, however, must be re-run for every protocol. In this work, we introduce the Neural Operator Quantum State (NOQS) as a foundation model for quantum dynamics. Rather than solving the Schrödinger equation for individual trajectories, our approach aims to \emph{learn the solution operator} that maps entire driving protocols to time-evolved quantum states. Once trained, the NOQS predicts time evolution under unseen protocols in a single forward pass, requiring no additional optimization. We validate NOQS on the two-dimensional Ising model with time-dependent longitudinal and transverse fields, demonstrating accurate prediction not only for unseen in-distribution protocols, but also for qualitatively different, out-of-distribution functional forms of driving. Further, a single NOQS model can be transferred between different temporal resolutions, and can be efficiently fine-tuned with sparse experimental measurements to improve predictions across all observables at negligible cost. Our work introduces a new paradigm for quantum dynamics simulation and provides a practical computational-experimental interface for driven quantum systems.
In this paper, machine learning and artificial neural network models are proposed for quantum noise classification in stochastic quantum dynamics. For this purpose, we train and then validate support vector machine, multi-layer perceptron and recurrent neural network, models with different complexity and accuracy, to solve supervised binary classification problems. By exploiting the quantum random walk formalism, we demonstrate the high efficacy of such tools in classifying noisy quantum dynamics using data sets collected in a single realisation of the quantum system evolution. In addition, we also show that for a successful classification one just needs to measure, in a sequence of discrete time instants, the probabilities that the analysed quantum system is in one of the allowed positions or energy configurations, without any external driving. Thus, neither measurements of quantum coherences nor sequences of control pulses are required. Since in principle the training of the machine learning models can be performed a-priori on synthetic data, our approach is expected to find direct application in a vast number of experimental schemes and also for the noise benchmarking of the already available noisy intermediate-scale quantum devices.
Estimation of the coin parameter(s) is an important part of the problem of implementing more robust schemes for quantum simulation using quantum walks. We present the estimation of the quantum coin parameter used for one-dimensional discrete-time quantum walk evolution using machine learning algorithms on their probability distributions. We show that the models we have implemented are able to estimate these evolution parameters to a good accuracy level. We also implement a deep learning model that is able to predict multiple parameters simultaneously. Since discrete-time quantum walks can be used as quantum simulators, these models become important when extrapolating the quantum walk parameters from the probability distributions of the quantum system that is being simulated.
We explore the capability of linear fully connected neural networks (FCNs) to learn and predict quantum dynamics governed by various equations of motion, including the time-dependent Schrödinger equation, quantum Liouville equation, and thermofield dynamics (TFD). Given the inherent linearity of these equations, linear FCNs naturally align with the fundamental structure of quantum propagators, making them a promising machine-learning approach for simulating quantum systems. Through numerical experiments on different model systems, including single-state and two-state anharmonic potentials, we demonstrate that linear FCNs without hidden layers achieve high accuracy while preserving population conservation. For finite-temperature dynamics, we find that learning TFD is more efficient than learning density matrix evolution due to the suppressed noise from the partial trace over the fictitious Hilbert space. However, as system dimensionality increases, the computational cost of linear FCNs becomes substantial due to scaling and sampling challenges, suggesting the necessity of more efficient network structures. Our results highlight the potential of physics-informed neural networks for quantum dynamics and point toward future developments in neural network architectures inspired by physical principles.