Mott insulators exhibit complex photoexcitation dynamics under intense optical driving, with potential implications for carrier multiplication beyond the Shockley-Queisser limit. Probing these nonequilibrium processes requires access to the thermodynamic limit, where the number of lattice sites becomes arbitrarily large, but conventional solvers are constrained to small systems due to the exponential growth of the Hilbert space. Fourier Neural Operators (FNOs), originally developed for solving partial differential equations, naturally accommodate inputs of varying resolution and are capable of capturing nonlocal effects. Here, we employ FNOs to learn the mapping from noise-perturbed ground-state momentum distributions to their post-pulse counterparts across a range of interaction strengths and driving parameters. Trained only on small lattices, the model generalizes zero-shot to much larger systems, producing physically reasonable momentum distributions well beyond the reach of numerical solvers. Specifically, the model can predict momentum distribution for a 1024x1024 system within a few seconds that matches the theoretical behavior of key observables, whereas direct numerical simulations have so far been restricted to edge sizes of ~30. These results demonstrate the potential of neural operators to directly access large-scale nonequilibrium dynamics, providing a new pathway toward the thermodynamic limit in strongly correlated materials.
Mott insulators exhibit complex photoexcitation dynamics under intense optical driving, with potential implications for carrier multiplication beyond the Shockley-Queisser limit. Probing these nonequilibrium processes requires access to the thermodynamic limit, where the number of lattice sites becomes arbitrarily large, but conventional solvers are constrained to small systems due to the exponential growth of the Hilbert space. Fourier Neural Operators (FNOs), originally developed for solving partial differential equations, naturally accommodate inputs of varying resolution and are capable of capturing nonlocal effects. Here, we employ FNOs to learn the mapping from noise-perturbed ground-state momentum distributions to their post-pulse counterparts across a range of interaction strengths and driving parameters. Trained only on small lattices, the model generalizes zero-shot to much larger systems, producing physically reasonable momentum distributions well beyond the reach of numerical solvers. Specifically, the model can predict momentum distribution for a 1024x1024 system within a few seconds that matches the theoretical behavior of key observables, whereas direct numerical simulations have so far been restricted to edge sizes of ~30. These results demonstrate the potential of neural operators to directly access large-scale nonequilibrium dynamics, providing a new pathway toward the thermodynamic limit in strongly correlated materials.
Time-periodic quantum systems exhibit a rich variety of far-from-equilibrium phenomena and serve as ideal platforms for quantum engineering and control. However, simulating their dynamics with conventional numerical methods remains challenging due to the exponential growth of Hilbert space dimension and rapid spreading of entanglement. In this work, we introduce Fourier neural operators (FNOs) as an efficient, accurate, and scalable framework for nonequilibrium quantum dynamics. Parameterized in Fourier space, FNO naturally captures temporal correlations and remains minimally dependent on discretization of time. We demonstrate the versatility of FNO through three complementary learning paradigms: reconstructing effective Floquet Hamiltonians, predicting expectation values of local observables, and learning quantum information spreading. For each learning task, FNO achieves remarkable accuracy, while attaining a significant speedup, compared to exact numerical methods. Moreover, FNO possesses capabilities beyond that of conventional methods, such as predicting all local observables from a subset of measurements without information about the Hamiltonian, as well as extrapolating beyond the time window provided by training data, enabling access to observables and operator-spreading dynamics that might be beyond the coherence time. By employing a spatially local basis, we argue that the computational cost of FNOs scales only polynomially with the system size. Our results establish FNO as a versatile and scalable computation framework that integrates numerical simulations and experimental data seamlessly, with direct implications for extracting meaningful physics from measurements by near-term quantum computers.
Density functional theory (DFT) is a cornerstone of computational chemistry and materials science, but its computational cost limits its use in large-scale and high-throughput applications. While machine learning has accelerated energy prediction for specific molecular classes, transferable prediction of electron density across diverse chemical spaces remains challenging. Here, we present a universal framework based on Fourier Neural Operators (FNOs) that directly learns the mapping from external potentials to electron density distributions. Unlike conventional approaches that rely on explicit atomic orbitals, basis sets, or handcrafted descriptors, the proposed method captures global electronic interactions and long-range correlations through operator learning in the spatial-frequency domain. Trained on datasets spanning multiple elements and molecular geometries, the model achieves zero-shot generalization to entirely unseen molecular systems and accurately predicts their electron densities without retraining. This transferability arises from the intrinsic ability of FNOs to represent global structure in continuous fields. Our work establishes neural operator learning as a promising route for fast, accurate, and transferable electronic structure prediction, with potential applications in high-throughput screening and chemical space exploration.
Abstract Fourier Neural Operators (FNOs) excel on functional data tasks, such as those originating from partial differential equations. This renders them effective at simulating the time evolution of quantum wavefunctions, a computationally challenging, yet coveted task. In this manuscript, we use FNOs to model the evolution of random quantum spin systems, chosen due to their representative dynamics. We find that traditional neural networks, such as U-Net, exhibit limited extrapolation beyond the training time interval, whereas FNOs capture the underlying time-evolution operator, generalizing to unseen times. Additionally, we apply FNOs to a compact set of Hamiltonian observables (~ poly( n )) instead of 2 n -component wavefunctions, greatly reducing the size of our inputs, outputs, and FNO models. This Hamiltonian observable-based method demonstrates that FNOs can distill information from high to low-dimensional spaces. We perform numerical experiments on a 20-qubit system, extrapolating Hamiltonian observables to twice the training time with a relative error of 5.8%. Notably, relative to numerical time-evolution methods, FNO achieves an inference speedup of approximately 10 4 × for 20-qubit systems, underscoring its computational efficiency. This extrapolation of observables past training times stands to fundamentally increase the simulatability of quantum systems beyond the limitations of quantum device coherence and tensor network circuit-depth.
The exact quantum dynamics of lattice models can be computationally intensive, especially when aiming for large system sizes and extended simulation times necessary to converge transport coefficients. By leveraging finite memory times to access long-time dynamics using only short-time data, generalized master equations can offer a route to simulating the dynamics of lattice problems efficiently. However, such simulations are limited to small lattices whose dynamics exhibit finite-size artifacts that contaminate transport coefficient predictions. To address this problem, we introduce a novel approach that exploits finite memory in both time and space to efficiently predict the many-body dynamics of dissipative lattice problems involving short-range interactions. This advance enables one to leverage the short-time dynamics of small lattices to nonperturbatively and exactly simulate arbitrarily large systems over long times. We demonstrate the strengths of this method by focusing on nonequilibrium polaron relaxation and transport in the dispersive Holstein model, successfully simulating lattice dynamics in one and two dimensions free from finite-size effects, thereby reducing the computational expense of such simulations by multiple orders of magnitude. Our method is broadly applicable and provides an accurate and efficient means to investigate nonequilibrium relaxation with microscopic resolution over mesoscopic length and time scales that are relevant to experiments.
Abstract Understanding the dynamics of nonequilibrium quantum many-body systems is an important research topic in a wide range of fields across condensed matter physics, quantum optics, and high-energy physics. However, numerical studies of large-scale nonequilibrium phenomena in realistic materials face serious challenges due to intrinsic high-dimensionality of quantum many-body problems and the absence of time-invariance. The nonequilibrium properties of many-body systems can be described by the dynamics of the correlator, or the Green's function of the system, whose time evolution is given by a high-dimensional system of integro-differential equations, known as the Kadanoff-Baym equations (KBEs). The time-convolution term in KBEs, which needs to be recalculated at each time step, makes it difficult to perform long-time numerical simulation. In this paper, we develop an operator-learning framework based on Recurrent Neural Networks (RNNs) to address this challenge. We utilize RNNs to learn the nonlinear mapping between Green's functions and convolution integrals in KBEs. By using the learned operators as a surrogate model in the KBE solver, we obtain a general machine-learning scheme for predicting the dynamics of nonequilibrium Green's functions. Besides significant savings per each time step, the new methodology reduces the temporal computational complexity from O(Nt3) to O(Nt) where Nt is the number of steps taken in a simulation, thereby making it possible to study large many-body problems which are currently infeasible with conventional KBE solvers. Through various numerical examples, we demonstrate the effectiveness of the operator-learning based approach in providing accurate predictions of physical observables such as the reduced density matrix and time-resolved photoemission spectra. Moreover, our framework exhibits clear numerical convergence and can be easily parallelized, thereby facilitating many possible further developments and applications.
This dissertation serves as a summary of my Ph.D. work numerically studying equilibrium and non-equilibrium properties of strongly-interacting one-dimensional (1D) boson systems. This work is motivated by the fact that 1D systems are realizable and highly controllable with ultracold atoms in optical lattice and atom chip experiments. We apply a recent worm algorithmic Monte Carlo approach developed for 1D continuous systems to study their equilibrium properties, both with and without an underlying lattice. We also apply an exact lattice approach based on the Bose-Fermi mapping to check our Monte Carlo results in the Tonks-Girardeau limit, and more importantly, to study far-from-equilibrium expansion dynamics of the systems.
We first study the scaling of one-particle correlations of the harmonically trapped Lieb-Liniger gas with changing temperature and interaction strength. Based on the universal behaviors of the density and momentum profiles, we are able to determine the effective parameters needed to fully characterize the system. We also find that the Tonks-Girardeau limit at low temperatures is the ideal regime for the experimental observation of the 1/k4 momentum tail.
An extra periodic lattice can drive the transition from superfluid to Mott insulator states. Exact and complete phase diagrams for such transitions are available only in the weak interacting and deep lattice limit, in which the system can be described using one-band Bose-Hubbard model. Beyond this limit, we use the worm algorithm in continuous space to map out the phase diagrams at various interaction strengths. We compare our phase diagrams with one-band Bose-Hubbard predictions and identify the regime where the one-band description breaks down. We introduce an inverse confined scattering solution to obtain effective Hubbard parameters, with which the Bose-Hubbard model provides correct results for strong interactions and deep lattices at unit filling.
In addition to the equilibrium properties, we also study the expansion dynamics of ultracold atoms in the hard-core limit. Experimentally, this is usually achieved by turning off confining potentials and letting atoms expand in optical lattices. Theoretical studies from initial ground states predicted the occurrence of fermionization of the momentum distribution after long expansion times. In addition, quasicondensation at finite momenta emerges when expanding from Mott insulating domains. Here, we develop a finite-temperature extension of the lattice approach for dynamics. We find the dynamical ferminoization of the momentum distributions at all temperatures. For expansion from initial Mott domains, we observe enhanced correlations reminiscent of dynamical quasicondensation. Surprisingly, we find the systems appear to cool down during the melting of the Mott domains. We use an emergent local Hamiltonian to understand these emergent phenomena.