Effective modeling and numerical spectral-based propagation schemes are proposed for addressing the challenges in time-dependent quantum simulations of systems ranging from atoms, molecules, and nanostructures to emerging nanoelectronic devices. While time-dependent Hamiltonian problems can be formally solved by propagating the solutions along tiny simulation time steps, a direct numerical treatment is often considered too computationally demanding. In this paper, however, we propose to go beyond these limitations by introducing high-performance numerical propagation schemes to compute the solution of the time-ordered evolution operator. In addition to the direct Hamiltonian diagonalizations that can be efficiently performed using the new eigenvalue solver FEAST, we have designed a Gaussian propagation scheme and a basis-transformed propagation scheme (BTPS) which allow to reduce considerably the simulation times needed by time intervals. It is outlined that BTPS offers the best computational efficiency allowing new perspectives in time-dependent simulations. Finally, these numerical schemes are applied to study the ac response of a (5,5) carbon nanotube within a three-dimensional real-space mesh framework.
Effective modeling and numerical spectral-based propagation schemes are proposed for addressing the challenges in time-dependent quantum simulations of systems ranging from atoms, molecules, and nanostructures to emerging nanoelectronic devices. While time-dependent Hamiltonian problems can be formally solved by propagating the solutions along tiny simulation time steps, a direct numerical treatment is often considered too computationally demanding. In this paper, however, we propose to go beyond these limitations by introducing high-performance numerical propagation schemes to compute the solution of the time-ordered evolution operator. In addition to the direct Hamiltonian diagonalizations that can be efficiently performed using the new eigenvalue solver FEAST, we have designed a Gaussian propagation scheme and a basis-transformed propagation scheme (BTPS) which allow to reduce considerably the simulation times needed by time intervals. It is outlined that BTPS offers the best computational efficiency allowing new perspectives in time-dependent simulations. Finally, these numerical schemes are applied to study the ac response of a (5,5) carbon nanotube within a three-dimensional real-space mesh framework.
A numerical method was proposed to propagate the quantum system with a time-dependent Hamiltonian. The propagator is composed of the time-independent-Hamiltonian-type multisplit forms, which becomes possible by realizing that the evolution operator has a close link with the time-shift operator. The present algorithm can achieve a high order of accuracy through the recurrence of multisplit forms with predetermined coefficients, and it has the advantage that only the Hamiltonian itself at different time instants is involved, while other currently available schemes need to perform either a time integration or a time differential of the Hamiltonian. This essential difference has made our proposal highly competitive. A one-dimensional model system is considered to verify the accuracy and efficiency of the numerical scheme.
This thesis investigates the grid-based Split Operator–Quantum Fourier Transform (SO-QFT) scheme as a general-purpose backbone for time-dependent quantum simulation. We develop an end-to-end algorithmic framework that unifies state preparation on discretised grids, SO-QFT time propagation, and the extraction of observables via autocorrelation functions, with a focus on efficient quantum circuit construction. Classical emulation is used to evaluate the performance and resource demands of this framework across diverse chemical contexts. For vibronic dynamics arising from conical intersections, we assemble a complete workflow that reproduces absorption spectra and population transfer for a well-characterised prototypical polyatomic system. On this basis, we estimate the required quantum resources by mapping grid size, time resolution and spectral precision to qubit budgets, circuit depths and measurement counts. The framework is further tailored to infrared spectroscopy through probabilistic encoding of non-unitary dipole operators and explicit treatment of higher-order Hamiltonian terms via multi-controlled operations. To minimise gate depth, we optimise time discretisation and employ approximation strategies for initial states and dipole truncations, while preserving the fidelity of computed spectra. Importantly, the resource scaling relations derived from the vibronic and infrared simulations generalise to comparable molecular systems, providing a transferable template for predicting resource estimates beyond the specific benchmark studied. Finally, the methodology is extended to Coulombic systems, whose singular potentials challenge grid-based representations. We improve the simulation fidelity by proposing two correction schemes: a Hamiltonian correction that accounts for grid-basis structure, and a modification of the initial wavefunction to better match the discretised Hamiltonian. Both corrections are compatible with compact quantum encodings, and their representation in truncated Fourier or Walsh series offers an additional reduction in circuit depth. Collectively, these studies advance the grid-based SO-QFT scheme from a promising construct to a systematically assessed framework, establishing methodological guidance, algorithmic variants and quantitative resource benchmarks for future fault-tolerant quantum applications.
The most widely used approach for simulating the dynamics of time-dependent Hamiltonians via quantum computation depends on the quantum-classical hybrid variational quantum time evolution algorithm, in which ordinary differential equations of the variational coefficients for determining time evolution are solved via classical simulations with a time discretization method. We here present a full-quantum approach, in which ordinary differential equations of the variational coefficients are transformed into static linear equations via the Chebyshev spectral discretization method and then solved via the quantum singular value transformation algorithm. Our full quantum algorithm avoids classical feedback, achieves exponential convergence for smooth Hamiltonians, and yields a quantum circuit depth that is independent of the number of time steps. We demonstrate two implementation strategies, with a global formulation designed for fault-tolerant architectures and a sequential formulation tailored to near-term devices, and validate the approach through numerical simulations of proton-hydrogen charge-transfer dynamics, a prototypical time-dependent quantum chemistry problem. This work establishes a systematic pathway from quantum-classical hybrid variational quantum algorithms to full-quantum solvers for general time-dependent Hamiltonians, particularly those whose dynamics admit compact variational descriptions, opening a route toward full quantum computational advantages in time-dependent simulations.
Time dependent linear and nonlinear equations govern the evolution of an extensive set of physical systems and processes describing, to enumerate just a few, Bose-Einstein condensates; soliton propagation in optical and photonic band-gap fibers; quantum control of atomic and molecular collisions and reactions; highly-compressed liquids; and dense and ultracold plasmas. While the media vary substantially, the basic computational procedures have many common features. We focus on the nonlinear Schrodinger equation and discuss two powerful approaches to its propagation: the Arnoldi/Lanczos(AL)l and Real Space Product Formula(RSPF)2. Both provide efficient systematic approximations to the short-time exponential propagator that moves the solution between time steps. We implement the former in a discrete variable representation (DVR)3 both in spatial grid and finite element forms and the latter in a spatial mesh with a finite difference representation of the kinetic energy operator. Both approaches require O(N) operations to propagate the wavefunction between time steps and handle multidimensional systems. We shall also draw connections with Liouville formulations used in quantum molecular dynamics simulations of large collections of atoms and molecules. After briefly outlining these formulations, we shall discuss some of the varied applications.
The time-dependent Schrödinger equation (TDSE) in real space is fundamental to understanding the dynamics of many-electron quantum systems, with applications ranging from quantum chemistry to condensed matter physics and materials science. However, solving the TDSE for complex fermionic systems remains a significant challenge, particularly due to the need to capture the time-evolving many-body correlations, while the antisymmetric nature of fermionic wavefunctions complicates the function space in which these solutions must be represented. We propose a general-purpose neural network framework for solving the real-space TDSE, Fermionic Antisymmetric Spatio-Temporal Network, which treats time as an explicit input alongside spatial coordinates, enabling a unified spatiotemporal representation of complex, antisymmetric wavefunctions for fermionic systems. This approach formulates the TDSE as a global optimization problem, avoiding step-by-step propagation and supporting highly parallelizable training. The method is demonstrated on five benchmark problems, achieving excellent agreement with reference solutions across all cases. These results demonstrate the method's accuracy and flexibility within the bound-state manifold across various dimensions and interaction regimes. While the current localized Ansatz inherently restricts the description of extensive ionization and continuum states, the method demonstrates the capability to stably simulate coherent multi-electron dynamics over extended time windows. Our framework offers a highly expressive alternative to traditional basis-dependent or mean-field methods, opening new possibilities for ab initio simulations of time-dependent quantum systems, with applications in quantum dynamics, molecular control, and ultrafast spectroscopy.
We consider computational methods for simulating the dynamics of molecular systems governed by the time-dependent Schrödinger equation. Solving the Schrödinger equation numerically poses a challenge due to its often highly oscillatory solutions, and to the exponential growth of work and memory with the number of particles in the system. Two different classes of problems are studied: the dynamics of the nuclei in a molecule and the dynamics of an electron in orbit around a nucleus. For the first class of problems we present new computational methods which exploit the relation between quantum and classical dynamics in order to make the computations more efficient. For the second class of problems, the lack of regularity in the solution poses a computational challenge. Using knowledge of the non-smooth features of the solution we construct a new method with two orders higher accuracy than what is achieved by direct application of a difference stencil.