Accurate time-dependent quantum dynamics of Coulombic systems on grid-based representations remains computationally demanding due to the singularity of the Coulomb potential, which necessitates extremely fine spatial grids to mitigate discretisation errors. We propose two complementary correction schemes that, under identical resource budgets, consistently outperform the uncorrected counterparts. The first scheme modifies the potential operator to incorporate grid-basis structure into its representation, while the second introduces a corrected initial wavefunction inspired by analytical solutions of softened Coulomb potentials. Applied to hydrogenic systems, these corrections deliver improved energy accuracy and time fidelity across long evolutions. Beyond classical simulations, the proposed framework aligns naturally with quantum computing architectures, where the corrected operators and states can be encoded through truncated Walsh and Fourier series expansions. A resource analysis for the representative 2D hydrogen system yields a circuit depth of $1.5\times10^{8}$ gates over 6,000 Trotter steps. This study thus establishes practical strategies toward high-accuracy Coulombic dynamics on both classical and emerging quantum platforms.
Accurate time-dependent quantum dynamics of Coulombic systems on grid-based representations remains computationally demanding due to the singularity of the Coulomb potential, which necessitates extremely fine spatial grids to mitigate discretisation errors. We propose two complementary correction schemes that, under identical resource budgets, consistently outperform the uncorrected counterparts. The first scheme modifies the potential operator to incorporate grid-basis structure into its representation, while the second introduces a corrected initial wavefunction inspired by analytical solutions of softened Coulomb potentials. Applied to hydrogenic systems, these corrections deliver improved energy accuracy and time fidelity across long evolutions. Beyond classical simulations, the proposed framework aligns naturally with quantum computing architectures, where the corrected operators and states can be encoded through truncated Walsh and Fourier series expansions. A resource analysis for the representative 2D hydrogen system yields a circuit depth of 1.5×108 gates over 6,000 Trotter steps. This study thus establishes practical strategies toward high-accuracy Coulombic dynamics on both classical and emerging quantum platforms.
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.
With quantum computers on the horizon, it is essential to understand what problems they can be used to solve. One potential application is simulating quantum systems, in particular, systems with many interacting electrons. However, there is a disparity between the computational resources required to study these problems and the capabilities of modern experimental quantum devices. In this thesis, we make progress toward closing this gap by reducing the cost of quantum simulation of quantum chemistry on a number of fronts. In the first half, we develop three improved quantum simulation algorithms by approximating time evolution under the Hamiltonian using a truncated Taylor series rather than the Trotter-Suzuki decomposition. We do this first for the molecular electronic structure Hamiltonian in two representations, obtaining asymptotic improvements in terms of both the number of orbitals and the required precision. We also apply the truncated Taylor series paradigm to quantum simulation in real space, where we consider simulation of general wave functions discretized on a uniform grid. In the second half, by working with Hamiltonians tailored specifically to quantum algorithms, we demonstrate that Trotter steps of the electronic structure Hamiltonian can be simulated in exactly N depth and with only linear connectivity, where N is the number of spin-orbitals. This resolves the problem of non-locality in fermion-to-qubit mappings, and has wide-ranging applications for variational and phase estimation-based quantum simulation of quantum chemistry. We further study this algorithm on an error-corrected quantum computer using the surface code. Benchmarking it alongside other product formula approaches, we show that quantum simulation of scientifically important instances of the electronic structure problem that are classically intractable can be performed with only a few hundred thousand physical qubits and fewer than one billion costly fault-tolerant gates. Finally, we study the effect of using randomized Hamiltonians in iterative phase estimation, and how this allows us to reduce the costs of simulation as well as to exploit knowledge of the ground state in how we design our simulation algorithm.
A quantum circuit for first-quantization-based Hamiltonian simulation, capable of running on a small-scale quantum computer emulator with fewer than 30 qubits, was constructed and demonstrated for one- and two-dimensional hydrogen-atom models, the simplest settings sufficient to check the proposed circuits without the confounding effects of electron-electron interaction and correlation. For these systems, Uniformly Controlled Rotation Z (UCRZ) circuits were found valuable for constructing a Coulomb potential circuit compact enough to fit on a small-scale emulator. Although this approach does not scale to larger models, we compare it against a conventional arithmetic-circuit implementation to identify the regime in which UCRZ is preferable. To deal with the Coulomb singularity, we propose regularizing it via a geometric average of the potential over a small circular region around the pole, avoiding divergence and minimizing the resulting energy error. To select parameters minimizing error for a given number of qubits, we determine the optimal spatial grid spacing from the trade-off between spatial discretization error and domain truncation error, and determine the time evolution step size not from the formal worst-case Trotter error bound, but from a sampling-theorem-based bound on the wave function’s maximum frequency component, avoiding unnecessary circuit depth without sacrificing accuracy. While the validity of a UCRZ-based circuit structure is restricted to the small-scale regime, the singularity treatment and parameter-selection procedure address the grid discretization itself and apply broadly to grid-based first-quantization simulations.
Efficiently simulating many-body quantum systems with Coulomb interactions is a fundamental question in quantum physics, quantum chemistry, and quantum computing, yet it presents unique challenges: the Hamiltonian is an unbounded operator; its Hilbert space dimension grows exponentially with particle number; and the Coulomb potential is singular, long-ranged, and non-smooth, violating the regularity assumptions of many prior state-of-the-art many-body simulation analyses. In this work, we establish rigorous error bounds for Trotter formulas applied to these systems. Our first main result shows that for general initial conditions in the domain of the Hamiltonian, second-order Trotter achieves a 1/4 convergence rate with explicit polynomial dependence of the error prefactor on the particle number. The polynomial dependence on system size suggests that the algorithm remains quantumly efficient, even without introducing any regularization of the Coulomb singularity. Such a worst-case rate has been observed in prior work for the hydrogen ground state, demonstrating its physical relevance. We further establish one-step local error lower bounds of order 5/4 for both formulas, showing that the local exponent is sharp. Our second main result identifies a set of physically meaningful conditions on the initial state under which the convergence rates improve, with the full first-order and second-order rates recovered at sufficiently high angular momentum. Our theoretical findings are consistent with prior numerical observations. Our third main result establishes improved convergence rates for many-body fermionic Coulomb systems with initial data of different Sobolev regularity. In particular, even without imposing any additional regularity beyond the natural domain of the Hamiltonian, both the first- and second-order Trotter formulas converge with rate 1/2, rather than the general 1/4 rate.
First-quantized, grid-based methods for chemistry modeling are a natural and elegant fit for quantum computers. However, it is infeasible to use today's quantum prototypes to explore the power of this approach because it requires a substantial number of near-perfect qubits. Here, we use exactly emulated quantum computers with up to 36 qubits to execute deep yet resource-frugal algorithms that model 2D and 3D atoms with single and paired particles. A range of tasks is explored, from ground state preparation and energy estimation to the dynamics of scattering and ionization; we evaluate various methods within the split-operator QFT (SO-QFT) Hamiltonian simulation paradigm, including protocols previously described in theoretical papers and our own techniques. While we identify certain restrictions and caveats, generally, the grid-based method is found to perform very well; our results are consistent with the view that first-quantized paradigms will be dominant from the early fault-tolerant quantum computing era onward.
We develop a time-dependent, grid-based framework for simulating infrared spectra that is specifically designed for quantum computers. The proposed circuit employs a probabilistic strategy for applying the non-unitary dipole operator and an Split Operator-Quantum Fourier Transform time evolution scheme. Using a vibrational model of the water molecule as a test system, our classical emulation results demonstrate accurate determination of fundamental and overtone band positions and intensities via Fourier-transformed dipole-dipole autocorrelation functions. We also identify the optimal time parameters that minimise gate depths while maintaining high fidelity. For further resource reduction, we validate the feasibility of utilising harmonic oscillator approximations in state preparation and dipole operator truncations. With its scalability to higher-dimensional normal mode spaces, this wavefunction-based approach establishes a robust foundation for studying IR spectra on future quantum hardware.