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Grid-based split operator-quantum fourier transform algorithms for time-dependent quantum simulation

Xiaoning Feng

Published Jan 1, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.5287/ora-gpjgodd84

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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.
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.
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We establish and analyse the performance and resource requirements of an end-to-end fault-tolerant quantum algorithm for computing the absorption spectrum and population dynamics of photoexcited pyrazine. The quantum circuit construction consists of initial state preparation using uniformly controlled rotations, the time-dependent Hamiltonian propagation based on the grid-based Split Operator Quantum Fourier Transform (SO-QFT) method, and cost-effective measurements including statistical and canonical phase estimation. We use classical emulations to validate the quantum resources required for the task, and propose generalised formulae for the qubit count and gate depth calculation. Simulating the vibronic dynamics of pyrazine in a low-dimensional abstraction requires 17-qubit circuits with a gate depth of O(105)\mathcal{O}(10^5)O(105), whereas a full-dimensional simulation of pyrazine in 24 modes requires at least 97-qubit circuits with a gate depth of O(106)\mathcal{O}(10^6)O(106). Our work provides a foundational framework for understanding high-dimensional wavepacket-based quantum simulations of photo-induced dynamics and vibronic spectra, anticipating future applications in the simulation of even larger molecular systems on fault-tolerant quantum computers.
Quantum computation is an attractive front for many problems that are intractable for computers today. One such problem is nonadiabatic quantum molecular dynamics, where quantized internal states coupling to parameterized modes result in a Hamiltonian resistant to oracle-based models and spectral decomposition. This dissertation applies diabatic Hamiltonian operators directly to the computational basis as first-quantized split-operator propagators, validated with dynamic observables including absorption and recurrence spectra, scattering cross-sections, population dynamics, and quantum scars. Circuits are derived and specified, with focused circuit optimization in multi-mode and multi-channel extensions, including multivariate potential energy terms and graph theoretic optimization from molecular symmetry. Resource estimation shows circuit depth advantage against QROM-loading architectures on a fault-tolerant scale, and a quantitative comparison against quantum signal processing variants confirms that a Trotter-based architecture retains a scalable T-gate advantage. Expanding beyond electronic states demonstrates that duality between finite basis and discrete variable representations permits congruent structural decompositions into quantum circuits, expanding the use of multi-channel dynamics far beyond chemistry.
The rapid advancement of quantum computing has created new opportunities for solving computationally intensive problems across diverse fields such as optimization, physical system simulation, and machine learning. However, practical implementation on Noisy Intermediate-Scale Quantum (NISQ) devices remains challenging due to hardware limitations including noise, decoherence, and restricted qubit connectivity. A fundamental constraint further complicating this effort is that all quantum operations must be unitary, while many real-world problems are naturally expressed in non-unitary forms. This discrepancy necessitates novel techniques for transforming non-unitary mathematical structures into unitary or Hamiltonian representations compatible with quantum hardware.This thesis proposes a unified framework for converting structured, sparse, or otherwise non-unitary classical matrices into unitary forms that can be executed efficiently on NISQ devices. The methodology introduces general transformation strategies such as linear combinations of unitaries, structured embeddings, and resource-aware circuit construction, enabling the deployment of quantum algorithms without incurring prohibitive overhead. Each technique is developed with an emphasis on preserving problem structure, minimizing circuit depth, and ensuring compatibility with IBM’s NISQ hardware. The proposed frameworks are validated through theoretical analysis and quantum simulations, with performance benchmarks based on gate count, circuit fidelity, convergence behavior, and scalability. This work demonstrates how non-unitary formulations arising in various computational contexts can be systematically adapted for quantum execution, advancing the practical utility of NISQ-era devices.
Efficient encoding of classical data into quantum circuits is a critical challenge that directly impacts the scalability of quantum algorithms. In this work, we present an automated compilation framework for resource-aware quantum data loading tailored to a given input vector and target error tolerance. By explicitly exploiting the trade-off between exact and approximate state preparation, our approach systematically partitions the total error budget between precision and approximation errors, thereby minimizing quantum resource costs. The framework supports a comprehensive suite of state-of-the-art methods, including multiplexer-based loaders, quantum read-only memory (QROM) constructions, sparse encodings, matrix product states (MPS), Fourier series loaders (FSL), and Walsh transform-based diagonal operators. We demonstrate the effectiveness of our framework across several applications, where it consistently uncovers non-obvious, resource-efficient strategies enabled by controlled approximation. In particular, we analyze a computational fluid dynamics workflow where the automated selection of MPS state preparation and Walsh transform-based encoding, combined with a novel Walsh-based measurement technique, leads to resource reductions of over four orders of magnitude compared to previous approaches. We also introduce two independent advances developed through the framework: a more efficient circuit for d-diagonal matrices, and an optimized block encoding for kinetic energy operators. Our results underscore the indispensable role of automated, approximation-aware compilation in making large-scale quantum algorithms feasible on resource-constrained hardware.