Development and analysis of variational quantum algorithms to solve engineering differential equations
File(s)
Author(s)
Hunout, Josephine
Type
Thesis
Abstract
Differential equations underpin a vast range of scientific and engineering domains, including structural mechanics, fluid dynamics, and financial modelling. Their intricate nature often renders classical numerical methods computationally demanding, requiring fine discretization or large basis expansions, which can lead to prohibitive costs. Quantum computing offers a fundamentally different paradigm, leveraging superposition and entanglement with the promise of computational speedup. However, in the current Noisy Intermediate-Scale Quantum (NISQ) era, fully fault-tolerant quantum computers remain decades away, motivating the development of hybrid strategies such as Variational Quantum Algorithms (VQAs), which train parametrised quantum circuits using classical optimizers.
This work investigates spectral VQAs for solving differential equations, extending the Chebyshev-encoded VQA of Kyriienko et al. and introducing a novel quantum spectral approach: the Lagrange-encoded VQA. This approach shifts the paradigm by incorporating entanglement directly within the encoding block while maintaining a shallow variational ansatz. Inspired by the Hadamard test structure, the design enables an efficient representation of the complete set of Lagrange polynomial basis functions using a limited number of gates and facilitates the use of the Hadamard test differentiation method, thereby reducing the computational load. VQA performance is analysed through the accuracy of the result in combination with the gate count and circuit count metrics, which serve as proxies for runtime and noise accumulation on NISQ devices.
The proposed VQA is benchmarked on multiple problems: (i) a first-order linear ODEs and (ii) the second-order ODEs damped mass-spring system, both comparing Lagrange- and Chebyshev-encoded VQAs, (iii) the Poisson equation, comparing the Lagrange-encoded VQA with the discretized approach of Sato et al., and (iv) a first-order ODE with nonlinear coefficients addressed with the Lagrange-encoded VQA. Across these cases, the Lagrange-encoded VQA consistently achieves comparable or improved accuracy while significantly reducing circuit depth and computational overload, making it a promising candidate for NISQ devices.
This work investigates spectral VQAs for solving differential equations, extending the Chebyshev-encoded VQA of Kyriienko et al. and introducing a novel quantum spectral approach: the Lagrange-encoded VQA. This approach shifts the paradigm by incorporating entanglement directly within the encoding block while maintaining a shallow variational ansatz. Inspired by the Hadamard test structure, the design enables an efficient representation of the complete set of Lagrange polynomial basis functions using a limited number of gates and facilitates the use of the Hadamard test differentiation method, thereby reducing the computational load. VQA performance is analysed through the accuracy of the result in combination with the gate count and circuit count metrics, which serve as proxies for runtime and noise accumulation on NISQ devices.
The proposed VQA is benchmarked on multiple problems: (i) a first-order linear ODEs and (ii) the second-order ODEs damped mass-spring system, both comparing Lagrange- and Chebyshev-encoded VQAs, (iii) the Poisson equation, comparing the Lagrange-encoded VQA with the discretized approach of Sato et al., and (iv) a first-order ODE with nonlinear coefficients addressed with the Lagrange-encoded VQA. Across these cases, the Lagrange-encoded VQA consistently achieves comparable or improved accuracy while significantly reducing circuit depth and computational overload, making it a promising candidate for NISQ devices.
Version
Open Access
Date Issued
2025-09-01
Date Awarded
2026-03-01
Copyright Statement
Attribution-NonCommercial 4.0 International Licence (CC BY-NC)
License URL
Advisor
Laizet, Sylvain
Iannucci, Lorenzo
Grant Number
Grant No. EP/W032643/1
Publisher Department
Department of Aeronautics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)