School of Mathematics and Statistics - Theses

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Start date: 2010 × Type: Masters Coursework thesis ×

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    A Variational Approach to Solving Differential Equations on a Quantum Computer
    Nelson, James ( 2021)
    As quantum computing technology continues to develop, quantum algorithms have the potential to outperform classical algorithms in future for certain computational tasks. One of these potential applications of quantum computing is numerically solving differential equations and integral equations. In this thesis the variational quantum linear solver algorithm (VQLS) proposed by Bravo-Prieto et al. is adapted to solve the Laplace and Poisson equations, ordinary differential equation initial value problems (IVPs) with both constant and function coefficients, and a Fredholm integral equation of the second type. As this thesis demonstrates, variational quantum algorithms for solving differential equations and integral equations such as VQLS are highly promising for use on Noisy Intermediate-Scale Quantum (NISQ) devices. The quantum circuit components of the VQLS algorithm were simulated using IBM Quantum's statevector simulator with the optimized solution circuits implemented on the IBM Quantum Guadalupe device (in most cases). A discussion of errors and convergence is also included. The application of VQLS to these differential equations and integral equations is novel and, to my knowledge, work using VQLS to solve the Poisson equation has only recently been published once by authors Liu et al. which was conducted concurrently and independently of this research.
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    Transport equations and boundary conditions for oscillatory rarefied gas flows
    Liu, Nicholas Zhixian ( 2018)
    A wide range of flow phenomena in everyday life can be modelled accurately using classical continuum theory: the Navier-Stokes equations with associated no-slip conditions. However, oscillatory flows generated by nanoscale devices violate the basic assumptions underpinning continuum theory. Study of such flows, under the assumption of small perturbations from equilibrium, requires analysis of the unsteady Boltzmann equation. At sufficiently high oscillation frequencies, the resulting flow can result in wave propagation. This thesis presents a rigorous asymptotic analysis of slightly rarefied wave motion. Particular emphasis is placed on the boundary layer structure, transport equations and associated slip boundary conditions, valid for a general curved oscillating boundary with a velocity-independent body force.