Quantum Machine Learning and Quantum Protocols for Solving Differential Equations

Abstract

Quantum devices are being developed to perform computation in an inherently non-classical way. These devices are fundamentally different from conventional computers and have unique properties due to effects such as superposition and entanglement. At the same time, quantum devices are prone to noise, posing limitations on the depth of calculations and scaling. It remains an important challenge for quantum computing to offer advantage in solving of currently intractable industrial problems.

One rising area of quantum algorithms is quantum machine learning. Building off classical machine learning, quantum machine learning concerns the training of quantum models to learn and recognise relationships in data. Another class is variational quantum algorithms which utilise an optimisation loop to train a trial solution involving quantum evaluations to solve a given problem. These classes of algorithm have possibility of advantage for problems with large amounts of data or large search spaces due to the wide range of functions expressible and data encodable because of the exponential working space.

A possible area of application is differential equations. Differential equations govern many areas of industrial and research interest, from aerodynamics to finance to chemistry, yet many instances remain difficult to solve classically. Throughout my research I have considered solving differential equations with quantum machine learning and variational approaches.

In my thesis I describe four algorithms that I have developed for solving differential equations, each with different strengths and weaknesses, quantum resource requirements and areas of applications. Particular techniques utilised are quantum models representing functions, the parameter shift rule, kernel methods and quantile mechanics. Additionally, I (in collaboration) develop a technique to transform between computational and Chebyshev space. This technique is utilised for developing the algorithm for efficient encoding of physics-informed constraints into quantum models. I conclude this thesis with an outlook into the nascent area of quantum scientific machine learning.