The Phase Ordering Problem for Quantum Circuit Compilation

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By leveraging the laws of quantum mechanics, quantum computing potentially allows us to perform tasks faster than a classical computer. Current devices are noisy, so algorithmsdesigned for fault-tolerant quantum computers cannot be reliably performed. Improvements made upon the resources required to execute such algorithms will enable larger tasks to beaccurately solved on a quantum computer, bringing us closer to quantum advantage. This proposal presents directions of research in programming language theory and circuit compilation, with the aim of designing new tools for quantum computing research and development.Although impressive progress has been made in developing new quantum algorithms and hardware, in practice quantum programs are still defined using circuits, which are akin to assembly code in classical computing. This is not a sensible way to design large-scale quantum applications, as it prevents the programmer from expressing their ideas in a modular, reusable manner. To fully take advantage of quantum advantage, it is necessary to develop semantics for classical-quantum programming. The algebraic framework proposed by Prof. Sam Staton [1], which characterises the unclonability of quantum information using linear type theory, is a good foundation for implementing a classical-quantum programming language which can use algebraic effects to separate the local classical computation from external quantum computation that permits only linear use of resources. This separation will enable the development of more advanced, less error-prone software that is easier to analyse for acompiler.Although ZX-calculus has enabled the development of new and effective rewrite rules for simplifying circuits [2], there is a lack of research in how to select and apply such rewriterules: many papers on circuit compilation propose new rewrite rules and combine them into a greedy routine without leveraging existing rewrite rules. In general, diagrammaticrewriting is a non-confluent operation, so the order in which the compilation phases are applied can severely impact the results. Meanwhile, this problem of phase ordering has beenmore thoroughly studied in the field of classical compilation [3]. My plan is to use pauliopt, a Pauli gadget based library developed by Dr. Stefano Gogioso and me during my master'sthesis, as a framework for implementing the new optimisation routines that I will develop over the course of the DPhil. As a starting point, I will use machine learning to find theoptimal compilation phase ordering of a given circuit. Then, I will incorporate other phase gadget rewrite rules such as the spider nest identities, which have been naively applied toachieve state-of-the-art T-count reduction.In my opinion, the development of open-source software is crucial to the field of quantum computing, so my papers that propose algorithms are accompanied with implemented code.[4, 5] To demonstrate their effcacy and benefits, the techniques developed over the course of the DPhil will be accompanied with implementation and experiments. In particular, Iwill focus on applying QML techniques to NLP, a field I have gained experience in from my work and undergraduate studies. With the excellent resources and supervision that Oxfordprovides, I believe I will be able to achieve state-of-the-art results and advance the field of quantum computing.My project falls within EPSRC Quantum Technologies area.