Reasoning in ZX calculus and quantum natural language processing

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Quantum computers have the potential to solve tasks intractable on classical computers, thereby revolutionising science and technology in various fields. However, quantum algorithms are still based on quantum circuits, a rather unintuitive low-level language without support for abstraction. One promising alternative comes from the quantum computing group at Oxford. Their diagrammatic languages allow for more intuitive approaches to quantum computing. In my DPhil, I will work with Aleks Kissinger, joint head at the quantum group at Oxford, and Bob Coecke, chief scientist at Quantinuum in Oxford, to explore two areas in quantum computing: (a) quantum circuit optimisation and (b) reasoning in quantum natural language processing - both utilising diagrammatic languages. As such, this DPhil falls within the EPSRC quantum technologies research area.The diagrammatic language developed at Oxford was initially created as a categorical view of quantum mechanics (Coecke & Kissinger, 2017). This language, with its intuitive visual representation, simplifies complex reasoning processes. A particularly useful application is ZX calculus - a diagrammatic language for quantum circuits developed by Coecke and Duncan (2008). It splits quantum gates into more fundamental, compositional elements. Using eight simple yet complete topological rules (Vilmart, 2019), ZX calculus is a potent, intuitive tool to reason over quantum circuits.The goal of quantum circuit optimisation is to rewrite quantum circuits for given requirements, such as routing conditions, or to reduce the number of (noisy) gates (Duncan et al., 2020). Current optimisation techniques often operate on greedy routines based on a subset of rewrite rules. In my DPhil, I will use my experience with machine learning to build on this work. Having more elaborate methods of applying circuit rewrites can improve performance drastically. Some early attempts to use simulated annealing and genetic algorithms by Krueger (2022) show good results for relatively small circuits. I will continue this endeavour by taking inspiration from proven optimisation techniques in other formal reasoning tasks such as theorem proving (e.g., Kaliszyk et al., 2018).While ZX calculus is for low-level quantum circuits, diagrammatic languages are also used for high-level tasks. DisCoCirc is a framework for natural language processing on quantum computers (Coecke, 2021). DisCoCirc models the meaning of entire texts by differentiating the syntactic interaction of words from their semantic representation. This separation creates more understandability and the potential for explicit reasoning.During my DPhil, I will explore reasoning in DisCoCirc from two directions: On the one hand, it is essential to understand better how neural-net-based DisCoCirc systems learn. For this, I will explore alternative representations of meaning for reasoning tasks. On the other hand, I will continue developing a formal understanding of reasoning by exploring individual rules and their interactions (as in Rodatz et al., 2021). Initially, I will explore the interaction between conjunction (proposed by Duneau (2021)), negation and later other operators, such as quantifiers.Using diagrammatic languages, reasoning on quantum circuits in ZX diagrams and in natural language become related. While quantum circuits are more structured, reasoning over quantum circuits can inspire reasoning in natural language, as, for example, in our work on conversational negation (Rodatz et al., 2021). Additionally, the DisCoCirc framework can gain computational speedups on quantum computers (Zeng & Coecke, 2016). Therefore, efficient optimisation techniques enable a larger variety of experiments on reasoning in natural language.The planned combination of two different topics in quantum computing is exciting and promising, given their shared approach of diagrammatic languages.