### DPhil Thesis

# Scaling
Quantum Compilation
with
Graph Rewriting #

Luca Mondada

Lady Margaret Hall

University of Oxford

A thesis submitted for the degree of

Doctor of Philosophy

Hilary 2025

## Abstract #

As the capabilities of quantum hardware advance and their architectures become more complex, quantum compilers must evolve to optimise increasingly large programs and support a growing range of computational primitives. In particular, the emergence of hybrid quantum-classical computations, required for instance by implementations of quantum error correcting protocols, poses significant challenges to established quantum compilation techniques.

This thesis argues that graph transformation systems (GTS) offer a principled foundation for compiler platforms that can express arbitrary hardware primitives and support computations over both quantum and classical data. To this end, it introduces a graph-based intermediate representation (IR), with support for linear types to model quantum data. This unifies graphical formalisms used for reasoning over quantum computations (such as the ZX calculus) with IR-based program transformation techniques from classical compiler design.

Building upon this foundation, the thesis tackles two critical scaling problems hindering the adoption of GTS in quantum compilers. First, it presents an efficient pattern matching algorithm based on a precomputed data structure that achieves query times independent of the number of transformation rules in the system. This removes a key bottleneck in quantum superoptimisers, which optimise quantum programs using tens to hundreds of thousands of rules.

Second, the thesis introduces a confluently persistent data structure that enables efficient exploration of the GTS state space of possible graph transformations. This factorised search space offers an exponential complexity advantage in search space size and traversal time compared to naively exploring all reachable graphs. The thesis also discusses the problem of extracting optimal programs from this data structure, relating it to Boolean satisfiability (SAT) problems.

Together, these contributions lay the groundwork for scalable, GTS-based quantum compilers and advance the integration of quantum and classical compilation techniques. The persistent data structure for graph rewriting also opens the door for concurrent graph rewriting in distributed systems, a technique that may have applications within the graph transformations field more broadly and merits further research.