Catalytic Embeddings of Quantum Circuits

If a set $\mathbb{G}$ of quantum gates is countable, then the operators that can be exactly represented by a circuit over $\mathbb{G}$ form a strict subset of the collection of all unitary operators. When $\mathbb{G}$ is universal, one circumvents this limitation by resorting to repeated gate approximations: every occurrence of a gate which cannot be exactly represented over $\mathbb{G}$ is replaced by an approximating circuit. Here, we introduce catalytic embeddings, which provide an alternative to repeated gate approximations. With catalytic embeddings, approximations are relegated to the preparation of a fixed number of reusable resource states called catalysts. Because the catalysts only need to be prepared once, when catalytic embeddings exist they always produce shorter circuits, in the limit of increasing gate count and target precision. In the present paper, we lay the foundations of the theory of catalytic embeddings and we establish several of their structural properties. In addition, we provide methods to design catalytic embeddings, showing that their construction can be reduced to that of a single fixed matrix when the gates involved have entries in well-behaved rings of algebraic numbers. Finally, we showcase some concrete examples and applications. Notably, we show that catalytic embeddings generalize a technique previously used to implement the Quantum Fourier Transform over the Clifford+$T$ gate set with $O(n)$ gate approximations.