Catalytic Embeddings of Quantum Circuits

TL;DR

This paper introduces catalytic embeddings as a novel method to improve quantum circuit approximations by using reusable resource states, leading to shorter circuits compared to traditional repeated approximation techniques.

Contribution

It establishes the foundational theory of catalytic embeddings, their structural properties, and practical methods for their design, extending previous approximation techniques.

Findings

Catalytic embeddings can produce shorter quantum circuits than repeated approximations.

Construction of catalytic embeddings reduces to designing a single fixed matrix for certain gate sets.

Applications include efficient implementation of the Quantum Fourier Transform over Clifford+$T$ gates.

Abstract

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.