Catalytic Embeddings of Quantum Circuits

Understand

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.