Catalytic $z$-rotations in constant $T$-depth

TL;DR

This paper demonstrates that using a catalyst state, any single-qubit $z$-rotation can be implemented with a constant $T$-depth, enabling efficient approximation of complex quantum gates.

Contribution

It introduces a method to reduce $T$-depth of $z$-rotations to 3 using catalyst states, with polynomial size in $ ext{log}(1/ ext{error})$, expanding the set of universal gates.

Findings

Catalyst states enable constant $T$-depth $z$-rotations.

Approximate multi-qubit gates like Toffoli and Fourier transform with arbitrary precision.

Catalyst states can be prepared efficiently in polynomial time.

Abstract

We show that the $T$-depth of any single-qubit $z$-rotation can be reduced to $3$ if a certain catalyst state is available. To achieve an $\epsilon$-approximation, it suffices to have a catalyst state of size polynomial in $\log(1/\epsilon)$. This implies that $\mathsf{QNC}^0_f/\mathsf{qpoly}$ admits a finite universal gate set consisting of Clifford+$T$. In particular, there are catalytic constant $T$-depth circuits that approximate multi-qubit Toffoli, adder, and quantum Fourier transform arbitrarily well. We also show that the catalyst state can be prepared in time polynomial in $\log (1/\epsilon)$.