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

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 $\log (1/\epsilon)$.

arXiv:2506.15147 [quant-ph]

(or arXiv:2506.15147v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2506.15147 Focus to learn more arXiv-issued DOI via DataCite

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