Exact catalytic constant T-depth circuits for Toffoli, addition, and QFT

Devise catalytic constant T-depth Clifford+T circuits that implement multi-qubit Toffoli, addition, and the quantum Fourier transform exactly, rather than approximately.

Background

The paper’s approach produces rotations whose phases are integer multiples of 2π/(2^n−1), which suffices for approximation but not for exact realization of certain standard subroutines.

The authors explicitly highlight that, as a consequence, circuits for multi-qubit Toffoli, addition, and QFT produced by their method are only approximately correct, and they pose the open problem of achieving exact implementations at constant T-depth with catalysts.