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

Develop compact and efficient catalytic constant T-depth Clifford+T circuits for multi-qubit Toffoli, addition (modular or non-modular), and the quantum Fourier transform.

Background

By leveraging catalyst states and phase kickback, the paper implies that many subroutines in QNC^0_f, including multi-qubit Toffoli, addition, and the quantum Fourier transform, can be realized in constant T-depth. However, directly instantiating these from general constructions would likely produce circuits that are too large for practical use.

The authors therefore explicitly pose the open problem of designing compact and efficient catalytic constant T-depth implementations for these core subroutines.