Distillation protocols for Fourier states in quantum computing

Fourier states are multi-qubit registers that facilitate phase rotations in fault-tolerant quantum computing. We propose distillation protocols for constructing the fundamental, n-qubit Fourier state with error O(2) at a cost of O(n logn) Toffoli gates and Clifford gates, or any arbitrary Fourier state using O(n) gates. We analyze these protocols with methods from digital signal processing. These results suggest that phase kickback, which uses Fourier states, could be the current lowest-overhead method for generating arbitrary phase rotations.