  We introduce state-of-the-art protocols to distill indistinguishable photons,
reducing distinguishability error rates by a factor of $n$, while using a
modest amount of resources scaling only linearly in $n$. Our resource
requirements are both significantly lower and have fewer hardware requirements
than previous works, making large-scale distillation experimentally feasible
for the first time. This efficient reduction of distinguishability error rates
has direct applications to fault-tolerant linear optical quantum computation,
potentially leading to improved thresholds for photon loss errors and allowing
smaller code distances, thus reducing overall resource costs. Our protocols are
based on Fourier transforms on finite abelian groups, special cases of which
include the discrete Fourier transform and Hadamard matrices. This general
perspective allows us to unify previous results on distillation protocols and
introduce a large family of efficient schemes. We utilize the rich mathematical
structure of Fourier transforms, including symmetries and related suppression
laws, to quantify the performance of these distillation protocols both
analytically and numerically. Finally, our work resolves an open question
concerning suppression laws for the $n$-photon discrete Fourier transform: the
suppression laws are exactly characterized by the well-known Zero Transmission
Law if and only if $n$ is a prime power.
