General protocols for the efficient distillation of indistinguishable photons

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.