Generic quantum Fourier transforms

Abstract

The quantum Fourier transform (QFT) is a principal ingredient appearing in many efficient quantum algorithms. We present a generic framework for the construction of efficient quantum circuits for the QFT by “quantizing” the highly successful separation of variables technique for the construction of efficient classical Fourier transforms. Specifically, we apply Bratteli diagrams, Gel'fand-Tsetlin bases, and strong generating sets of small adapted diameter to provide efficient quantum circuits for the QFT over a wide variety of finite Abelian and non-Abelian groups, including all families of groups for which efficient QFTs are currently known and many new families as well. Moreover, our method provides the first subexponential-size quantum circuits for the QFT over the linear groups GL k ( q ), SL k ( q ), and the finite groups of Lie type, for any fixed prime power q .