On the computational complexity of the general discrete fourier transform
Theoretical Computer Science
Fast generalized Fourier transforms
Theoretical Computer Science
Fast Fourier analysis for abelian group extensions
Advances in Applied Mathematics
Quantum computation of Fourier transforms over symmetric groups
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Adapted diameters and the efficient computation of Fourier transforms on finite groups
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
Normal subgroup reconstruction and quantum computation using group representations
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem
Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures
Quantum algorithms for solvable groups
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Quantum mechanical algorithms for the nonabelian hidden subgroup problem
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
An improved quantum Fourier transform algorithm and applications
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
The power of basis selection in fourier sampling: hidden subgroup problems in affine groups
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Quantum fourier transform over symmetric groups
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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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 GLk(q), SLk(q), and the finite groups of Lie type, for any fixed prime power q.