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
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
The hidden subgroup problem and permutation group theory
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
The Symmetric Group Defies Strong Fourier Sampling
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Quantum algorithms for Simon's problem over general groups
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
The quantum Schur and Clebsch-Gordan transforms: I. efficient qudit circuits
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Quantum algorithms for Simon's problem over nonabelian groups
ACM Transactions on Algorithms (TALG)
Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Finding conjugate stabilizer subgroups in PSL(2; q) and related groups
Quantum Information & Computation
Permutational quantum computing
Quantum Information & Computation
How a Clebsch-Gordan transform helps to solve the Heisenberg hidden subgroup problem
Quantum Information & Computation
Quantum expanders from any classical Cayley graph expander
Quantum Information & Computation
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The quantum Fourier transform (QFT) is the principal ingredient of most 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 use 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 group families for which efficient QFTs are currently known and many new group families. 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.