On the computational complexity of the general discrete fourier transform
Theoretical Computer Science
Fast generalized Fourier transforms
Theoretical Computer Science
Quantum computation of Fourier transforms over symmetric groups
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Quantum Wavelet Transforms: Fast Algorithms and Complete Circuits
QCQC '98 Selected papers from the First NASA International Conference on Quantum Computing and Quantum Communications
AAECC-13 Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Algorithms for quantum computation: discrete logarithms and factoring
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
AAECC-13 Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
How a Clebsch-Gordan transform helps to solve the Heisenberg hidden subgroup problem
Quantum Information & Computation
Efficient quantum algorithms for the hidden subgroup problem over semi-direct product groups
Quantum Information & Computation
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An algorithm is presented allowing the construction of fast Fourier transforms for any solvable group on a classical computer. The special structure of the recursion formula being the core of this algorithm makes it a good starting point to obtain systematically fast Fourier transforms for solvable groups on a quantum computer. The inherent structure of the Hilbert space imposed by the qubit architecture suggests to consider groups of order 2n first (where n is the number of qubits). As an example, fast quantum Fourier transforms for all 4 classes of nonabelian 2-groups with cyclic normal subgroup of index 2 are explicitly constructed in terms of quantum circuits. The (quantum) complexity of the Fourier transform for these groups of size 2n is O(n2) in all cases.