Fast Quantum Fourier Transforms for a Class of Non-abelian Groups
<|reference_start|>Fast Quantum Fourier Transforms for a Class of Non-abelian Groups: 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 2^n first (where n is the number of qubits). As an example, fast quantum Fourier transforms for all 4 classes of non-abelian 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 2^n is O(n^2) in all cases.<|reference_end|>
@article{pueschel1998fast,
    title={Fast Quantum Fourier Transforms for a Class of Non-abelian Groups},
    author={Markus Pueschel, Martin Roetteler, and Thomas Beth (Universitaet
  Karlsruhe)},
    journal={Proceedings 13th International Symposium on Applied Algebra,
  Algebraic Algorithms and Error-Correcting Codes (AAECC'99), Honolulu, Hawaii,
  Springer LNCS, pp. 148-159, 1999},
    year={1998},
    archivePrefix={arXiv},
    eprint={quant-ph/9807064},
    primaryClass={quant-ph cs.ET}
}