Fast Quantum Fourier Transforms for a Class of Non-abelian Groups

Details Fast Quantum Fourier Transforms for a Class of Non-abelian Groups Fast Quantum Fourier Transforms for a Class of Non-abelian Groups

1998-07-22

arxiv.org/abs/quant-ph/9807064v1

Physics

Quantum Physics

16 pages, LaTeX2e

Scientific paper

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.

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