Fast Quantum Fourier Transforms for a Class of Non-abelian Groups
- Cite this paper as:
- Püschel M., Rötteler M., Beth T. (1999) Fast Quantum Fourier Transforms for a Class of Non-abelian Groups. In: Fossorier M., Imai H., Lin S., Poli A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1999. Lecture Notes in Computer Science, vol 1719. Springer, Berlin, Heidelberg
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.
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