Semiclassical Fourier Transform for Quantum Computation

Shor's algorithms for factorization and discrete logarithms on a quantum computer employ Fourier transforms preceding a final measurement. It is shown that such a Fourier transform can be carried out in a semi-classical way in which a ``classical'' (macroscopic) signal resulting from the measurement of one bit (embodied in a two-state quantum system) is employed to determine the type of measurement carried out on the next bit, and so forth. In this way the two-bit gates in the Fourier transform can all be replaced by a smaller number of one-bit gates controlled by classical signals. Success in simplifying the Fourier transform suggests that it may be worthwhile looking for other ways of using semi-classical methods in quantum computing.