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 twostate 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. Recently Shor [1, 2] has shown that a quantum computer [3], if it could be built, would be capable of solving certain problems, such as factoring long numbers, much more rapidly than is possible using currently available algorithms on a conventional computer. This has stimulated a lot of interest in the subject [4, 5, 6], and various proposals have been made for actually constructing such a computer [7, 8, 9, 10, 11]. The basic idea is that bits representing numbers can be embodied in two-state quantum systems, for example, in the spin degree of freedom of a spin half particle, and the computation proceeds by manipulating these bits using appropriate gates. It turns out that quantum computations can be carried out using circuits employing one-bit gates, which produce a unitary transformation on the twodimensional Hilbert space representing a single bit, together with two-bit gates producing appropriate unitary transformations on a four-dimensional Hilbert space [12, 13, 14, 15]. One-bit gates should be much easier to construct than two-bit gates, since, for example, an arbitrary unitary transformation on the spin degree of freedom of a spin half particle can be produced by subjecting it to a suitable time-dependent macroscopic magnetic field. On the other hand, a two-bit gate requires that one of the bits influence the other in a Electronic mail: [email protected] Electronic mail: [email protected]