The role of super-fast transforms in speeding up quantum computations

We present the role that spectral methods play in the development of the most impressive quantum algorithms, such as the polynomial time number factoring algorithm by Shor. While the fast transform algorithms reduce the number of operations needed in obtaining the transforms from O(2²ⁿ) to O(n²ⁿ), quantum transforms are in comparison super-fast. The quantum Fourier transform can be performed in O(n²) time, while the specific cases of Walsh-Hadamard and Chrestenson transforms require only O(n) operations. We derive quantum Fourier transform over non-binary quantum digits using the Chrestenson gate, which can serve as a basic block for quantum transforms