  We show that many well-known signal transforms allow highly efficient
realizations on a quantum computer. We explain some elementary quantum circuits
and review the construction of the Quantum Fourier Transform. We derive quantum
circuits for the Discrete Cosine and Sine Transforms, and for the Discrete
Hartley transform. We show that at most O(log^2 N) elementary quantum gates are
necessary to implement any of those transforms for input sequences of length N.
