Suppose that a quantum circuit with K elementary gates is known for a unitary matrix U, and assume that $U^m$ is a scalar matrix for some positive integer m. We show that a function of U can be realized on a quantum computer with at most ${O(mK+m}^2\ln m)$ elementary gates. The functions of U are realized by a generic quantum circuit, which has a particularly simple structure. Among other results, we obtain efficient circuits for the fractional Fourier transform.

• Received 20 August 2002

DOI:https://doi.org/10.1103/PhysRevA.67.010302