  The design of efficient quantum circuits is an important issue in quantum
computing. It is in general a formidable task to find a highly optimized
quantum circuit for a given unitary matrix. We propose a quantum circuit design
method that has the following unique feature: It allows to construct efficient
quantum circuits in a systematic way by reusing and combining a set of highly
optimized quantum circuits. Specifically, the method realizes a quantum circuit
for a given unitary matrix by implementing a linear combination of representing
matrices of a group, which have known fast quantum circuits. We motivate and
illustrate this method by deriving extremely efficient quantum circuits for the
discrete Hartley transform and for the fractional Fourier transforms. The sound
mathematical basis of this design method allows to give meaningful and natural
interpretations of the resulting circuits. We demonstrate this aspect by giving
a natural interpretation of known teleportation circuits.
