Every quantum gate can be decomposed into a sequence of single-qubit gates and controlled-NOTs. In many implementations, single-qubit gates are relatively 'cheap' to perform compared to C-NOTs (for instance, being less susceptible to noise), and hence it is desirable to minimize the number of C-NOT gates required to implement a circuit.
I will consider the task of constructing a generic isometry from m qubits to n qubits, while trying to minimize the number of C-NOT gates required. I will show a lower bound and then give an explicit gate decomposition that gets within a factor of about two of this bound.
Through Stinespring's theorem this points to a C-NOT-efficient way to perform an arbitrary quantum operation. I will then discuss the case of quantum operations in more detail.
