  We study the implementation of quantum channels with quantum computers while
minimizing the experimental cost, measured in terms of the number of
Controlled-NOT (C-NOT) gates required (single-qubit gates are free). We
consider three different models. In the first, the Quantum Circuit Model (QCM),
we consider sequences of single-qubit and C-NOT gates and allow qubits to be
traced out at the end of the gate sequence. In the second (RandomQCM), we also
allow external classical randomness. In the third (MeasuredQCM) we also allow
measurements followed by operations that are classically controlled on the
outcomes. We prove lower bounds on the number of C-NOT gates required and give
near-optimal decompositions in almost all cases. Our main result is a
MeasuredQCM circuit for any channel from m qubits to n qubits that uses at most
one ancilla and has a low C-NOT count. We give explicit examples for small
numbers of qubits that provide the lowest known C-NOT counts.
