Quantum circuits for quantum channels

We study the implementation of quantum channels with quantum computers while minimizing the experimental cost, measured in terms of the number of controlled-NOT (CNOT) 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 CNOT 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 CNOT 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 CNOT count. We give explicit examples for small numbers of qubits that provide the lowest known CNOT counts.