This is a quantum circuit. M represents the act of making a measurement on the first two qubits. The circuit is supposed to transfer the state $|\psi\rangle = a |0\rangle + b |1\rangle$ ($a, b \in \Bbb C$) from the first qubit to the third qubit, involving a classical operation to select a quantum gate. Depending on the measurement outcome of the first two qubits, what quantum gate should be applied in place of ‘...’, such that the transfer is successful? The measurement results of M and the quantum gate to be applied in each case.
For this circuit a purely classical and a purely quantum circuit is possible or not?