A two-live-qubit programme

Define commuting operators on the first two carriers:

They are independent and have one-dimensional joint eigenspaces. Let

Direct conjugation gives

Write the computational readout of positions 1 and 2 after $B$ as $(t,s)$, so

are the eigenvalues of $Q_{12}$ and $P_{12}$, respectively.

After the third carrier arrives, put

The product $C_{ts}$ is implemented by one Pauli gate, up to phase.

The programme is:

1. receive $q_1,q_2$;

2. apply $H_2,{\mathrm{CX}}_{1\to 2},{S_1}^{†},H_1$;

3. destructively read both qubits, retaining bits $t,s$;

4. receive $q_3$;

5. apply ${S_3}^{†},H_3,C_{ts}$;

6. forget $t,s$, and return $q_3$.