C4: a matching two-slot decoder

Define

They commute and have rank-one joint eigenspaces. Let

Then

Let the destructive computational outcomes after $B_{12}$ be $(t,s)$. Their eigenvalues are

Set

The last product is one classically selected Pauli, up to phase.

The chronological programme is:

1. receive carriers $1,2$;

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

3. destructively read both carriers and retain $t,s$;

4. receive carrier $3$;

5. apply ${S_3}^{†},H_3,C_{ts}$;

6. forget the two record bits and return carrier $3$.