Theorem: C4: exact executable recovery

This programme recovers the message exactly with every reference. Each of its four measurement histories has probability $1/4$, independent of the message-reference state.

Its resource bounds are

with no evolution-oracle calls, no auxiliary entanglement, and $p_{\mathrm{succ}}=1$. Consequently,

Proof. Put $⟨{\beta }_{ts}|=⟨t,s|B_{12}$. The full branch map, including the inaccessible remnant, is

The logical representatives are $L_X=-P_{12}{Z}_3$ and $L_Z=-Q_{12}{Y}_3$. Also,

The correction signs therefore give

The $Z$ equation makes the two logical basis inputs have their corresponding output basis values. The $X$ equation makes their remaining environment vectors equal. Hence

for a fixed branch vector. The reduction on $12$ is $I_4/4$, so ${\Vert {\chi }_{ts}\Vert }^2=1/4$. Linearity proves branchwise reference preservation.

There are two live input qubits before the early readout. Both are consumed before the third arrives. Thereafter one is live. Four early gates, two late basis-change gates, and one Pauli correction give seven gates. The simultaneous early readout adds one round.