Theorem: Streaming decoder correctness

The programme implements the same channel $\mathcal{D}$ as the batch decoder. On encoded inputs each of its four histories has probability $1/4$, independent of the logical-reference input.

Its resources satisfy

Proof. Let $⟨{\beta }_{ts}|=⟨t,s|B_{12}$. The branch map on the full encoded state is

Since $X_A=-P_{12}{Z}_3$ and $Z_A=-Q_{12}{Y}_3$, the eigenvalue equations imply

and similarly $W_{ts}Z=Z_3{W}_{ts}$. Here $R_3{Z}_3{R_3}^{†}=X_3$, $R_3{Y}_3{R_3}^{†}=Z_3$, and the correction supplies the required signs.

The same intertwining argument as in Lemma lem:factor gives

for a fixed branch vector. The first-two-qubit reduction is $I_4/4$, so ${\Vert {\chi }_{ts}\Vert }^2=1/4$. Thus every branch preserves the logical-reference state, and the complete channel is exact recovery. Theorem thm:B3 then identifies it with $\mathcal{D}$ on the whole accessible input space.

Before the first readout two qubits are live. They are both consumed before the third arrives. Thereafter only one qubit is live. Four early gates, two late basis-change gates, and at most one correction give $G\le 7$. The early readout adds one round to the seven sequential gate rounds. No ancilla or retained quantum record is used.