Theorem: Sequential local route for the standard qubit code

Initialize $q$ in $|0⟩$, apply ${\mathsf{Enc}}_1(X_1,q)$, then ${\mathsf{Enc}}_2(X_2,q)$. The resulting state is

The destructive decoder (eq:decoderterm) satisfies

and hence

The only route from the first input to the second encoding event in this implementation is the carrier. No separate retained earlier-bit wire is required.

Proof. The first rotation takes the $+Z$ Bloch vector to

Conjugation by ${\sigma }_x$ preserves its $X$ component and reverses its $Z$ component. This gives (eq:qubitencoding).

The destructive branch maps are

Their effects are

Evaluation on ${\rho }_x$ gives the stated probability.

In both encoding macros the current input is consumed as a guard and the carrier is used once on the executed branch. There is no output wire for an earlier-bit copy.