A single logical cell with an occupied-vacuum flag

Let a memory cell and each outgoing cell contain $\mathbb{C}|\mathrm{vac}⟩\oplus C$. At step $j$, a specified unitary rotates

for all logical labels $i$, using the complementary rotation on the opposite occupied-vacuum states and the identity elsewhere. Choose $0\le {\theta }_j\le \pi /2$. Define

The amplitude of first transfer at step $s$ is $\sin {\theta }_s{\prod }_{\ell <s}\cos {\theta }_{\ell }$. Orthogonal transfer locations imply that, when $q_j<1$, these amplitudes define an isometry $A_j:C\to R_j$ after normalization.