A complete finite schedule

Let $X$ be a finite site set with finite-dimensional local Hilbert spaces. All persistent quantum memories, classical controllers, environments retained for later use, and record registers must be included in the declared schedule.

At tick $r$, let ${\mathcal{B}}_r$ be a partition of $X$, and specify a channel

on its fixed block spaces. Adaptive programs can be represented only after their controllers and conditional operations have been included in the specification.

For endpoint $Y\subseteq X$, define backwards ancestor sets:

Write $\mathcal{A}(S)$ for operators supported on $S$.