Definition: Decoder decomposition and frontier

A declared decomposition is a finite acyclic collection of modules. Module $v$ consumes $i_v$ named input qubits, produces $o_v$ named output qubits, and has a certified local peak $c_v$. Classical dependencies are specified separately. There is no quantum fanout.

For a legal completed-module set $J$ and delivered input prefix $k$, let $F(J,k)$ be the live quantum frontier. Executing an enabled module has laboratory peak at most

and replaces its input names by its output names. A delivery increases the frontier by one.