Definition: Compiled recorded circuit

A compiled tower is a finite event sequence on $Q$, initialized in a pure lab state and fresh zero-state ancillas, with evolving pairwise disjoint copy classes $K_1,\dots ,K_N$, each having a designated root. The admissible events are:

1. (T1)

   Designation: a qubit outside all classes becomes the root of a new singleton class. This is bookkeeping, not a gate.

2. (T2)

   Perfect copy: a CNOT from a class member to a fresh ancilla in $|0⟩$; the target is added to the source's class.

3. (T3)

   Record-controlled processing: a unitary on qubits outside all classes, controlled in the computational basis by class members.

4. (T4)

   Residual processing: a unitary acting only on qubits outside all classes.

After designation or copy creation, a class member may appear only as a (T2) source or a (T3) control.

Write

The final pure state is $|\Psi ⟩$, with $\rho =|\Psi ⟩⟨\Psi |$.