Definition: Finite coherent-oracle instance

A finite instance consists of:

1. an $n$-qubit message register $M$, $d=2^n$;

2. a finite circuit skeleton $A$ on $w\ge n$ qubits, with $g$ ordinary gates in $\{H,T,T^{†},\mathrm{CNOT}\}$, and $q$ oracle calls;

3. a fixed address length $m$ and one supplied function $f:\{0,1{\}}^m\to \{0,1\}$;

4. coherent access to

   $$O_f|x,b⟩=|x,b\oplus f(x)⟩;$$

5. a pure zero initialization of the $w-n$ workspace qubits;

6. designated message outputs, an ownership declaration for every other output, and a target unitary channel $\mathcal{U}={\mathrm{Ad}}_U$.

The resulting unitary circuit is $V_f$, and its logical channel is

after the declared output regrouping.