A bare unitary oracle does not supply controlled powers for free

Let the target be one-dimensional and compare $U=I$ with $U^{\mathrm{\prime }}=-I$. An uncontrolled call to $U^{\mathrm{\prime }}$ differs from an uncontrolled call to $U$ by a scalar phase on the target, so every fixed sequence of uncontrolled calls has the same physical outcome probabilities after any final measurement; global phase cancels from a vector's rank-one projector.

On a control qubit, however, $C(I)=I$ while $C(-I)=\mathrm{diag}(1,-1)=Z$. On input $\mathrm{\mid }+⟩=(\mathrm{\mid }0⟩+\mathrm{\mid }1⟩)\mathrm{/}\sqrt{2}$, the first returns $\mathrm{\mid }+⟩$ and the second returns $\mathrm{\mid }-⟩=(\mathrm{\mid }0⟩-\mathrm{\mid }1⟩)\mathrm{/}\sqrt{2}$; a computational-basis-independent $X$-basis measurement distinguishes them with certainty. Thus controlled access contains information absent from the uncontrolled phase-equivalence class.

The cost theorem [F2] counts supplied controlled powers or repetitions of controlled $Q$ only under the corresponding access declaration. In repeated-unitary mode the controlled-$Q$ constituents must be charged; in power-query mode each supplied power carries its declared cost vector, including any implementation constituents. This witness proves that neither mode's controlled access is free from a bare uncontrolled oracle.