One standard query implements one phase query

Given: The zero-initialized target, the standard bit query, and the exact one-qubit gate matrices in the ordered computational basis.

Standard bit-flip query convention: one query applies $\mathrm{\mid }x⟩\mathrm{\mid }y⟩\mapsto \mathrm{\mid }x⟩\mathrm{\mid }y\oplus f(x)⟩$.

Binary phase-query convention: $P_f\mathrm{\mid }x⟩=(-1{)}^{f(x)}\mathrm{\mid }x⟩$.

Phase kickback from the standard bit query: a target in $\mathrm{\mid }-⟩$ converts one standard query to the phase query and returns unchanged.

Exact one-qubit gates and rotations: $X\mathrm{\mid }0⟩=\mathrm{\mid }1⟩$ and $H\mathrm{\mid }1⟩=(\mathrm{\mid }0⟩-\mathrm{\mid }1⟩)\mathrm{/}\sqrt{2}$.

Finite circuit resource accounting: the declared gate count counts each $X$ and $H$ occurrence once, separately from black-box calls.

Nonoracle quantum gate cost: query calls and gates outside those calls are separate resource coordinates, and the latter count preparation gates.

For the explicit preparation from $\mathrm{\mid }0⟩$ and its two-gate count, the selected nonoracle operation model has a finite gate alphabet containing and permitting $X$ and $H$ as callable gates, each occurrence is counted once, and the standard query is charged separately as one oracle call.

The phase identity in [F3] for an already prepared $\mathrm{\mid }-⟩$ target does not require [A1]; [A1] is needed only for the displayed preparation from $\mathrm{\mid }0⟩$ and its gate count.