Reversible binary dot-product query

Input: An integer $n\ge 0$, ordered $n$-bit registers $\mathrm{\mid }a⟩,\mathrm{\mid }x⟩$, a one-bit target $\mathrm{\mid }z⟩$, and one distinct clean work bit $r=0$.

Output: The reversible map

$$\mathrm{\mid }a⟩\mathrm{\mid }x⟩\mathrm{\mid }z⟩\mathrm{\mid }0{⟩}_r⟼\mathrm{\mid }a⟩\mathrm{\mid }x⟩\mathrm{\mid }z\oplus (a\cdot x)⟩\mathrm{\mid }0{⟩}_r\mathrm{.}$$

Gate model: Use an operation model whose declared finite nonoracle gate alphabet contains and permits Toffoli and CNOT as callable gates; each listed occurrence counts once under Finite circuit resource accounting. This procedure makes no black-box oracle call.

Instructions: For $i=0,1,\dots ,n-1$, apply Toffoli with controls $a_i,x_i$ and target $r$. Apply CNOT from $r$ to $z$. Then apply those same Toffoli gates in reverse order, from $i=n-1$ down to $0$. All gates act on the ordered computational-basis registers of Finite registers and their ordered computational bases; their basis actions are fixed in Toffoli and Fredkin gates and CNOT and Control-Target Order. For $n=0$, the forward and reverse loops are empty and the XOR fold is $0$.

The procedure uses $2n$ Toffoli occurrences and one CNOT occurrence, with one clean work bit. It has no failure branch.