Toffoli and Fredkin gates

All bits are in ${\mathbb{F}}_2=\{0,1\}$, with addition denoted by $\oplus$ and multiplication by juxtaposition. On three distinct ordered computational-basis wires, define the Toffoli gate $T$ and Fredkin gate $F$ by

$$T\mathrm{\mid }a,b,t⟩=\mathrm{\mid }a,b,t\oplus ab⟩,$$

and

$$F\mathrm{\mid }0,a,b⟩=\mathrm{\mid }0,a,b⟩,F\mathrm{\mid }1,a,b⟩=\mathrm{\mid }1,b,a⟩\mathrm{.}$$

These basis actions extend linearly. Wire order is exactly the order displayed; the first two wires of $T$ are controls and the third is its target, while the first wire of $F$ is its control and the other two are conditionally swapped. Negative-control variants mean conjugating the chosen positive control by $X$ before and after the gate. These are fixed arity-three labels in the declared finite gate alphabet, not implicit access to arbitrary multicontrol gates. Their reversibility is established in X, CNOT, Toffoli, and Fredkin are involutions ↗.