computational physics - Quantum XOR: How do you generalize it?

Consider the classical XOR Gate:

Given a 2 bit system: $G = [u_1, u_2]$

$$XOR(G) = (u_1 + u_2) \ mod \ 2$$

Is the following a good generalizaiton of a Quantum XOR Gate:

Given a 2-qubit system: $Q = a_1|00] + a_2|01] + a_3|10] + a_4|11]$

We now write:

$$XOR(Q) = a_1|0] + a_2|1] + a_3|1] + a_4|0] = (a_1 + a_4)|0] + (a_2 + a_3)|1]$$

Notice that if we set $a_i = 1$ and the rest to $0$ that the quantum XOR behaves exactly like a classical XOR but it also has the ability to generalize to fractional and complex $a_i$

Answer

Usually, a XOR quantum gate is implemented by the function :

$XOR(Q) = a_1|00 \rangle + a_2|01 \rangle + a_4|10\rangle + a_3|11 \rangle$

The first bit is conserved, while the second bit is the result of an XOR operation between the first and second bit.

For instance, if we have the combination $|11\rangle$, this means, after the transformation : $1$ for the first bit (conserved bit) and $1$ XOR $1 = 0$ for the second bit, so $|11\rangle$ transforms in $|10\rangle$