Classical Computation without NOT
You're confusing classical and quantum computation, so let me ignore the quantum aspects for now. If you forbid the unary NOT gate then you can use a binary NOT gate, say $g(a,b) = \lnot a$. You can also simulate NOT using natural gates: $XOR(a,1) = \lnot a$ (and this gate can even be made reversible!). So this kind of restriction is not really meaningful.

It is more meaningful to ask what happens if all gates are monotone, that is, they satisfy $g(x_1,\ldots,x_n) \leq g(y_1,\ldots,y_n)$ whenever $x_i \leq y_i$ (here $\text{FALSE}\leq\text{TRUE}$). Such circuits are known as monotone circuits, and they can only compute monotone functions. One natural complete basis is $\{\text{AND},\text{OR}\}$. You can show that this basis is complete using DNFs.