What is the fastest addition algorithm on a turing machine?
(This should be a comment but I can't comment yet)

Just to clarify, when you have an integer $n$, it's represented on $\operatorname{size}(n)=\log_2 n$ bits.

So here, you are given $n_1$ and $n_2$ and you want to compute $n_1+n_2$. Let $f(n_1, n_2)$ be the number of steps needed to compute that. You seem to be claiming that $f(n_1, n_2)=O\left((n_2)^2\right)=O\left(\left(2^{\operatorname{size}(n_2)}\right)^2\right)$, which is obviously false. Now if you change it to $f(n_1, n_2)=O\left((\operatorname{size}(n_2))^2\right)=O\left((\log_2 n_2)^2\right)$, it becomes a little bit more plausible.

But that still can't be the complexity because if you have $n_1=2^{k}-1$ and $n_2=2^0$, your algorithm will at least read $n_1$, which takes $O\left(\operatorname{size}(n_1)\right)=O\left(\log_2 n_1\right)=O(k)$ while $O\left((\log_2 n_2)^2\right)=O(1)$ in these cases. The complexity should depend on the size of $n_1$ one way or another.