Replacing n with 2n in asymptotic bounds
In mathematics we have the substitution principle. If a certain statement is true, then it also holds if you replace all occurrences of $n$ by some other expression, as long as that other expression has the same type as $n$. In particular, you can substitute $2n$ for $n$ and deduce that $|p-q|_1 \lll 2^{-\Omega(2n)}$. You can then conclude that $|p-q|_1 \lll 2^{-\Omega(n)}$, using the definition of $\Omega$.