Let $p(x)$ be a fixed distribution over a discrete space.

Let $A, C > 0$ be constants. Let $\epsilon > 0$. Can we find an example of a distribution $q_{\epsilon}$ such that $\mathrm{KL}(p||q_{\epsilon}) < \epsilon$, but $E[(\log (p(x)/q_\epsilon(x)))^2] \ge A * \epsilon^C$?

$q_{\epsilon}$ should be geometric, if possible (in which case, the discrete space is the natural numbers).

Perhaps I could also get a clue if I knew whether there is a name for the quantity $E[(\log (p(x)/q_\epsilon(x)))^2]$.

All expectations are taken with respect to $p$. So, for example,

$E[(\log (p(x)/q_\epsilon(x)))^2] = \sum_x p(x) \left(\log (p(x)/q_\epsilon(x))\right)^2$

and

$\mathrm{KL}(p||q_{\epsilon}) = \sum_x p(x) \log \frac{p(x)}{q_{\epsilon}(x)}$

I think that the only chance of solving this is by creating a $q_\epsilon$ such that $\log(p/q_\epsilon)$ is either extremely negative or extremely positive. Then, when taking the squares, it would diverge. But I don't see how this could happen when the KL divergence is smaller than $\epsilon$, because, for example, when the KL divergence is exactly 0, $p(x)=q_{\epsilon}(x)$ for all $x$, and we could not have this effect of $\log p(x)/q(x)$ being extremely small or big.

I think I can show that for the geometric case, unfortunately, there is no such construction, but I am not sure. I would be okay with other examples, as long as the space is discrete.