Quantum complexity of LWE
We already know that $P$ $ \subseteq $ $BPP$ $\subseteq $ $BQP$. So, since the problems involved in the reductions are "hard", at first glance, such a LWE solver might have some impact on the following famous questions:


Is $P$ equal to $NP$?
Is $NP$ contained in $BQP$ ?


But analyzing it more carefully, we see that the first one is not really related to that setting, because, by Regev's results, given a quantum algorithm to LWE, we have a quantum algorithm to the lattice problems. But both $P$ and $NP$ are defined with respect to "classical" algorithms.

The second question makes more sense because because at least it involves quantum algorithms: if we had a quantum polynomial-time algorithm for LWE and some $NP$-Hard problem that reduces to LWE, then we would have a quantum polynomial-time algorithm for any $NP$ problem (therefore, $NP$ would be a subset of $BQP$).

However, the detail now is that the lattice problems involved in Regev's quantum reduction are not known to be NP-Hard.

That occurs because, roughly speaking, Regev's reduction states that a quantum solver for LWE with parameters $n$, $q$, and $\alpha$, where $0 &lt; \alpha &lt; 1$, gives us a quantum solver for the approximate GapSVP and SIVP over lattices of dimension $n$ with approximation factor $\gamma \approx \frac{n}{\alpha} $.

So, even if the LWE solver worked for $\alpha$ arbitrarily close to $1$, the approximation factor would still be greater than $n$. But we are far away from proving that those lattice problems are NP-Hard for linear approximation factors. And there are even negative results for the hardness of those approximate problems for $\gamma \ge \sqrt{n}$.

Therefore, with the current reductions we know, a quantum polynomial-time algorithm for LWE would (obviously) put LWE inside $BQP$, but would not impact questions 1 and 2.

Of course, from a practical point of view, that is a huge breakthrough.  On one hand those approximate versions of the lattice problems are not known to be NP-Hard, on the other hand, we don't know efficient algorithms to solve them.