Suppose we have $2n+1$ variables $\mathbf{x}=[x_0,x_1,x_2,\ldots,x_{2n}]^\mathsf{T}$ and $n$ symmetric matrices $(\mathbf{A}_1,\mathbf{A}_2,\ldots,\mathbf{A}_n)$ of dimension $(n+2)\times(n+2)$ and having coefficients in $\mathbb{Q}$. Define $\mathbf{u}={[1,x_0,x_1,x_2,\ldots,x_n]}^\mathsf{T}$, consider a system of $n$ quadratic equations $$\begin{cases}\mathbf{u}^\mathsf{T}\mathbf{A}_1\mathbf{u}=x_{n+1}^2\\\mathbf{u}^\mathsf{T}\mathbf{A}_2\mathbf{u}=x_{n+2}^2\\\vdots\\\mathbf{u}^\mathsf{T}\mathbf{A}_n\mathbf{u}=x_{2n}^2\end{cases}$$
It is known that the solutions $\mathbf{x}$ of the above system form a $n+1$ dimensional Fano variety, which is rationally connected and is thus supposed to have a lot of rational points in $\mathbb{Q}$ (or at least in a finite extension of $\mathbb{Q}$).
However, instead of knowing how many rational solutions there are, I'd rather want to get some explicit rational solutions in $\mathbb{Q}$, provided that there is any.
This can be done by applying a generic rational point finding algorithm, which is somewhat useful when $n$ is small. However, the complexity grows exponentially with $n$ so it quickly becomes infeasible even for moderately large $n$ (e.g., $n=100$). I can also try to compute a Gröbner basis, but the complexity is even worse (double exponential) so it is not likely to give any help either.
I would like to know that, apart from those completely generic methods (exhaustive search, or Gröbner basis), is there any more efficient algorithm in finding rational solutions, that can take advantage of the fact that the defining equations are limited to quadratic forms and the resulting variety is rationally connected/Fano?
I am aware that the degree alone will not play an essential role as it is possible to convert any system of polynomials into system of quadratics, but I believe that being rationally connected really should mean something, or otherwise it will seem really strange to me, that despite of the variety being supposed to have a lot of rational points, and the rational points on it possess a particularly nice structure, I will have no way to do better in finding them than doing a generic search on a variety of generic type.
The algorithm is not required to be deterministic, nor is it required to prove the existence or non-existence of a rational solution. It would be acceptable as long as when there does exist rational solutions, it has a non-trivial success rate to find one (by non-trivial success rate I mean a success rate guaranteed to be larger than a positive absolute constant independent of $n$). It is not required to have completely polynomial complexity either. It would be acceptable as long as it runs in sub-exponential time and is capable of handling reasonably large $n$ like $n=100$.
If such an algorithm (of finding rational points in sub-exponential time) does not exist, then at least I would like to know an algorithm that, given two rational solutions $\mathbf{x}_1,\mathbf{x}_2$ of the above system, it constructs an explicit rational curve on the variety passing through those given points. Such a rational curve must exist (due to the very definition of being "rationally connected"), but instead of knowing "it must exist" I'd rather want to know an algorithm to explicitly compute it.
Does anyone has ideas on this? Any materials and references on explicit computations in rationally connected/Fano varieties are also much appreciated.