Testing systems of real quadratic equations for approximate solutions

Consider systems of equations $q_i(x)=0$, where $q_i: {\Bbb R}^n \longrightarrow {\Bbb R}$, $i=1, \ldots, m$, are quadratic forms. Our goal is to tell efficiently systems with many non-trivial solutions or near-solutions $x \ne 0$ from systems that are far from having a solution. For that, we pick a delta-shaped penalty function $F: {\Bbb R} \longrightarrow [0, 1]$ with $F(0)=1$ and $F(y) < 1$ for $y \ne 0$ and compute the expectation of $F(q_1(x)) \cdots F(q_m(x))$ for a random $x$ sampled from the standard Gaussian measure in ${\Bbb R}^n$. We choose $F(y)=y^{-2}\sin^2 y$ and show that the expectation can be approximated within relative error $0< \epsilon < 1$ in quasi-polynomial time $(m+n)^{O(\ln (m+n)-\ln \epsilon)}$, provided each form $q_i$ depends on not more than $r$ real variables, has common variables with at most $r-1$ other forms and satisfies $|q_i(x)| \leq \gamma \|x\|^2/r$, where $\gamma >0$ is an absolute constant. This allows us to distinguish between "easily solvable" and "badly unsolvable" systems in some non-trivial situations.