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We give a {\em deterministic} algorithm for approximately computing thefraction of Boolean assignments that satisfy a degree-$2$ polynomial thresholdfunction. Given a degree-2 input polynomial $p(x_1,\dots,x_n)$ and a parameter$\eps > 0$, the algorithm approximates \[ \Pr_{x \sim \{-1,1\}^n}[p(x) \geq 0]\] to within an additive $\pm \eps$ in time $\poly(n,2^{\poly(1/\eps)})$. Notethat it is NP-hard to determine whether the above probability is nonzero, soany sort of multiplicative approximation is almost certainly impossible evenfor efficient randomized algorithms. This is the first deterministic algorithmfor this counting problem in which the running time is polynomial in $n$ for$\eps= o(1)$. For "regular" polynomials $p$ (those in which no individualvariable's influence is large compared to the sum of all $n$ variableinfluences) our algorithm runs in $\poly(n,1/\eps)$ time. The algorithm alsoruns in $\poly(n,1/\eps)$ time to approximate $\Pr_{x \sim N(0,1)^n}[p(x) \geq0]$ to within an additive $\pm \eps$, for any degree-2 polynomial $p$.  As an application of our counting result, we give a deterministic FPTmultiplicative $(1 \pm \eps)$-approximation algorithm to approximate the $k$-thabsolute moment $\E_{x \sim \{-1,1\}^n}[|p(x)^k|]$ of a degree-2 polynomial.The algorithm's running time is of the form $\poly(n) \cdot f(k,1/\eps)$.

Deterministic Approximate Counting for Degree-$2$ Polynomial Threshold Functions