Pretty much the same as this. Clearly we may assume that all $x_i$ are non-negative. Then, notice that for all $x,y\ge 0$, we have $2\lceil xy \rceil - \lfloor x^2 \rfloor - \lfloor y^2 \rfloor < 2(xy+1) - (x^2+y^2-2) \le 4$, and since this is an integer, we get $2\lceil xy \rceil - \lfloor x^2 \rfloor - \lfloor y^2 \rfloor \le 3$. Also, if $\lfloor x^2 \rfloor$ and $\lfloor y^2 \rfloor$ have the same parity, then $2\lceil xy \rceil - \lfloor x^2 \rfloor - \lfloor y^2 \rfloor \le 2$. Assume $a$ of $\lfloor x_i^2 \rfloor$ is odd and the remaining $31-a$ of them are even. Then, $\sum_{i\neq j}{\lceil x_ix_j \rceil }-30\left(\sum_{i=1}^{31} {\lfloor x_i^2 \rfloor } \right) = \sum_{1\le i<j\le 31} \left(2\lceil x_ix_j \rceil - \lfloor x_i^2 \rfloor - \lfloor x_j^2 \rfloor\right) \le 3\cdot a(31-a) + 2\cdot \left(\binom{a}{2}+\binom{31-a}{2}\right) \le 1170$. Equality holds for example when $16$ of $x_i<\sqrt{2}$ are close to $\sqrt{2}$ and $15$ of $x_i<\sqrt{3}$ are close to $\sqrt{3}$.