Given a system of linear equations $\ell_i(x)=\beta_i$ in an $n$-vector $x$ of 0-1 variables, we compute the expectation of $\exp\left\{- \sum_i \gamma_i \left(\ell_i(x) - \beta_i\right)^2\right\}$, where $x$ is a vector of independent Bernoulli random variables and $\gamma_i >0$ are constants. The algorithm runs in quasi-polynomial $n^{O(\ln n)}$ time under some sparseness condition on the matrix of the system. The result is based on the absence of the zeros of the analytic continuation of the expectation for complex probabilities, which can also be interpreted as the absence of a phase transition in the Ising model with a sufficiently strong external field. As an example, we consider the problem of "smoothed counting" of perfect matchings in hypergraphs.

翻译：考虑到一个线性方程系统$\ell_i(x)\ ⁇ beta_i美元, 以美元为单位, 以0-1变量计算, 我们计算出一个线性方程的预期值, 以0-1 美元计算, 以0-1 变量计算, 我们计算出 $\ exm\ left\\\\\ sum_ i\ gamma_i\ left (x) -\ beta_ i- right)\\\\\\\\ right\ $, 美元是独立的Bernoulli 随机变量的矢量, 美元\ gamma_ i > 0 美元是恒定的。 算法在系统矩阵的某种稀疏状态下运行。 其结果是, 对复杂概率的预期没有分析结果持续零, 这也可以被解释为 Ising 模型中没有具有足够强的外部域的阶段性过渡。 例如, 我们考虑在超镜中“ 移动计算” 完美匹配的“ ” 问题 。