We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the restriction amounts to requiring that the shape of the circuit is invariant under simultaneous row and column permutations of the matrix. We establish unconditional exponential lower bounds on the size of any symmetric circuit for computing the permanent. In contrast, we show that there are polynomial-size symmetric circuits for computing the determinant over fields of characteristic zero.

翻译：我们引入了对称算术电路，即具有自然对称性限制的算术电路。在计算定义于变量矩阵上的多项式（如行列式或积和式）的电路背景下，这种限制要求电路的形状在矩阵的行列同时置换下保持不变。我们建立了计算积和式的任何对称电路规模的无条件指数下界。相比之下，我们证明了在特征为零的域上计算行列式存在多项式规模的对称电路。