Abstract: The Gaussian Elimination with Partial Pivoting (GEPP) is a classical algorithm for solving systems of linear equations. Although in specific cases the loss of precision in GEPP due to roundoff errors can be very significant, empirical evidence strongly suggests that for a {\it typical} square coefficient matrix, GEPP is numerically stable. We obtain a (partial) theoretical justification of this phenomenon by showing that, given the random n n standard Gaussian coefficient matrix A, the {\it growth factor} of the Gaussian Elimination with Partial Pivoting is at most polynomially large in n with probability close to one. This implies that with probability close to one the number of bits of precision sufficient to solve Ax b to m bits of accuracy using GEPP is m O(logn), which improves an earlier estimate m O(log2n) of Sankar, and which we conjecture to be optimal by the order of magnitude. Abstract: Linear reversible circuits represent a subclass of reversible circuits with many applications in quantum computing.