Determinant computation is equivalent to matrix powering
It states that matrix powering is computationaly equivalent to computation.
From another angle, Coppersmith–Winograd algorithm for matrix multiplication has complexity $\mathcal O(n^{2.373})$ and the same complexity is for determinant computation by fast multiplication.
The result comes from Triangularization and inversion via fast multiplication by James R. Bunch and John E. Hopcroft