A quantum algorithm for computing the determinant of a unitary matrix $U\in U\left(N\right)$ is given. The algorithm requires no preparation of eigenstates of $U$ and estimates the phase of the determinant to $t$-binary digits accuracy with $O(N{log}^{2}N+t^2)$ operations and $tN$ controlled applications of $U^{2^m}$ with $m=0,...,t-1$. For an orthogonal matrix $O\in O\left(N\right)$ the algorithm can determine with certainty the sign of the determinant using $O\left(N{log}^{2}N\right)$ operations and $N$ controlled applications of $O$. An extension of the algorithm to contractions is discussed.

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• Quantum algorithms & computation