Quantum determinant estimation

A quantum algorithm for computing the determinant of a unitary matrix U U (N ) 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 O(N ) the algorithm can determine with certainty the sign of the determinant using O(N log 2 N ) operations and N controlled applications of O.An extension of the algorithm to contractions is discussed.