Recently, our group has developed an alternative optimization protocol for choosing the optimal phase of the adiabatic states. Basically, instead of evaluating the matrix logarithm Tr{| log U| 2 } 2 Ns times, we minimize a polynomial function of U (Tr(3U 2 -16U) for a real overlap matrix U) using Jacobi sweeps. Note that the target function, Tr(3U 2 -16U), is a second order approximation of Tr{| log U| 2 } around U = I. Explicitly, the OP protocol can be implemented at each time step as follows: