The problem of reconstructing the phases of a unitary matrix with prescribed moduli is of a broad interest to people working in many applications, e. g in the circuit theory, phase shift analysis, multichannel scattering, computer science (e. g in the theory of error correcting codes, design theory). We propose efficient algorithms for computing Hermitian unitary matrices for given symmetric bistochastic matrices A(n × n) for n=3 and n=4. We mention also some results for matrices of arbitrary size n.