  The main objective of this paper is to find algorithms accompanied by
explicit error bounds for phase retrieval from noisy magnitudes of frame
coefficients when the underlying frame has a low redundancy. We achieve these
goals with frames consisting of $N=6d-3$ vectors spanning a $d$-dimensional
complex Hilbert space. The two algorithms we use, phase propagation or the
kernel method, are polynomial time in the dimension $d$. To ensure a successful
approximate recovery, we assume that the noise is sufficiently small compared
to the squared norm of the vector to be recovered. In this regime, the error
bound is inverse proportional to the signal-to-noise ratio. Upper and lower
bounds on the sample values of trigonometric polynomials are a central
technique in our error estimates.
