  This note shows that we can recover a complex vector x in C^n exactly from on
the order of n quadratic equations of the form |<a_i, x>|^2 = b_i, i = 1, ...,
m, by using a semidefinite program known as PhaseLift. This improves upon
earlier bounds in [3], which required the number of equations to be at least on
the order of n log n. We also demonstrate optimal recovery results from noisy
quadratic measurements; these results are much sharper than previously known
results.
