Solving Quadratic Equations via PhaseLift when There Are About As Many Equations As Unknowns

This note shows that we can recover any complex vector x0 ∈ C exactly from on the order of n quadratic equations of the form |〈ai,x0〉| = bi, 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. Further, we show that exact recovery holds for all input vectors simultaneously, and also demonstrate optimal recovery results from noisy quadratic measurements; these results are much sharper than previously known results.