  Suppose ${\bf x}$ is any exactly $k$-sparse vector in $\mathbb{C}^{n}$. We
present a class of phase measurement matrix $A$ in $\mathbb{C}^{m\times n}$,
and a corresponding algorithm, called SUPER, that can resolve ${\bf x}$ up to a
global phase from intensity measurements $|A{\bf x}|$ with high probability
over $A$. Here $|A{\bf x}|$ is a vector of component-wise magnitudes of $A{\bf
x}$. The SUPER algorithm is the first to simultaneously have the following
properties: (a) it requires only ${\cal O}(k)$ (order-optimal) measurements,
(b) the computational complexity of decoding is ${\cal O}(k\log k)$ (near
order-optimal) arithmetic operations.
