  In this paper we consider a system of quadratic equations |<z_j, x>|^2 = b_j,
j = 1, ..., m, where x in R^n is unknown while normal random vectors z_j in R_n
and quadratic measurements b_j in R are known. The system is assumed to be
underdetermined, i.e., m < n. We prove that if there exists a sparse solution
x, i.e., at most k components of x are non-zero, then by solving a convex
optimization program, we can solve for x up to a multiplicative constant with
high probability, provided that k <= O((m/log n)^(1/2)). On the other hand, we
prove that k <= O(log n (m)^(1/2)) is necessary for a class of naive convex
relaxations to be exact.
