As a generalization of the standard phase retrieval problem,we seek to reconstruct symmetric rank-
1 matrices from inner products with subclasses of positive semidefinite matrices. For such subclasses, we
introduce random cubatures for spaces of multivariate polynomials based on moment conditions. The inner
products with samples from sufficiently strong random cubatures allow the reconstruction of symmetric rank-
1 matrices with a decent probability by solving the feasibility problem of a semidefinite program.