  We consider the task of recovering two real or complex $m$-vectors from
phaseless Fourier measurements of their circular convolution. Our method is a
novel convex relaxation that is based on a lifted matrix recovery formulation
that allows a nontrivial convex relaxation of the bilinear measurements from
convolution. We prove that if the two signals belong to known random subspaces
of dimensions $k$ and $n$, then they can be recovered up to the inherent
scaling ambiguity with $m >> (k+n) \log^2 m$ phaseless measurements. Our method
provides the first theoretical recovery guarantee for this problem by a
computationally efficient algorithm and does not require a solution estimate to
be computed for initialization. Our proof is based Rademacher complexity
estimates. Additionally, we provide an ADMM implementation of the method and
provide numerical experiments that verify the theory.
