  In this paper we introduce an approximate method to solve the quantum cavity
equations for transverse field Ising models. The method relies on a projective
approximation of the exact cavity distributions of imaginary time trajectories
(paths). A key feature, novel in the context of similar algorithms, is the
explicit separation of the classical and quantum parts of the distributions.
Numerical simulations show accurate results in comparison with the sampled
solution of the cavity equations, the exact diagonalization of the Hamiltonian
(when possible) and other approximate inference methods in the literature. The
computational complexity of this new algorithm scales linearly with the
connectivity of the underlying lattice, enabling the study of highly connected
networks, as the ones often encountered in quantum machine learning problems.
