  Finding a succinct representation to describe the ground state of a
disordered interacting system could be very helpful in understanding the
interplay between the interactions that is manifested in a quantum phase
transition. In this work we use some elementary states to construct recursively
an ansatz of multilayer wave functions, where in each step the higher-level
wave function is represented by a superposition of the locally "excited states"
obtained from the lower-level wave function. This allows us to write the
Hamiltonian expectation in terms of some local functions of the variational
parameters, and employ an efficient message-passing algorithm to find the
optimal parameters. We obtain good estimations of the ground-state energy and
the phase transition point for the transverse Ising model with a few layers of
mean-field and symmetric tree states. The work is the first step towards the
application of local and distributed message-passing algorithms in the study of
structured variational problems in finite dimensions.
