  We propose the variational quantum cavity method to construct a minimal
energy subspace of wave vectors that are used to obtain some upper bounds for
the energy cost of the low-temperature excitations. Given a trial wave function
we use the cavity method of statistical physics to estimate the Hamiltonian
expectation and to find the optimal variational parameters in the subspace of
wave vectors orthogonal to the lower-energy wave functions. To this end, we
write the overlap between two wave functions within the Bethe approximation
which allows us to replace the global orthogonality constraint with some local
constraints on the variational parameters. The method is applied to the
transverse Ising model and different levels of approximations are compared with
the exact numerical solutions for small systems.
