  For the variational quantum eigensolver we propose to generate trial
wavefunctions from a small amount of selected Pauli terms of the problem
Hamiltonian. Two different approaches, one inspired by the quantum approximate
optimization algorithm and the other by imaginary-time evolution, are proposed
and studied in detail. Using numerical calculations, we study the efficiency of
these trial wavefunctions for finding the ground-state energy of three
molecules: H2, LiH and H2O. We find that only a small number of Pauli terms are
needed to reach chemical accuracy, leading to short-depth quantum circuits with
a small number of variational parameters. For the LiH molecule, the quantum
circuit consists of 36 two-qubit gates, 45 one-qubit gates, and four
variational parameters, with a favorable scaling for larger molecules.
