Hybrid Quantum-Classical Eigensolver Without Variation or Parametric
  Gates
The use of near-term quantum devices that lack quantum error correction, for
addressing quantum chemistry and physics problems, requires hybrid
quantum-classical algorithms and techniques. Here we present a process for
obtaining the eigenenergy spectrum of electronic quantum systems. This is
achieved by projecting the Hamiltonian of a quantum system onto a limited
effective Hilbert space specified by a set of computational bases. From this
projection an effective Hamiltonian is obtained. Furthermore, a process for
preparing short depth quantum circuits to measure the corresponding diagonal
and off-diagonal terms of the effective Hamiltonian is given, whereby quantum
entanglement and ancilla qubits are used. The effective Hamiltonian is then
diagonalized on a classical computer using numerical algorithms to obtain the
eigenvalues. The use case of this approach is demonstrated for ground sate and
excited states of BeH$_2$ and LiH molecules, and the density of states, which
agrees well with exact solutions. Additionally, hardware demonstration is
presented using IBM quantum devices for H$_2$ molecule.
Pejman Jouzdani Stefan Bringuier