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 eigen-energy spectrum of electronic quantum systems. This is
achieved by projecting the Hamiltonian of a quantum system onto a limited effective Hilbert space
specied by a set of computational basis. 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 in order to obtain the eigenvalues. The use case of this approach is
demonstrated for ground-sate and excited states of BeH2 and LiH molecules, and the density of
states, which agrees well with exact solutions. Additionally, hardware demonstration is presented
using IBMQ cloud for H2 molecule.