  We propose quantum-selected configuration interaction (QSCI), a class of
hybrid quantum-classical algorithms for calculating the ground- and
excited-state energies of many-electron Hamiltonians on noisy quantum devices.
Suppose that an approximate ground state can be prepared on a quantum computer
either by variational quantum eigensolver or by some other method. Then, by
sampling the state in the computational basis, which is hard for classical
computation in general, one can identify the electron configurations that are
important for reproducing the ground state. The Hamiltonian in the subspace
spanned by those important configurations is diagonalized on classical
computers to output the ground-state energy and the corresponding eigenvector.
The excited-state energies can be obtained similarly. The result is robust
against statistical and physical errors because the noisy quantum devices are
used only to define the subspace, and the resulting ground-state energy
strictly satisfies the variational principle even in the presence of such
errors. The expectation values of various other operators can also be estimated
for obtained eigenstates with no additional quantum cost, since the explicit
eigenvectors in the subspaces are known. We verified our proposal by numerical
simulations, and demonstrated it on a quantum device for an 8-qubit molecular
Hamiltonian. The proposed algorithms are potentially feasible to tackle some
challenging molecules by exploiting quantum devices with several tens of
qubits, assisted by high-performance classical computing resources for
diagonalization.
