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We propose quantum-selected configuration interaction (QSCI), a class ofhybrid quantum-classical algorithms for calculating the ground- andexcited-state energies of many-electron Hamiltonians on noisy quantum devices.Suppose that an approximate ground state can be prepared on a quantum computereither by variational quantum eigensolver or by some other method. Then, bysampling the state in the computational basis, which is hard for classicalcomputation in general, one can identify the electron configurations that areimportant for reproducing the ground state. The Hamiltonian in the subspacespanned by those important configurations is diagonalized on classicalcomputers to output the ground-state energy and the corresponding eigenvector.The excited-state energies can be obtained similarly. The result is robustagainst statistical and physical errors because the noisy quantum devices areused only to define the subspace, and the resulting ground-state energystrictly satisfies the variational principle even in the presence of sucherrors. The expectation values of various other operators can also be estimatedfor obtained eigenstates with no additional quantum cost, since the expliciteigenvectors in the subspaces are known. We verified our proposal by numericalsimulations, and demonstrated it on a quantum device for an 8-qubit molecularHamiltonian. The proposed algorithms are potentially feasible to tackle somechallenging molecules by exploiting quantum devices with several tens ofqubits, assisted by high-performance classical computing resources fordiagonalization.

Quantum-Selected Configuration Interaction: classical diagonalization of Hamiltonians in subspaces selected by quantum computers