Quantum-Selected Configuration Interaction: Exact diagonalization of Hamiltonians in a subspace selected by quantum computers

In this work, we propose a new hybrid quantum-classical algorithm for calculating the ground- and excited-state energies of molecular Hamiltonians. For calculating the ground-state energies on noisy intermediate-scale quantum (NISQ) devices, variational quantum eigensolver (VQE) is most widely used. However, although VQE is designed to work on the noisy devices, it still suffers the physical and statistical errors. Most critically, the effect of the noises spoils the "variational" nature of VQE; due to the errors, the resulting energy is not guaranteed to be higher or equal to the exact ground-state energy, which makes it hard to assess the quality of the VQE calculation.

We propose a class of algorithms to find the lowest energy eigenstates of Hamiltonians in truly variational way. Suppose that an approximate ground state can be prepared on a quantum computer either by VQE or by other methods. Then, sampling from the state identifies the electron configurations that are important for the ground state. One can then classically diagonalize the truncated Hamiltonian in the subspace spanned by those important basis states to get the ground-state energy and the corresponding eigenvector. One can also iteratively continue the process to get the excited-state energies. We verified our proposal by numerical simulations and also by experiment using 8 qubits on an actual device.