Auxiliary-field quantum Monte Carlo method with quantum selected configuration interaction

We propose using the wave function generated by the quantum selected configuration interaction (QSCI) method as the trial wave function in phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC). In the QSCI framework, electronic configurations are sampled from the quantum state realized on a quantum computer. These configurations serve as basis states for constructing an effective Hamiltonian, which is then diagonalized to obtain the corresponding eigenstate. Using this wave function, ph-AFQMC is performed to recover the dynamical electron correlation across the whole orbital space. The use of the QSCI trial wave function is expected to improve the feasibility of the quantum-classical (QC) hybrid quantum Monte Carlo approach [Nature, 603, 416 (2022)]. We call this integrated approach QC-QSCI-AFQMC, or QSCI-AFQMC for short. This method is validated across several molecular systems. For H2O and a linear H4 chain, we achieved chemical accuracy in most investigations relative to full configuration interaction while utilizing superconducting quantum computers at Osaka University and RIKEN. Additionally, the application of QSCI-AFQMC to the O-H bond dissociation in an organic molecule highlights the complementary synergy between capturing static correlation on quantum hardware and incorporating dynamical correlation via classical post-processing. For the N2, when QSCI-AFQMC is executed with a noiseless simulator, it ranks among the most accurate methods compared to various multireference electronic structure theories. Although the proposed method is demonstrated using small active spaces on current quantum devices, the concept is not limited to few-qubit problems. The QSCI-AFQMC can compete with state-of-the-art classical computational techniques, particularly in larger active spaces, displaying considerable potential for resolving classically intractable problems in quantum chemistry.