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A quantum eigensolver is designed under a multilayer cluster mean-field (CMF) algorithm by partitioning a quantum system into spatially-separated clusters. For each cluster, a reduced Hamiltonian is obtained after a partial average over its environment cluster. The products of eigenstates from different clusters construct a compressed Hilbert space, in which an effective Hamiltonian is diagonalized to determine certain eigenstates of the whole Hamiltonian. The CMF method is numerically verified in multispin chains and experimentally studied in a fully-connected three-spin network, both yielding an excellent prediction of their ground states.