  A quantum eigensolver is designed under a multi-layer 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 multi-spin chains and
experimentally studied in a fully-connected three-spin network, both yielding
an excellent prediction of their ground states.
