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Systems evolving under the influence of competing two-body and three-bodyinteractions, are of particular interest in exploring the stability ofequilibrium states of strongly interacting many-body system. We present asolvable model based on qubit networks, which allows us to investigate theintricate influence of these couplings on the possible asymptotic equilibriumstates. We study the asymptotic evolution of finite qubit networks under twoand three-qubit interactions. As representatives of three-qubit interactions wechoose controlled unitary interactions (cu-interactions) with one and twocontrol qubits. It is shown that networks with purely three-qubit interactionsexhibit different asymptotic dynamics depending on whether we deal withinteractions controlled by one or two qubits. However, when we allowthree-qubit interactions next to two-qubit interactions, the asymptotics isdictated by two-qubit interactions only. Finally, we prove that thesimultaneous presence of two types of three-qubit interactions results in theasymptotic dynamics characteristic for two-qubit cu-interactions.

Dominant couplings in qubit networks with controlled interactions