Complete eigenstates of N identical qubits arranged in regular polygons

We calculate the eigenstates corresponding to coherent single and multiple excitations of an array of N identical qubits arranged on the vertices of a regular polygon. We assume only that the coupling occurs via an exchange interaction which depends on the separation between the qubits. We include the interactions between all pairs of qubits, and our results are valid for arbitrary distances relative to the radiation wavelength. Our results also give the total decay rates from each calculated eigenstate, thereby allowing easy identification of long-lived states useful for quantum information processing.