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Ising interaction between two qubits composed of the highest magnetic quantum number states through magnetic dipole-dipole interaction

In quantum information processing, one of the most useful interaction between qubits is the Ising type interaction. We propose a scheme to implement the exact Ising interaction through magnetic dipole-dipole interaction. Although magnetic dipolar interaction is Heisenberg type in general, this interaction can bring about the exact mathematical form of the Ising interaction if qubit levels are chosen among the highest magnetic quantum number states. Real physical systems to which our scheme can be applied include rotational states of molecules, hyperfine states of atoms, or electronic states of nitrogen-vacancy centers in diamond. We analyze the feasibility of our scheme for these systems. For example, when the hyperfine levels of rubidium 87 atoms are chosen as qubits and the distance of the two atoms is 0.1 micrometer, controlled-Z gate time will be 8.5 ms. We suggest diverse search and study to achieve optimal implementation of this scheme.