As a nuclear spin model of scalable quantum register, the one-dimensional chain of the magnetic atoms with nuclear spins 1/2 substituting the basic atoms in the plate of nuclear spin free easy-axis 3D antiferromagnet is considered. It is formulated the generalized antiferromagnet Hamiltonian in spin-wave approximation (low temperatures) considering the inhomogeneous external magnetic field, which is directed along the easy axis normally to plane of the plate and has a constant gradient along the nuclear spin chain. Assuming a weak gradient, the asymptotic expression for coefficients of unitary transformations to the diagonal form of antiferromagnet Hamiltonian is found. With this result the expression for indirect interspin coupling, which is due to hyperfine nuclear electron coupling in atoms and the virtual spin wave propagation in antiferromagnet ground state, was evaluated. It is shown that the inhomogeneous magnetic field essentially modifies the characteristics of indirect interspin coupling. The indirect interaction essentially grows and even oscillates in relation to the interspin distance when the local field value in the middle point of two considered nuclear spin is close to the critical field for quantum phase transition of spin-flop type in bulk antiferromagnet or close to antiferromagnetic resonance. Thus, the external magnetic field, its gradient, microwave frequency and power can play the role of control parameters for qubit states. Finally, the one and two qubit states decoherence and longitudinal relaxation rate are caused by the interaction of nuclear spins with virtual spin waves in antiferromagnet ground state are calculated.
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