The Tavis-Cummings interaction plus global J_z restricts unitaries on n>2 qubits due to an accidental symmetry; adding J_z^2 breaks the symmetry and achieves semi-universality for symmetry-respecting operations up to center constraints.
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Global Control with the Tavis-Cummings Interaction
The Tavis-Cummings interaction plus global J_z restricts unitaries on n>2 qubits due to an accidental symmetry; adding J_z^2 breaks the symmetry and achieves semi-universality for symmetry-respecting operations up to center constraints.