Mathieu Control of the Effective Coupling in Superconducting Qubits

A common challenge in superconducting quantum circuits is the trade-off between strong coupling and computational subspace integrity. We present Mathieu control, which uses a non-resonant two-photon drive to create a selective nonlinear frequency shift. This shift modifies interactions while preserving qubit states, enabling continuous tuning of the ZZ coupling, including full suppression, and integrating single- and two-qubit gates with low leakage. For a qubit-coupler-qubit device, it allows independent ZZ control, facilitating a programmable Heisenberg (XXZ) Hamiltonian. Extended to a five-qubit chain, the system can be reconfigured to simulate dynamics of quantum magnetic phases. Mathieu control thus provides a framework for high-fidelity quantum logic and programmable simulation.