Optimal reset of Zeeman disk qubits

Qubits in planar quantum disks can be controlled using magnetic fields. This provides a practical mechanism for qubit reset during operation. Then, the question arises: What reset protocol is the optimal one in terms of least dissipated heat? To answer this question, we solve the Euler-Lagrange equation for the dissipation rate of Zeeman disks with both soft and hard confinement. Exploiting the conserved Hamilton function reduces the complexity from solving nonlinear differential equations to purely algebraic ones. We also discuss practical constraints imposed by the maximal achievable drive. Finally, an approximate analytical protocol is devised and shown to be highly accurate.