We study the dynamics of a two-level crossing model with a polynomial modification of the linear Landau–Zener tunneling. For a cubic modification, we express the non-adiabatic transition amplitudes analytically via the bi-confluent Heun functions. We find a closed-form series expression of the transition probability at the long time limit, and derive tractable approximate formulas to the state populations in a large part of the parameter space. The analytical approximations are validated by comparison with numerical results. Finally, we discuss the application to fast high fidelity quantum gates in quantum computing.