Dirac four-potential tunings-based quantum transistor utilizing the Lorentz force

We propose a mathematical model of $\mathit{\text{quantum}}$ transistor in which bandgap engineering corresponds to the tuning of Dirac potential in the complex four-vector form. The transistor consists of $n$-relativistic spin qubits moving in $\mathit{\text{classical}}$ external electromagnetic fields. It is shown that the tuning of the direction of the external electromagnetic fields generates perturbation on the potential temporally and spatially, determining the type of quantum logic gates. The theory underlying of this scheme is on the proposal of the intertwining operator for Darboux transfomations on one-dimensional Dirac equation amalgamating the $\mathit{\text{vector-quantum}}\mathit{\text{gates}}\mathit{\text{duality}}$ of Pauli matrices. Simultaneous transformation of qubit and energy can be accomplished by setting the $\{\mathit{\text{control,}}\mathit{\text{cyclic}}\}$-operators attached on the coupling between one-qubit quantum gate: the chose of $\mathit{\text{cyclic}}$-operator swaps the qubit and energy simultaneously, while $\mathit{\text{control}}$-operator ensures the energy conservation.