The rise of graphene opens a door to qubit implementation, as discussed in the recent proposal of valley pair qubits in double quantum dots of gapped graphene (G. Y. Wu, N.-Y. Lue, and L. Chang, arXiv:1104.0443). The work here presents the comprehensive theory underlying the proposal. It discusses the interaction of electrons with external magnetic and electric fields in such structures. Specifically, it examines a strong, unique mechanism, i.e., the analog of the first-order relativistic effect in gapped graphene. This mechanism is state mixing free and allows, together with electrically tunable exchange coupling, a fast, all-electric manipulation of qubits via electric gates, in the time scale of ns. The work also looks into the issue of fault tolerance in a typical case, yielding at 10 K a long qubit coherence time [similar to O(ms)].