Manipulation of two spin qubits in a double quantum dot using an electric field

We propose purely electric manipulation of spin qubits by means of the spin-orbit interaction (SOI) {\it without magnetic field or magnets} in a double-quantum dot. All the unitary transformations can be constructed by the time-dependent Dzyaloshinsky-Moriya interaction between the two spins, which arises from the Rashba SOI modulated by electric field. As a few demonstrations, we study both analytically and numerically the three operations, i.e., (A) the spin initialization, (B) the two-spin rotation in the opposite directions, and (C) the two-spin rotation in the same direction. The effects of the relaxation and the feasibility of this proposal are also discussed.