  The key to realizing fault-tolerant quantum computation for semiconductor
quantum-dot singlet-triplet (ST) qubits is to operate both the single- and
two-qubit gates with high fidelity. The feasible way includes operating the
qubit near the transverse sweet spot (TSS) to reduce the leading order of the
noise, as well as adopting the merit of the global property of the geometric
phase. Here, we propose the implementation of the universal geometric gates for
the ST qubits in the double quantum dot (DQD). The single-qubit non-cyclic
geometric gate which is using the non-cyclic evolution path of the state to
obtain the geometric phase can be achieved via introducing the AC-driven
detuning operating near the TSS. We perform numerical simulation under the
realistic $1/f$ noise environment in DQD. We find that the non-cyclic geometric
gate can surpass its dynamical counterpart with a relatively high fidelity of
99.81\%. Meanwhile, we have used the dipole coupling between the DQDs and the
cavity resonator which is maximized at the TSS to construct the geometric
(holonomic) entangling geometric gate. We have shown that the fidelity for this
two-qubit gate can be as high as 99.63\%. Our results suggests that taking
advantage of the non-cyclic geometric gates together with using the TSS can be
effective to obtain high-fidelity ST qubits.
