  Semiconductor double quantum dot hybrid qubits are promising candidates for
high-fidelity quantum computing. However, their performance is limited by
charge noise, which is ubiquitous in solid-state devices, and phonon-induced
dephasing. Here we explore methods for improving the quantum operations of a
hybrid qubit, using strong microwave driving to enable gate operations that are
much faster than decoherence processes. Using numerical simulations and a
theoretical method based on a cumulant expansion, we analyze qubit dynamics in
the presence of $1/f$ charge noise, which forms the dominant decoherence
mechanism in many solid-state devices. We show that, while strong-driving
effects and charge noise both reduce the quantum gate fidelity, simple
pulse-shaping techniques effectively suppress the strong-driving effects.
Moreover, a broad AC sweet spot emerges when the detuning parameter and the
tunneling coupling are driven simultaneously. Taking into account
phonon-mediated noise, we find that it should be possible to achieve $X_{\pi}$
gates with fidelities higher than $99.9\%$.
