  Low-frequency $1/f^\alpha$ charge noise significantly hinders the performance
of voltage-controlled spin qubits in quantum dots. Here, we utilize fractional
calculus to design voltage control pulses yielding the highest average
fidelities for noisy quantum gate operations. We focus specifically on the
exponential voltage control of the exchange interaction generating two-spin
$\mathrm{SWAP}^k$ gates. When stationary charge noise is the dominant source of
gate infidelity, we derive that the optimal exchange pulse is long and weak,
with the broad shape of the symmetric beta distribution function with parameter
$1-\alpha/2$. The common practice of making exchange pulses fast and
high-amplitude still remains beneficial in the case of strongly nonstationary
noise dynamics, modeled as fractional Brownian motion. The proposed methods are
applicable to the characterization and optimization of quantum gate operations
in various voltage-controlled qubit architectures.
