  Spin-spin exchange interactions between semiconductor spin qubits allow for
fast single and two-qubit gates. During exchange, coupling of the qubits to a
surrounding phonon bath may cause errors in the resulting gate. Here, the
fidelities of exchange operations with semiconductor double quantum dot spin
qubits in a Si-SiGe heterostructure coupled to a finite temperature phonon bath
are considered. By employing a master equation approach, the isolated effect of
each spin-phonon coupling term may be resolved, as well as leakage errors of
encoded qubit operations. As the temperature is increased, a crossover is
observed from where the primary source of error is due to a phonon induced
perturbation of the two electron spin states, to one where the phonon induced
coupling to an excited orbital state becomes the dominant error. Additionally,
it is shown that a simple trade-off in pulse shape and length can improve
robustness to spin-phonon induced errors during gate operations by up to an
order of magnitude. Our results suggest that for elevated temperatures within
200-300 mK, exchange gate operations are not currently limited by bulk phonons.
This is consistent with recent experiments.
