Phonon-induced exchange gate infidelities in semiconducting Si-SiGe spin qubits

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 tradeoff 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 $200--300\phantom{\rule{0.16em}{0ex}}\mathrm{mK}$, exchange gate operations are not currently limited by bulk phonons. This is consistent with recent experiments.