  Quantum computers require high fidelity quantum gates. These gates are
obtained by routine calibration tasks that eat into the availability of
cloud-based devices. Restless circuit execution speeds-up characterization and
calibration by foregoing qubit reset in between circuits. Post-processing the
measured data recovers the desired signal. However, since the qubits are not
reset, leakage -- typically present at the beginning of the calibration -- may
cause issues. Here, we develop a simulator of restless circuit execution based
on a Markov Chain to study the effect of leakage. In the context of error
amplifying single-qubit gates sequences, we show that restless calibration
tolerates up to 0.5% of leakage which is large compared to the $10^{-4}$ gate
fidelity of modern single-qubit gates. Furthermore, we show that restless
circuit execution with leaky gates reduces by 33% the sensitivity of the ORBIT
cost function developed by J. Kelly et al. which is typically used in
closed-loop optimal control~[Phys. Rev. Lett. 112, 240504 (2014)]. Our results
are obtained with standard qubit state discrimination showing that restless
circuit execution is resilient against misclassified non-computational states.
In summary, the restless method is sufficiently robust against leakage in both
standard and closed-loop optimal control gate calibration to provided accurate
results.
