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Fault tolerant quantum computing requires quantum gates with high fidelity.Decoherence reduces the fidelities of quantum gates when the operation time istoo long. Optimal control techniques can be used to decrease the operation timein theory, but generally do not take into account the realistic nature ofuncertainty regarding the system parameters. We apply robust optimal controltechniques to demonstrate that it is feasible to reduce the operation time ofthe cross-resonance gate in superconducting systems to under 100\,ns withtwo-qubit gate fidelities of $\mathcal{F}>0.99$, where the gate fidelity willnot be coherence limited. This is while ensuring robustness for up to 10\%uncertainty in the system, and having chosen a parameterization that aides inexperimental feasibility. We find that the highest fidelity gates can beachieved in the shortest time for the multi-level qubits compared with atwo-level qubit system. This suggests that non-computational levels may beuseful for achieving shorter cross-resonance gate times with high fidelity. Theresults further indicate a minimal control time for experimentally feasiblepulses with the inclusion of robustness and the maximum amount of uncertaintyallowable to achieve fidelities with $\mathcal{F}>0.999$.

Minimal Time Robust Two Qubit Gates in Circuit QED