Robustness analysis: (a) The randomized benchmarking analysis reveals the decay of the average fidelity in dependence of the number of computational gates l for a set of SAGQG (orange) and a set of dynamic quantum gates (blue). The average probability of error per gate are ε SAGQG g 

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... probability over the set of universal gates employing randomized benchmarking 35 . Based on the application of randomly assembled sequences of a set of logical gates, randomized benchmarking allows for a good estimation of the error scaling given a long sequence of quantum gates, as relevant for viable applications in longer quantum algorithms. Fig. 3a presents the average fidelity as a function of the number of computational gates l. For the SAGQG we obtain an average probability of error per gate of ε SAGQG g = 0.0013(3), whereas an identical analysis for a set of dynamic quantum gates represented by π and π/2-pulses reveals an average probability of error of ε dynamic g = ...

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... parameter specifications. The latter is particularly relevant for the common experimental case where the Rabi frequency (for practical reasons) obeys a maximum bound max t (Ω S (t, Ω 0 , ∆ 0 )) ≤ Ω max (for parameter dependences see Methods section). Given such a practical maximum bound Ω max for the experimentally achievable Rabi frequency, in Fig. 3b we show a contour plot of the numerically determined ...

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... of the SAGQG we explicitly vary the gate time parameter τ within a non-optimal range of τ reaching from 0.5 · t π to 1.5 · t π (whereas the theoretical min Ω 0 ,∆ 0 (τ min ) = t π ) for three sets of parameters A, B, and C (Ω 0 = {1.5, 1.5, 2} MHz and ∆ 0 = {1.5, 6, 8} MHz). The min Ω 0 ,∆ 0 (τ min ) value for each parameter set is marked in Fig. 3c as a vertical, dashed line of matching colors, respectively. Fig. 3c shows the extracted quantum gate fidelity F of the Pauli-X gate in dependence of τ. We observe that even for τ smaller than the calculated threshold τ min (indicated by vertical dashed lines) the quantum gate fidelity F remains close to one. Only for τ < t π ≈ 71 ns ...

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... a non-optimal range of τ reaching from 0.5 · t π to 1.5 · t π (whereas the theoretical min Ω 0 ,∆ 0 (τ min ) = t π ) for three sets of parameters A, B, and C (Ω 0 = {1.5, 1.5, 2} MHz and ∆ 0 = {1.5, 6, 8} MHz). The min Ω 0 ,∆ 0 (τ min ) value for each parameter set is marked in Fig. 3c as a vertical, dashed line of matching colors, respectively. Fig. 3c shows the extracted quantum gate fidelity F of the Pauli-X gate in dependence of τ. We observe that even for τ smaller than the calculated threshold τ min (indicated by vertical dashed lines) the quantum gate fidelity F remains close to one. Only for τ < t π ≈ 71 ns is the fidelity dropping. These results proof the tolerance of the ...

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... the following we examine the fidelity performance of the SAGQG with respect to variations in the gate evolution time. This is important for two reasons: 1) In order to most efficiently exploit the coherence time of the qubit, we need to investigate the theoretical velocity limits and experimental performance of the SAGQG and aim for fast quantum gate performance. 2) We experimentally examine the fault-tolerance of the SAGQG with respect to experimental parameter imperfections. In particular we analyse the SAGQG performance outside its optimal parameter specifications. The latter is particularly relevant for the common experimental case where the Rabi frequency (for practical reasons) obeys a maximum bound max t (Ω S (t, Ω 0 , ∆ 0 )) ≤ Ω max (for parameter dependences see Methods section). Given such a practical maximum bound Ω max for the experimentally achievable Rabi frequency, in Fig. 3b we show a contour plot of the numerically determined ...

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... an experimental robustness analysis of the SAGQG we explicitly vary the gate time parameter τ within a non-optimal range of τ reaching from 0.5 · t π to 1.5 · t π (whereas the theoretical min Ω 0 ,∆ 0 (τ min ) = t π ) for three sets of parameters A, B, and C (Ω 0 = {1.5, 1.5, 2} MHz and ∆ 0 = {1.5, 6, 8} MHz). The min Ω 0 ,∆ 0 (τ min ) value for each parameter set is marked in Fig. 3c as a vertical, dashed line of matching colors, respectively. Fig. 3c shows the extracted quantum gate fidelity F of the Pauli-X gate in dependence of τ. We observe that even for τ smaller than the calculated threshold τ min (indicated by vertical dashed lines) the quantum gate fidelity F remains close to one. Only for τ < t π ≈ 71 ns is the fidelity dropping. These results proof the tolerance of the SAGQG to perform stably over a large range of timing parameter variations and give evidence for the intrinsic robustness of the SAGQG against timing imprecision and concomitant mismatches in the driving field ...

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... an experimental robustness analysis of the SAGQG we explicitly vary the gate time parameter τ within a non-optimal range of τ reaching from 0.5 · t π to 1.5 · t π (whereas the theoretical min Ω 0 ,∆ 0 (τ min ) = t π ) for three sets of parameters A, B, and C (Ω 0 = {1.5, 1.5, 2} MHz and ∆ 0 = {1.5, 6, 8} MHz). The min Ω 0 ,∆ 0 (τ min ) value for each parameter set is marked in Fig. 3c as a vertical, dashed line of matching colors, respectively. Fig. 3c shows the extracted quantum gate fidelity F of the Pauli-X gate in dependence of τ. We observe that even for τ smaller than the calculated threshold τ min (indicated by vertical dashed lines) the quantum gate fidelity F remains close to one. Only for τ < t π ≈ 71 ns is the fidelity dropping. These results proof the tolerance of the SAGQG to perform stably over a large range of timing parameter variations and give evidence for the intrinsic robustness of the SAGQG against timing imprecision and concomitant mismatches in the driving field ...

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... of the performance of the superadiabatic geometric gates is obtained via standard quantum process tomography (QPT) 31 measurements, which allows to reconstruct the full experimental quantum process matrix χ exp and therefore to determine the quantum gate fidelity F = Tr χ exp χ 0 32 , where χ 0 is the theoretically anticipated process matrix (for details on the experimental QPT procedure see Supplementary Information and reference 10 ). Due to their dynamic nature and finite time duration the QPT pulses are susceptible to errors and we obtain the corrected quantum gate fidelity value˜Fvalue˜ value˜F = F/F ID by normalization with the fidelity of the identity operation. We determine the experimental gate fidelities of the SAGQG to be˜Fbe˜ be˜F SAGQG Besides the fidelity of the individual, logical gates, we additionally assess the average error probability over the set of universal gates employing randomized benchmarking 35 . Based on the application of randomly assembled sequences of a set of logical gates, randomized benchmarking allows for a good estimation of the error scaling given a long sequence of quantum gates, as relevant for viable applications in longer quantum algorithms. Fig. 3a presents the average fidelity as a function of the number of computational gates l. For the SAGQG we obtain an average probability of error per gate of ε SAGQG g = 0.0013(3), whereas an identical analysis for a set of dynamic quantum gates represented by π and π/2-pulses reveals an average probability of error of ε dynamic g = 0.023(8), i.e. the geometric-phase based SAGQG performs one order of magnitude better than its dynamic-phase based standard gate (see Supplementary Information for details). Our results suggest that the SAGQG is significantly more resilient with respect to the type of noise and parameter imperfections present in our experimental system than the standard realization of dynamic phase-based quantum gates. This experimental finding strongly supports the long time conjectured robustness 3 of geometric phase-based quantum gates, which owes to the fact that geometric phases and holonomies are global features which are intrinsically robust with respect to locally occurring parameter imperfections and noise that leave the state-space area enclosed by the states trajectory on the respective projective space ...