Path toward manufacturable superconducting qubits with relaxation times exceeding 0.1 ms

As the superconducting qubit platform matures towards ever-larger scales in the race towards a practical quantum computer, limitations due to qubit inhomogeneity through lack of process control become apparent. To benefit from the advanced process control in industry-scale CMOS fabrication facilities, different processing methods will be required. In particular, the double-angle evaporation and lift-off techniques used for current, state-of-the-art superconducting qubits are generally incompatible with modern-day manufacturable processes. Here, we demonstrate a fully CMOS compatible qubit fabrication method, and show results from overlap Josephson junction devices with long coherence and relaxation times, on par with the state-of-the-art. We experimentally verify that Argon milling—the critical step during junction fabrication—and a subtractive-etch process nevertheless result in qubits with average qubit energy relaxation times T1 reaching 70 µs, with maximum values exceeding 100 µs. Furthermore, we show that our results are still limited by surface losses and not, crucially, by junction losses. The presented fabrication process, therefore, heralds an important milestone towards a manufacturable 300 mm CMOS process for high-coherence superconducting qubits and has the potential to advance the scaling of superconducting device architectures.


INTRODUCTION
Superconducting circuits have emerged as a leading candidate for realizing a scalable quantum computing platform. The improvements in qubit coherence times 1 and gate fidelities 2-5 enabled the demonstration of quantum simulators 6,7 , small-scale quantum algorithms 8,9 , and even the elusive demonstration of quantum supremacy 10,11 . State-of-the-art coherence times routinely reach 50 µs, with specific cases exceeding 100 µs [12][13][14][15] . These have been exclusively fabricated using aluminum (Al) double-angle evaporation and lift-off techniques on sapphire or high-resistivity silicon (Si) substrates. Double-angle evaporated junctions enabled the fabrication of Noisy Intermediate Scale Quantum (NISQ) processors with an intermediate scale number of qubits 10,11 , only recently exceeding 100 16 . To further advance the technological state-of-the-art, the design and device fabrication inevitably become increasingly complex 17,18 . This imposes tighter constraints on the design parameters of qubits, readout resonators, on-chip filters 19,20 , and tunable couplers 21,22 . To facilitate this upscaling, versatile, reliable, and reproducible fabrication processes are needed.
The large-scale implementation of superconducting qubits is inherently hindered by the variability in Josephson energy of the double-angle evaporated junctions 23,24 . This is primarily a consequence of the variability in fabricated junction area, which results from the angle dependence across the wafer during metal deposition [24][25][26] . Another limitation is that polymer masks that are typically used during processing restrict the thermal budget. This further limits the choice of superconductor, limiting the potential optimization space for qubit improvement 27,28 . Furthermore, the fabrication itself requires dedicated evaporation tools with tilt capability. Double-angle evaporation with a required lift-off step introduces resist contaminations 29 and reduces reproducibility in larger diameter wafers 25 . This fabrication technique is therefore considered incompatible with advanced complementary metaloxide semiconductor (CMOS) manufacturing, where large-scale integration of devices is instead generally based on subtractive etch, sputtering deposition, and advanced optical lithography 30 . To overcome these limitations, alternative junction fabrication techniques are being actively investigated.
Several alternatives to double-angle evaporated junctions exist, including finMET 31 , trilayer 30,32 , and overlap [33][34][35][36][37] Josephson junctions. These are compatible with manufacturable CMOS processes. The advantages of overlap junctions compared to other alternatives are the reduced structural complexity and, therefore, a smaller number of fabrication steps. This reduces the processing-induced sources of loss and parameter variability. Recent work has shown promising overlap junction qubit performance, with energy relaxation T 1 times up to 80 µs 33,36,38 . However, these qubits were still created using a CMOSincompatible lift-off process. Qubits with state-of-the-art coherence times, fabricated with a fully manufacturable CMOS process, i.e., without the use of lift-off techniques, sapphire, or high resistivity (above~10 kΩ·cm) Si substrates, are yet to be demonstrated.
Overlap junctions have two electrodes that are defined in two patterning cycles, with a vacuum break in between which results in the uncontrolled growth of native metal oxide. After the first cycle, the native oxide needs to be removed by in-situ argon (Ar) milling to enable controlled subsequent junction oxidation. This step is critical since Ar milling has been linked to superconducting device performance degradation 15,39,40 . Strong Ar milling can lead to amorphized Si 41 and sapphire 29 substrate layers, which were found to limit resonator quality factors and qubit coherence times. The milling has also been reported to introduce additional loss at metal-metal interfaces 15 . This indicates that the Ar milling could compromise the junction's integrity and composition. However, the impact of Ar milling on the junction performance is still unexplored, and at present, it is unknown if it inherently limits the overlap junction qubit lifetimes.
In our work, we perform an in-depth characterization of Al overlap junctions, fabricated with wet and dry subtractive-etch processes. Intrinsic Si substrate coupons with 3 kΩ·cm (3k) and 20 kΩ·cm (20k) specific resistivity are used from 300 and 100 mm wafers, respectively. Results for fixed frequency transmon qubits with overlap junctions show time-averaged energy relaxation T 1 times of 50-70 µs, with individual measurements surpassing 100 µs, and average Pauli gate fidelity of 99.94%. By comparing quality factors (Q = 2πT 1 f) of qubits with different capacitor geometries, we find that the total loss (1/Q) scales with the metal-air surface oxide participation ratio, suggesting that the qubits are limited by the surface losses residing at the capacitor pads instead of the junction. The dominant loss likely originates from the subtractive-etch-specific sidewall residues and native surface oxides. Our study reveals that the overlap junctions do not limit the T 1 coherence time up to at least 131 µs for qubits at 3 GHz.

Overlap junction fabrication and room-temperature characterization
The process used to fabricate transmon qubits with overlap Josephson junctions is based on previously published work 33,38 and is described in detail in the Methods section and Supplementary Fig. 1. A 70 nm Al film is sputtered on a hydrofluoric (HF) acid-cleaned, high-resistivity Si substrate. Qubit capacitor pads, readout resonators, ground plane, and overlap junction bottom electrode (BE) are defined simultaneously using either a dry or wet subtractive etch. Fabricating these structures in a single layer minimizes the number of interfaces. Next, native oxides from the entire surface including BE are removed using an optimized Ar milling step inside the deposition tool. This step impacts the BE-air, Si-air, and junction interfaces. Following the native oxide removal, a controlled dynamic oxidation is performed to create the junction barrier. Next, a 50 nm thick top Al layer is deposited. In the final step, the top electrode (TE) is patterned with a subtractive dry etch. The targeted overlap junction areas are in the range of 0.03-0.07 µm 2 . In addition to the overlap junction, the TE forms a large stray junction (~10 × 20 µm) used as galvanic contact to the qubit's capacitor pad (Fig. 1a: blue interface). A scanning tunneling electron microscope (STEM) image of a fabricated overlap junction is shown in Fig. 1b. Devices fabricated with the described process exhibit record-high qubit energy relaxation times for overlap Josephson junction qubits, exceeding 100 µs (Fig. 1c). Furthermore, randomized benchmarking 42 yields average single qubit gate fidelities of F 1q = (99.940 ± 0.003)%, which are very close to the coherence limit of F 1q,inc = 99.96% (see Methods section and Fig. 1d). The uncertainty on the values represents the standard deviation (SD) on the fit.
We perform both wet and dry subtractive etching to define the BE. Wet etch processes are selective to Si and leave the substrate's top-surface intact. It can be argued that they yield better qubit coherence times 14 and resonator quality factors 43 compared to dry-etched devices, as the latter can be affected by an increase in Si substrate roughness and sidewall residues 14 . Nevertheless, the recent demonstration of record-high relaxation times of dryetched qubits 12 indicates that these problems can be mitigated. Dry etch processes also provide better junction dimension control and reproducibility, making it the preferred technique in state-ofthe-art CMOS processing 30 .
To characterize fabrication performance, resistances of test Josephson junction arrays with different junction areas are measured at room-temperature. Arrays consist of either 30 or 60 individual test junctions and junction areas range between 0.03 and 0.125 µm 2 , as determined from SEM images. The optimized fabrication process results in 99.8% junction yield for 3k dry etch samples. The resistance's relative standard deviations (RSD) range between 2-5% ( Supplementary Fig. 2), which is comparable to the state-of-the-art double-angle evaporated Josephson junctions [23][24][25]44 . The reported RSD is notably lower than the RSD estimated for previously reported overlap junctions 33 . Based on RSD analysis 24 , we conclude that the junction parameter variation is dominated by the variation in junction area ( Supplementary  Fig. 2).

Overlap Josephson junction qubit performance
High-coherence overlap junction qubits are characterized at 10 mK in a conventional dilution-refrigerator setup (see Methods Section and Supplementary Fig. 3). Double-pad transmon qubits (TM) show typical T 1 = 59 µs, Ramsey decoherence time of T 2 * = 63 µs and spin-echo decoherence time of T 2 e = 59 µs (Fig. 2a). We extract a mean of T 1 = 58 µs, with observed values reaching up to 100 µs, from repeated measurements over a span of 11 h for a dry-etched TM qubit fabricated on a 3k Si substrate (Figs. 1c and 2b). The notable T 1 variation has been previously attributed to the presence of fluctuating TLS defects 45,46 and quasiparticle tunneling through the Josephson junction 47 (Supplementary Fig. 6). We note that our reported record values surpass previously reported relaxation times for overlap junction qubits 33,36,38 .
The impact of etching processes and substrate resistances on the qubit lifetimes is studied next. We collect T 1 statistics from devices fabricated with different process combinations: wet etch on 20k wafers (wet-20k), dry etch on 20k wafers (dry-20k) and dry etch on 3k wafers (dry-3k). The transmon qubits consistently achieve average T 1 lifetimes between 50-70 µs. The violin plot (Fig. 2c) represents kernel distribution estimates from measured T 1 times for each device. We observe no significant difference between the etch processes and wafer resistivities. This agrees with SEM images, where no visual difference is observed between junctions fabricated by the wet and dry etch processes ( Supplementary Fig. 4). The T 1 independence on the substrate resistivity also agrees with our estimation of negligible microwave losses from similar 3k substrates (Q i~8 M) of previous resonator measurements 48 , setting an upper limit of the qubit T 1 > 400 µs at 3 GHz. Within the measurement uncertainty, the qubit lifetimes, therefore, appear not to be compromised by the dry etch manufacturable process on 3k wafer substrates.
To further investigate the sources of loss in overlap junction qubits, we design and fabricate two additional Xmon type qubits 49 (XM1, XM2) with reduced gaps between the center electrode pad and the ground plane ( Supplementary Fig. 5). By engineering the geometry and gap size, we control the participation ratios (fraction of the device's total electric field energy stored in the volume of lossy dielectric materials) of surface interfaces such as the substrate-air (SA) and metal-air (MA) interface [49][50][51] . For each design, we estimate participation ratios with electrostatic simulations (see Methods section and Supplementary Fig. 5). The Josephson junction's dimensions (Supplementary Table 1), and its associated interface participation ratios are similar and uncorrelated with the qubit design. This allows us to distinguish between losses originating from the junction and its vicinity (junction losses) and from the qubit capacitor pads (capacitor losses).
For a proper comparison of qubit lifetimes of devices at different transition frequencies, Q-factors, calculated from mean qubit relaxation times (corrected for Purcell decay 52 ) and their transition frequency (Q = 2πT 1 f) 46,53,54 , are compared to the MA participation ratio of different qubit designs. The MA participation ratio was chosen since its interface was found to dominate microwave loss 51 and generally scales with other interface participation ratios (Supplementary Table 2). We find that the reciprocal of the qubit's Q-factors (the qubit's total loss) scales with the qubit MA participation ratio, as indicated by the linear fit of the qubit data in Fig. 2d. The linear fit agrees with the on-chip  Fig. 2d). This confirms that qubits and resonators share a common dominant loss source, associated with surface interfaces. For qubits, the dominant loss, therefore, originates from the interfaces at capacitor pads.

coplanar waveguide resonator (RES) datapoints (open symbols in
From the y-axis intercept of the linear fit, we can extract a hypothetical residual loss of 1/Q = (0.41 ± 0.17) × 10 −6 , corresponding to a postulated, junction-limited T 1 % 131 þ115 À41 μs for a qubit transition frequency of 3 GHz. The uncertainty is estimated from the 95% confidence bound of 1/Q. The residual loss is associated with the junction and constitutes a combination of losses coming from the Josephson junction and interfaces at its vicinity 41,52,55 , the stray junction 56 , or quasiparticle tunneling 47 . While quasiparticle tunneling signatures have been observed in T 1 measurements ( Supplementary Fig. 6), we cannot quantitatively determine separate contributions from the listed loss sources to the T 1 limit.

Interface characterization
We attempt to correlate the origin of the qubit losses and the possible effect of Ar milling by examining interface morphology and atomic compositions with high-resolution cross-sectional transmission electron microscopy (TEM) and energy-dispersive Xray spectroscopy (EDS), respectively. The characterization results depicted in Fig. 3 are of a dry-etched qubit device (see Methods section for details).
The fabricated Josephson junction barriers have a typical thickness of 2-3 nm (Fig. 3a), as is also commonly observed for angle-evaporated junctions 57 . No structural damage is seen on top of the BE due to Ar milling (Fig. 3a, b). However, traces of Ar can be detected at the barrier, which extends~10 nm into the BE (Fig. 3e). Similarly, Si atom traces can be found at the barrier, likely originating from the resputtered Si during the Ar milling (Fig. 3d). An atomic concentration of less than 4% for both elements is observed inside the junction (Supplementary Fig. 7). Impurities can lead to two-level system (TLS) defect losses in the junction 55,56,58 . In our experiment however, they cannot be distinguished from other residual junction losses. High-resolution TEM images of overlap junctions fabricated with dry and wet etch processes show no discernible difference ( Supplementary Fig. 4).
At the Si-TE interface, an amorphous Si layer of~5 nm is visible (Fig. 1a: green interface, Fig. 3f). This is caused by Ar milling, which has been shown to damage the Si substrate 41 . This is also in agreement with traces of Ar found in this layer (Fig. 3j). Damaged Si layer has been reported to host TLS defects, which lead to microwave losses 41 . At the Si-air interface however, no amorphous Si layer can be observed (Fig. 1a: yellow interface, Fig. 3g, l) and no Ar can be detected (Fig. 3j, o). We conclude that the amorphous Si is consumed by the TE etch, except below the TE. The potentially lossy amorphous Si layer constitutes only a small fraction of the total qubit footprint ( Supplementary Fig. 5) and likely contributes to the residual junction losses. Near the TE sidewalls, we observed crust residues ( Fig. 3g and Supplementary  Fig. 4) that consist of oxygen (O) (Fig. 3h) and Si atoms (Fig. 3i). This crust arises from Si deposition on the sidewall and top of the resist during subtractive dry etching 59 (Fig. 1a: red interface). Similar to the amorphous Si layer, this crust can lead to residual microwave losses, however, only in the area surrounding the TE.
A less than 1 nm thin substrate-BE interface layer (Fig. 1a: orange interface, Fig. 3k) shows barely detectable traces of O impurities (Fig. 3m). This indicates that the substrate surface passivation with HF prior to the BE metal deposition successfully prevents oxide growth. The BE shows a distinct metal-air interface profile (Fig. 1a: brown interface), which we attribute to a combination of Ar milling, TE etching and residual BE crusts in case of BE dry etching. Residues of the BE crust and TE Al may be found at the sidewalls due to the anisotropic nature of the Ar milling and TE dry etch (Fig. 3l-n). The absence of Ar signal on the BE-air interface (Fig. 3o) indicates that the TE etch removes the amorphized layer from both the BE-air and Si-air interface.

DISCUSSION
As shown above, qubit devices are limited by the losses associated with the qubit capacitor pads. These losses can originate from crust residues, Al sidewall residues, and native oxides near the BE structures. Since coherence times of wet and dry-etched qubits do not significantly differ (Fig. 2c), we can exclude the BE crust remnants from the dominant loss sources. Other contributions cannot be confidently separated at this stage. However, advanced surface treatment steps can be devised to reduce its effects in future implementations. Residual losses associated with the Josephson junction encompass loss from (i) quasiparticle tunneling, (ii) the stray junction, (iii) crusts and native oxides in the vicinity of the junction, (iv) the amorphous Si interface underneath the TE, and (v) the junction barrier. Quasiparticle tunneling losses can be alleviated by proper filtering of infrared 60 and high-energy radiation 61 or including quasiparticle trapping sites on chip 62 . Stray junction loss can be avoided by using a bandage process to ensure a proper galvanic contact between the TE and capacitor pad 15,24,41,63 . Similarly, as discussed for capacitor losses, crust and native oxides in the vicinity of the junction could be further reduced by developing dedicated surface cleaning processes. On the other hand, the amorphous Si and junction impurities are loss sources specific to Ar milling and can therefore not be easily eliminated. The effect of amorphous Si below the TE could be reduced by minimizing the TE surface area. Potential losses from junction contaminations can only be alleviated by further optimizing the Ar milling step. However, within our measurement uncertainty, their effect currently cannot be distinguished from other losses. To further investigate the junction losses, different qubit designs, such as 3D transmons 50 , merged-element transmons 55 , or finMETs 31 could be explored.
We have fabricated and measured record-high coherence times for qubits based on overlap junctions, with mean T 1 values comparable to qubits fabricated with double-angle evaporated junctions. We demonstrated a qubit fabrication process that is fully compatible with CMOS fabrication requirements and tools. By studying different qubit designs, we infer that qubit limiting losses are located at the capacitor surface interfaces and that our overlap junctions are not limiting the qubit relaxation times up to at least 131 þ115 À41 μs. Further advancements in qubit lifetimes could be achieved by developing specialized surface treatments. The presented qubit fabrication process paves the way for reproducible and well-controlled qubit integration in 300 mm wafer fabrication facilities, expediting the upscaling of quantum computers and other superconducting Josephson junction devices.

METHODS Overlap Josephson junction fabrication process
The substrate coupons (3 × 3 cm) for the qubit fabrication process are diced from two types of intrinsic Si wafers: 300 mm Si wafer with resistivity 3000 Ω.cm and 100 mm Si wafer with resistivity >20,000 Ω.cm labeled 3k and 20k, respectively. After a close coupled wet etch of the native Si oxide with hydrofluoric acid, a 70 nm of Al (bottom electrode-BE layer) is deposited by magnetron rf sputtering (Pfeiffer Spider 630) at a rate of 0.5 nm/s. Exposure of cleaned silicon surface to air did not exceed 2 min. No atomic impurities from the Ar processing gas could be detected in the EDS measurement of Al films (Fig. 3e, j, o). A 300 mm industry-grade ebeam tool is used to define structures in the PMMA positive resist layer (PMMA AR-P 630-670). The pattern is transferred to the BE layer either by dry or wet etch. For the room-temperature wet etch process we use OPD5262 etchant based on 2% TMAH. Dry etching is done with inductively coupled plasma (ICP) using a SiCl 4 /He chemistry (Oxford PlasmaLabSys-tem100). The dry etch process also forms a shallow recess into the Si substrate. During dry etching in the presence of PMMA resist, a Si-rich sidewall crust forms on the BE and resist layer sidewalls. A post-etch residue remover EKC265 is used to simultaneously reduce the presence of resist and minimize sidewall crusts. After resist strip, the native Al oxide on top of the BE is removed by a coupon-wide Ar milling step in the load lock of the sputtering system. Ar milling at rf powers of 100 W yields an Al etch rate of~1 nm/min. The duration of the Ar mill step is determined as the time needed to obtain a short contact between the TE and BE in a standard JJ fabrication flow without the controlled oxidation step. The Josephson junction (JJ) barrier layer is formed by controlled dynamic oxidation at constant pressure. Without vacuum break, 50 nm of Al is deposited to form the top electrode. The subsequent TE is patterned using dry etching. Process steps for the TE patterning are similar to those for the BE patterning with the exception of using a negative resist (ma-N 2400 series) to minimize the e-beam writing time. The full process flow is visualized in Supplementary Fig. 1. A list of all fabricated samples in this study is found in Supplementary Table 1.

Material characterization
TEM, STEM, and EDS measurements were performed at the Materials Characterization and Analysis center at the IMEC microelectronics research institute. Before characterization, the samples were coated with a spin-on carbon (SOC) layer. Lamellae with a thickness of <50 nm were cut with a focused ion beam (FIB) using Helios 450. TEM, STEM, and EDS measurements were performed on Titan Cubed Themis 300 STEM with a 200 kV source. For this study, TEM, EDS, atomic-resolution HAADF-STEM, and ABF-STEM were used to investigate interfaces between the substratetop electrode, substrate-bottom electrode, and AlO x barrier between the top and bottom Al electrodes.

Qubit measurement setup
Overlap qubits are measured with a standard dilution-refrigerator setup ( Supplementary Fig. 3). Input lines are thermalized with 20 dB attenuators at three different temperature stages. High-frequency noise above the measurement frequency range is filtered before reaching the qubit with a low-pass filter (VLF-8400 + ) with cutoff frequency of 8.4 GHz. Each superconducting qubit chip is placed and wire bonded to the nonmagnetic gold-plated copper PCB enclosed in an O-free copper sample holder. The sample holder is thermalized to the mixing chamber plate in a dilution refrigerator. The sample holder is surrounded by the copper radiation shield as well as two cryo-perm shields to minimize magnetic fields at the sample. Output signal lines are thermalized with three isolators (LNF-ISC4_8A) with a total reverse isolation of~60 dB and a 4-8 GHz band-pass filter (KBF-4/8-2S). The signal is amplified with a HEMT amplifier (LNF-LNC4_8C) at the 4 K stage and ultra-low noise amplifier (LNA-30-04000800-07-10P) at room temperature.
At room temperature, pulsed signals are generated and acquired using the Keysight Quantum Engineering Toolbox: M3202A AWGs and M3102A Digitizer. The qubit excitation and readout pulse are combined at roomtemperature and applied to the qubit's feedline. No dedicated charge line or flux line were used to excite or bias qubits.
Three qubit designs were used in this study, two Xmon (XM1, XM2) 49 styles and a two-pad transmon (TM) 14 qubit. Qubit frequencies were targeted in the range of 3-4 GHz and readout resonator frequencies are in the range of 5-6 GHz. Qubit anharmonicities are between 200 and 250 MHz.

Numerical simulations of the participation ratios
The electric field losses are dependent on and are proportional to the participation ratios of the dielectric regions in the devices. The participation ratio p i in a dielectric i is defined as the fraction of electrical energy contained in the dielectric with respect to the total electrical energy in the device, as follows 64 : where ∈ i and V i correspond to the dielectric constant and volume of the i th dielectric, respectively, and |E V | correspond to the absolute value of the electric field in the dielectric in an incremental volume dV. To calculate the participation ratio's p i , we first estimate the electric fields by solving the Poisson's equation in the devices using a commercial TCAD electrostatic simulator 65 , and then subsequently incorporate the fields into Eq. (1). We invoke the following approximations for calculating the electric fields in TCAD. First, we omit the Josephson junction region in the electrostatic model, as we are primarily interested in the losses in the capacitor region of the superconducting qubits. Second, with the geometry of the capacitor being largely symmetric along certain axes (see Supplementary Fig. 5a), we perform a two-dimensional electrostatic simulation, rather than a full 3-dimensional calculation. This further offers the advantage of reduced computational time and larger numerical accuracies for the simulations with an extremely refined mesh at interfaces in the device. Third, we only consider metal-air (MA) and substrate-air (SA) oxide interfaces in the simulations, while omitting the metal-substrate (MS) oxide interface. This is also reasonable considering that there is no substantial oxide formation between the metal and substrate, as indicated in Fig. 3c, m. Furthermore, uncertain parameters are the exact thickness of the oxide at the different interfaces and their dielectric constant. From TEM images, we estimate an average thickness of the SA and MA interfaces of 4 and 5 nm, respectively. The geometry of the structures used in the simulations is very similar to that of the experimental devices, and we also employ commonly chosen values for the dielectric constants, i.e., 11.9, 3.9, and 5.0 for the silicon substrate, silicon dioxide, and metal-air oxide interfaces, respectively. Based on the above technique, approximations, and parameters, we estimate the participation ratios in the capacitor regions for the different designs (resonator, Xmons, and transmon), which are shown in Supplementary Table 2. These obtained participation ratios have further been used to illustrate the scaling trend of 1/Q for the four devices in Fig. 2d. The linear scaling with the participation ratio calculations in Fig. 2d indeed confirms that the major limiting region for the qubit or resonator lifetimes (T 1 ) is the capacitor.

Randomized benchmarking: measurement and analysis
The average physical gate error r g and gate fidelity F 1q have been measured for device C2 (Fig. 1d). A random sequence of Clifford gates of varying lengths n is applied to a qubit initialized in the ground state. At the end of the sequence, an inverting gate is added to create an overall identity operation and the final qubit state is measured to determine the fidelity. The measurement is repeated 80 times for each sequence length. Cosine pulses with a duration of 50 ns are used in the experiment. The averaged sequence fidelity is fitted to F = Ap n + B, where the parameters A and B depend on the state preparation and measurement errors 42 . From p, we determine r g as the error per Clifford r Clifford normalized by the average number of physical Clifford generator gates (1.875): where d = 2 m is the dimensionality of the Hilbert space, which is equal to 2 for a single qubit. The average physical gate fidelity is obtained from: The coherence limit on the measured fidelity (F 1q,inc ) is estimated with qubit parameters: T 1 = 50 µs and T 2 * = 60 µs 66 .