Strongly anharmonic flux-tunable transmon based on InAs-Al 2D heterostructure

Domain: physics
Published: 2025-12-15
Editorial outcome: major_revision

Abstract
The gatemon qubits, made of transparent superconducting-semiconducting Josephson junctions, typically have even weaker anharmonicity than the opaque AlOx-junction transmons. However, flux-frustrated gatemons can acquire a much stronger anharmonicity, originating from the interference of the higher-order harmonics of the supercurrent. Here we investigate this effect of enhanced anharmonicity in split-junction gatemon devices based on InAs-Al 2D heterostructure. We find that anharmonicity in excess of 100% can be routinely achieved at the half-integer flux sweet-spot without any need for electrical gating or excessive sensitivity to the offset charge noise. We verified that such intrinsically large anharmonicity enables our devices to be driven coherently with raw Rabi frequencies exceeding 100 MHz, without any pulse shaping, simplifying implementation and control compared to traditional gatemons and transmons. Furthermore, by analyzing a relatively high-resolution spectroscopy of the device transitions as a function of flux, we were able to extract fine details of the current-phase relation, to which transport measurements would hardly be sensitive. The strong anharmonicity of our anharmonic tunable transmons, along with their bare-bones design, can prove to be a precious resource that transparent superconducting-semiconducting junctions bring to quantum information processing.

Introduction
Electrically-gated transmon qubits (gatemons) have been demonstrated as potentially useful devices for building large scale quantum processors 1–15. A typical gatemon device is made with a gate electrode close to a hybrid superconductor-semiconductor Josephson junction (superconducting-semiconducting JJ). The voltage on the gate electrode changes the electron density in the semiconductor weak link and thereby tunes the Josephson energy EJ of the qubit. The advantage of electrical gating is that it can be used for controlling the quantum dynamics by rapidly changing the qubit frequency, similarly to switching transistors in conventional semiconductor circuits. Just like transmons, however, gatemons suffer from a suppressed anharmonicity of the qubit transition, as they operate in a regime where the Josephson phase-difference is localized near the bottom of the Josephson potential well 16–20. Increasing the anharmonicity, which we define by

η = |f12 − f01| / f01, (1)

would require increasing the charging energy EC, which would exponentially increase the qubit’s sensitivity to the offset-charge noise. Furthermore, the relatively high transparency of superconducting-semiconducting junctions further reduces anharmonicity in comparison to opaque AlOx transmon junctions 21. The weak anharmonicity of gatemons would limit gatemon-based quantum processors in the same way it limits transmons, that is single-qubit and two-qubit gate operations will have to be done sufficiently slowly as to prevent leakage outside the computational space 22,23.

Replacing the gatemon junction with a split junction and applying a flux bias through the loop can dramatically increase the qubit anharmonicity in comparison to an equivalent split-junction transmon device 2,24. This effect originates from a destructive interference of the first Josephson harmonics and a constructive interference of the typically much weaker second Josephson harmonics, bringing their contribution on par 25–31. As a result, the Josephson potential energy, while remaining a 2π-periodic function, acquires a more complex shape involving local minima, which can lead to much more anharmonic spectra. By contrast, higher harmonics are suppressed in conventional tunnel junctions, and hence flux bias merely rescales the regular Josephson potential, with weak effect on the anharmonicity 32.

So far, higher harmonics of the supercurrent has been mainly explored in the context of parity-protected qubits, which would emerge when the first harmonic is completely canceled. Achieving such cancellation, however, requires precise control of the junction parameters via electrical gating, which, in practice, has proven challenging 24,25,28. In this paper, we describe a more immediate impact of the supercurrent interference in two transparent junctions by focusing on the half-flux quantum sweet spot, where the anharmonicity is already dramatically enhanced but the sensitivity to the 1/f flux noise is zero to first order. Such a device presents an opportunity to improve on the main drawback of transmons - the weak anharmonicity – using the unique property of mesoscopic superconducting-semiconducting junctions.

Results
We investigate the anharmonic flux-tunable transmon qubits on an InAs quantum well (QW) proximitized by epitaxial Aluminum (epi-Al) 33,34 at the Φ = 0.5Φ0 flux sweet spot. For simplicity we model the weak link JJs with their effective number of channel N and a characteristic channel transparency T. The SQUID loop formed by two weak link JJs is shunted by a capacitance C in between the superconducting island and the ground. The qubit potential is tuned by the applied magnetic flux Φ in the SQUID loop 35. We denote the phase across the first JJ as φ and the phase of the second JJ is 2πΦ/Φ0 − φ. As seen in Fig. 1a, the superconducting island is subject to voltage fluctuations modeled by a capacitively coupled electrode with an offset voltage Vg, which is equivalent to an offset charge of ng = CgVg/2e 36. The Hamiltonian of the qubit is given by the sum of the charging energy and the Josephson energy U(φ):

H = 4EC(n − ng)2 + UJ(φ), (2)

where n and φ are the number of the Cooper pairs and the phase operators, satisfying the commutator [n, φ] = i. The Josephson energy has a simple structure for the tunnel junctions, UJ(φ) = EJ(1 − cosφ). In a transmon with conventional AlOx JJs, the same scale determines the barrier height, the curvature and quartic terms at the potential minimum, resulting in competition between transmon sensitivity to charge noise and anharmonicity.

The current transport in a JJ with a semiconducting weak link is enabled by Andreev Bounds States (ABSs) formed by conduction channels in the weak link 37. When the JJ has a superconducting phase bias φ, the energy of an ABS is given by

U(φ, T) = Δ sqrt(1 − T sin2(φ/2)), (3)

where Δ is the superconducting gap and T is the conduction channel transparency. Owning to a finite lateral width in our JJs, multiple transverse channels contribute to the supercurrent across each JJ based on the lateral confinement. As such, the SQUID energy of a flux-tunable gatemon is determined by the sum over the contribution from all the ABSs in both the JJs:

USQUID(φ) = sum_{k=1}^M U(φ, T1(k)) + sum_{l=1}^N U(2πΦ/Φ0 − φ, T2(l)), (4)

where T1(1), ..., T1(M) are the channel transparencies in the first JJ with M total conduction channels and T2(1), ..., T2(N) are the channel transparencies in the second JJ with N total conduction channels. This model yields a good fit to the spectra of flux-tunable gatemon qubits with InAs nanowire JJs when the number of channels in the JJs is small 26,27,38. When the number of conduction channel is large, Eq. (4) can be approximated by grouping the channel transparencies into one or a few characteristic transparencies or by determining the leading Fourier harmonics of the Josephson potential energies. In this study, we also compare the approximation methods mentioned above for modeling anharmonic flux-tunable transmon qubits with InAs-based 2DEG weak-link JJs by examining the quality of their fits to the measured qubit spectra.

The flux-tunable gatemon qubit biased at Φ = 0 is equivalent to the typical gatemon qubit with similar qubit potential to a transmon qubit, as shown in Fig. 1b. When biased at Φ = 0.5Φ0, the bottom of the potential changes to a flattened double-well shape near the minima of the well, drastically increasing the qubit anharmonicity. The double-well shape of the qubit potential arises from the destructive interference of the cosφ terms and constructive interference of the cos(2φ) terms in the two JJs’ potentials. The qubit states see a high potential barrier that suppresses the tunneling probability to adjacent wells. Meanwhile, the flux-tunable gatemon potential has reduced curvature of the potential near its minima so the qubit has a large anharmonicity up to a few hundred percent. As seen in Fig. 1c, the simulated flux-tunable gatemon |0> and |1> states are insensitive to the offset charge, but the qubit is strongly anharmonic.

Although the trade-off between the charge dispersion and the anharmonicity still applies to the anharmonic tunable transmon qubit, the charge dispersion can be further suppressed by engineering the asymmetry of the SQUID. As shown in Fig. 1d, as the qubit anharmonicity increases, the charge dispersion linewidth increases by one to two orders of magnitude from a conventional transmon with <5% anharmonicity. The loss in the suppression of charge dispersion can be compensated with a more asymmetric design of the SQUID, which is equivalent to increasing the first Josephson harmonic in the SQUID current-phase relation (CPR). The charge dispersion sees four orders of magnitude suppression going from Ic1/Ic2 = 0.7 to Ic1/Ic2 = 0.4, while the anharmonicity >100% remains achievable. The ability to simultaneously achieve suppressed charge dispersion and large anharmonicity for flux-tunable gatemon devices at the Φ = 0.5Φ0 flux sweet spot promises better qubit performance than the conventional transmon counterpart.

An equivalent circuit diagram and false color scanning electron microscopy images of the investigated anharmonic flux-tunable transmon device are shown in Fig. 2a-e. We measure three nominally identical devices, all of which show similar spectra. Our typical device consists of a λ/4 resonator capacitively coupled to a common transmission line for microwave control and readout. A T-shaped Al island etched into the surrounding ground plane on an InP substrate provides the qubit shunt capacitance, which has an estimated charging energy EC ∼ 180 MHz from electrostatic simulations. The qubit, comprising of two weak link JJs in a SQUID loop, is connected to the T-shaped island and the ground plane via Al leads. Each individual JJ is formed by etching away a L ∼ 250 nm long and W ∼ 800 nm wide segment of the epi-Al. The qubit is readout using the λ/4 resonator with a resonance frequency of fr = 6.01 GHz and Q-factor ∼ 400, and measured in a dilution refrigerator at < 50 mK using standard complex reflectance measurements, as shown in detail schematically in supplementary information 1.

We begin by measuring the normalized phase response of the readout resonator as a function of the magnetic flux Φ, while sweeping the resonator drive frequency fr. The resonator response for each device, indicated by the black dashed lines in Fig. 2f-h and shown in detail in supplementary information 2, exhibits a vacuum Rabi splitting with a cavity-qubit coupling strength of g = 122 MHz for device I, g = 101 MHz for device II and g = 170 MHz for device III, as shown in supplementary information 8. These values scale with the respective charge matrix elements and are indicative of qubit states hybridizing with the readout resonator mode. To further probe the qubit transitions directly, we perform two-tone spectroscopy, where the readout resonator drive tone is fixed at its resonant frequency for each flux, while a second drive tone fq is applied to excite the qubit transitions.

When biased at Φ = 0, each device in Fig. 2f-h shows a single qubit transition with transmon-like behavior, characterized by a decreasing |0> - |1> transition frequency f01 as Φ increases. In contrast, at Φ = 0.5Φ0, we observe a rich spectrum featuring multiple flux-dependent transitions that strongly anticross symmetrically around 0.5Φ0. The qubit anharmonicity shows a strong increase with increasing flux, changing from traditional gatemon and transmon-like values of η ≃ 2% at Φ = 0 to η ≃ 96% for device I, η ≃ 730% for device II, and η ≃ 125% for device III at Φ = 0.5Φ0 (see Table 1). In addition, the |0> - |1> transition shows a first-order insensitivity to flux noise at Φ = 0.5Φ0 (details in supplementary information 3), consistent with a flux ‘sweet spot’ at half-flux quanta.

In order to understand the observed spectrum, we simulate the measured qubit transitions by determining the eigenenergies of the qubit Hamiltonian in Eq. (2). The qubit transition frequencies extracted from Fig. 2f-h are shown in Fig. 3a-c with blue solid curves. To account for the large number of transparent conduction channels in superconducting-semiconducting JJs, we model the Josephson potential by a single characteristic channel transparency for each JJ. This approach is justified for atomically clean JJs, where channel-to-channel variations are expected to be minimal. However, in practice, only a fraction of the total channels (∼ 10% in our devices) is highly transmissive due to mesoscopic imperfections such as disorder and selective mode coupling 34,39–41. Under this assumption, the Josephson energy USQUID(φ) in Eq. (4) simplifies to USQUID(φ) = NU(φ, T1) + NU(2πΦ/Φ0 − φ, T2), where T1 and T2 are the channel transparencies of the first and second JJ with N1 and N2 conduction channels, respectively. The black dashed curves in Fig. 3a-c show the calculated qubit transition frequencies with the fit parameters presented in Table 1, consistent with the short-junction limit and quasi-ballistic regime estimated in supplementary information 11 for our devices. In addition, our model indicates a strong suppression of the charge matrix element, as shown in supplementary information 4, for the |0> - |1> qubit transition at 0.5Φ0. This suppression diminishes the dipole coupling of the qubit to charge noise and dielectric loss channels, thereby enhancing the T1 relaxation time at the half-flux sweet spot.

To quantitatively assess the accuracy of the single transparency model, we plot the normalized residual (fmodel − fmeas)/fmeas vs Φ for the lowest-energy |0> - |1> transition in Fig. 3d-f. As shown by the black filled squares, we observe a residual of < 2% at Φ = 0 and no more than 4% at Φ = 0.5Φ0 for device I and II, while device III remains below 5% for all Φ. For comparison, we consider a higher-harmonic model in which the Josephson potential energy for each JJ is approximated by its leading Fourier harmonic contributions: U = sum Ek cos(kφ), where i ∈ [1, 2] refers to the first and second JJ. The normalized residual, determined using the four leading harmonic terms k = 4 (see supplementary information 5 for k = 1 and k = 2) for each JJ, exhibits an oscillating behavior with residual ≤ 10% for device I and ≤ 5% for devices II and III (blue hollow squares). These results indicate that the single transparency model provides the most accurate representation of the flux-tunable gatemon spectra despite using fewer fitting parameters. We observe that further increasing the number of characteristic transparencies does not significantly reduce the residual, as demonstrated in supplementary information 6.

The extracted fit parameters allows us to determine the CPR of the individual JJs and the SQUID loop. For each JJ, we use the Josephson potential: Ii(φi) = Φ0/2π ∂/∂φi U(φi) with i ∈ [1, 2] for JJ1 and JJ2 respectively, to extract the normalized CPR I/Ic for devices I and II in Fig. 4a. We observe highly skewed non-sinusoidal CPRs, reminiscent of higher-order harmonic contributions due to the interference of supercurrents carried by 2-electrons, 4-electrons and higher charges across the JJs. Consequently, for the SQUID loop CPR at Φ = 0.5Φ0, we sum over the individual JJ contributions to obtain:

Isq(φ) = I1(φ) + I2(2πΦ/Φ0 − φ) (5)

Isq/Ic,sq extracted for devices I and II is shown in Fig. 4b. We further extract the higher-order harmonic contributions for each JJ’s CPR using: I(φ) = sum An sin(nφ). Figure 4c shows the normalized nth-order harmonic term |An/A1| for devices I and II, determined up to the leading sixteen terms (n = 16) with an amplitude baseline threshold set at |An/A1| = 10−6. We observe a gradual fall-off in the harmonic amplitudes with increasing n. In addition, while some JJs exhibit similar CPRs, others show substantial deviations, resulting in distinct harmonic content. This diversity reflects the sensitivity of the CPR to microscopic transport parameters of the JJs. The harmonic analysis provides complementary information to standard transport measurements, allowing one to identify deviations from purely sinusoidal behavior and engineer the Josephson potential landscape including the realization of multi-well potentials, thereby tailoring qubit properties for novel circuit architectures and optimized performance.

We now demonstrate the basic operations of the flux-tunable gatemon qubit by performing Rabi oscillation measurements and Ramsey interferometry using time-domain manipulation and readout. For each measurement, microwave pulses with frequency f ≈ f01 are applied via the cavity readout feedline, after an initialization time of 100 μs to allow the qubit to relax to the ground state |0>. For Rabi measurements, we apply the drive pulse for a time duration τRabi, followed by reading out the qubit state via the cavity. The microwave pulse induces Rabi oscillations between the qubit states |0> and |1>, as shown in Fig. 5a, where we plot the probability of qubit state |1>, P|1> as a function of τRabi and the applied drive power P. We observe that the Rabi frequency increases for larger drive power P. To explicitly demonstrate this, we extract the Rabi frequency fRabi for each power P by fitting the Rabi oscillations in Fig. 5a with a sinusoidal function with exponentially decaying envelope. The Rabi frequency shows a linear dependence in Fig. 5b, with Rabi frequencies as large as > 100 MHz enabling coherent manipulation and gate operations of the flux-tunable gatemon qubit much faster than traditional transmons and gatemons.

We first measure the energy relaxation time T1 to estimate the qubit coherence times. The Rabi oscillations data in Fig. 5a allow us to calibrate the pulse amplitudes and durations for the corresponding rotations around the x-axis on the Bloch sphere. To measure T1, we first apply a Rπ pulse to excite the qubit to state |1>, followed by a waiting time delay τ before readout. The qubit probability P|1>, plotted in Fig. 5c, as a function of the time delay τ shows an exponential decay due to the qubit relaxation, yielding T1 ∼ 230 ns for device I. Similarly, to measure Ramsey dephasing time T*, two Rπ/2 pulses slightly detuned from the qubit frequency, δf ∼ 40 MHz and separated by a time delay τRamsey are applied before readout. The resulting cavity readout response as a function of the time delay τRamsey, shown in Fig. 5d, demonstrates Ramsey fringes, consistent with the qubit state acquiring a phase φ = 2πδfτRamsey while precessing around the z-axis of the Bloch sphere. We obtain T* ≃ 39 ns by fitting to a sinusoidal function with an exponential decay envelope.

Methods
The anharmonic tunable transmon qubit device is fabricated on a stacked heterostructure grown on 500 μm thick Fe-doped InP substrate. An In0.53Al0.47/In0.52Al0.48 superlattice and a graded InxAl1−xAs layer are further grown on the substrate to act as a buffer layer, followed by 4 nm of InGaAs, 4 nm of InAs and 10 nm of InGaAs, which defines the quantum well structure with InAs as the 2DEG layer. The wafer is then capped with a 25 − 30 nm thick layer of Al deposited in-situ with molecular beam epitaxy.

The coplanar waveguide (CPW) resonators are fabricated on a 6 mm by 5 mm rectangular wafer with a center conductor width of 35 μm and a trench gap width of 20 μm using e-beam lithography. The capping Al layer is then removed with 50 °C Transene Aluminum Etchant Type D, followed by a wet-etch of the III-V layers at room-temperature for 3-4 min, resulting in a trench depth of 350 − 450 nm. The III-V layer etchant is a mixture of phosphoric acid (85%), hydrogen peroxide (30%) and DI water with a volumetric ratio of 1:2:60 respectively. The qubit SQUID, fabricated in the same step as the CPW resonators, has a dimension of 50 μm by 50 μm for devices 1 and 2, and 22 μm by 22 μm for device 3. We then etch the Al for 6 seconds in a separate e-beam lithography step to form the Josephson junctions. The device is finally packaged in a copper cavity via Al bond wires and measured in a dilution refrigeration at < 50 mK using standard reflectance measurements.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This paper presents a detailed study of flux-tunable hybrid Josephson junction qubits, demonstrating strong anharmonicity without the need for electrostatic gating. Using spectroscopic techniques, the authors extract the current-phase relation (CPR) of the hybrid junctions and showcase coherent qubit control at the half-flux sweet spot, including the observation of fast Rabi oscillations. The findings on CPR are particularly relevant to the superconducting-semiconducting hybrid community. While the pronounced anharmonicity is a key advantage for qubit control and gate fidelity, its potential impact is tempered by relatively limited coherence times, which may constrain the practical utility of these qubits.\nI find that there is a significant issue with a main claim of the paper. The authors state that the gate time they have achieved using their qubit (≤10 ns) is significantly faster than existing transmons, which is not true. The typical anharmonicity of a transmon is ~250 MHz, which sets the speed limit for when leakage to non-computational states will occur. While it is true that Rabi rates above 100 MHz in AlOx-based transmons will exhibit leakage, multiple groups have used the DRAG protocol to make sub-10 ns gates in standard AlOx-based transmons. I cannot support publication of this paper until the authors clarify their claim regarding their gate time relative to existing transmons.\nThe following comments should also be addressed in their manuscript:\n1. Since the authors do not use any electrostatic gating in their hybrid superconductor-semiconductor Josephson junctions, the term “gateless gatemon” appears somewhat redundant. Referring to the device simply as a flux-tunable transmon with hybrid junctions more accurately and clearly reflects the nature of the work.\n2. In Figure 1(d), the measured anharmonicity for devices 1 and 2 is marked with dots. However, given the already dense presentation of the graph, these indicators are difficult to understand as data and may be misinterpreted as markers.\n3. In Figure 2(h), a transition is visible above the 0–1 transition and below the 0–2 transition, but the authors do not provide any explanation for this feature. Clarifying its origin would be helpful.\n4. The authors model the Andreev bound state (ABS) energy using the short junction approximation. Given the junction length of 250 nm, which is relatively long, have the authors considered alternative models beyond this approximation? As highlighted in Fatemi et al. (SciPost Phys. 18, 091), such junctions may deviate significantly from the short junction regime.\n5. The authors should also provide additional motivation for why they expect all of the channels in their junction to have the same transparency.\n6. The authors could provide a discussion on the importance of the higher-order harmonic contributions to the current. I assume a numerical derivative is performed on the potentials that they found to get the current-phase relation. The extracted current-phase relation is skewed so the authors then expand it in a sine series and find the higher-order harmonic contributions. It would be useful to understand how these higher-order contributions could provide information about the junction or guidance for future designs that was not previously available.\n7. Have the authors performed T2E (echo coherence time) versus flux measurements? Such data would provide further insight into dephasing mechanisms and flux noise sensitivity in these devices.\n8. The conclusion mentions suppressed charge matrix elements as a key result, but this point is only addressed in the supplementary material. Given its importance, incorporating a brief discussion in the main text would strengthen the narrative and highlight its relevance to qubit coherence. It would also be useful to plot the expected T1 times vs flux for the quality factor quoted in the text in the same plot as their measured T1 times.", "reviewer_2": "The manuscript reports on a implementation of gatemon qubits using split-junction devices based on InAs/Al 2D heterostructures. The key advance is the demonstration of significantly enhanced anharmonicity (>100%) at the half-integer flux sweet spot, enabling fast and potentially more robust gate operations without requiring gate voltages. The authors also use high-resolution spectroscopy to extract detailed features of the current-phase relation, offering insight beyond standard transport techniques.\nThe results are timely and relevant to the community working on hybrid superconducting-semiconducting qubits and may present a significant step toward leveraging transparent junctions in scalable quantum hardware. However, there are several points that should be addressed before the manuscript can be considered for publication.\nComments:\n- The phrase \"anharmonicity in excess of 100%\" in the abstract is confusing. Anharmonicity is defined formally in Eq. (4), but the term is used repeatedly before that point in the manuscript. The authors should clarify this early in the text, preferably when the concept is first introduced. For instance, is 100% referring to the qubit frequency itself, or relative to the fundamental harmonic level spacing?\n- In Fig. 1(d), the mapping between plotted lines and the left/right y-axes is unclear. The caption and figure labeling should be revised to indicate clearly which data correspond to which axis. As it stands, this figure is difficult to interpret.\n- The transition from Eq. (2) to Eq. (3) introduces the concept of multiple channels in the Josephson junction, but the text does not explain this development adequately. Please expand this section to clarify the physical picture, assumptions, and implications of including multiple conduction channels.\n- The coupling strength gg is introduced in connection with the spectroscopy data, but its precise definition is missing. This is crucial, as multiple conventions exist in the literature. For example, whether the Hamiltonian term is written as hbar g(adagger sigma- + a sigma+) or with hbar g/2, or hg etc which all leads to numerical differences. Please state the exact convention used.\n- Furthermore, how is g extracted? Is it taken directly from the vacuum Rabi splitting at resonance, or obtained via a full fit of the qubit-resonator avoided crossing as a function of flux? For devices with broad linewidths, a full fit generally provides a more reliable estimate, and this should be clarified.\n- The extracted values of g differ significantly between the three devices, despite nominally identical qubit designs and similar resonator frequencies. The authors should comment on this variation. Is it due to fabrication variation, junction asymmetry, or some other source? If it is due to the different matrix elements from the various junction transparencies, then it should be mentioned clearly.\n- There appears to be a missing T2 in the inline equation before Table 1, which makes the expression incomplete. Please correct this.\n- The manuscript claims suppression of the charge matrix element near the sweet spot. However, this does not seem to be supported by experimental evidence in the current work. In the supplemental material, a theoretical calculation is shown, but the flatness of T1-times across the sweet spot implies no observable suppression. Do the authors see any change in Rabi frequency that might indicate such suppression? If so, they should quantify this, but note that such a measurement would require a careful calibration of the entire drive line including PCB and device response. Without this, the claim should be moderated.\nThe work presents interesting and potentially impactful experimental results that merit publication in Nature Communications after the above issues have been addressed. I recommend revision to clarify the points aobve.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "The authors of the manuscript “Liu et al., Strongly-anharmonic gateless gatemon qubits based on InAs/Al 2D heterostructure,” study the properties of transmon devices in which the qubit Josephson junction has been replaced by an asymmetric DC SQUID with semiconductor Josephson junctions (Sm-JJs). The Sm-JJs are characterized by a current-phase relation (CPR) that contains higher harmonics in phase, φ. When the SQUID flux is fixed at half of a flux quantum (Φ0/2), the supercurrent interference partially cancels the cos ⁡φ-component of the SQUID energy-phase relation. This results in a modified transmon spectrum with increased anharmonicity while, at the same time, retaining an acceptable charge dispersion linewidth. The authors report measurements of the qubit dispersion as a function of SQUID flux on three different devices. Next, they fit the transmon spectra, replacing the Josephson energy term by an ansatz for the flux/phase-dependent energy of a SQUID with two junctions. They compare two different approaches. The first one uses a parametrization of the junction energy-phase relation that assumes N identical channels of transparency Ti and gap Δ in the short channel limit (following Beenakker, Ref 37). The other model constitutes of a series of cos ⁡φ-harmonics up to eighth order. Both models yield fits of roughly similar accuracy.\nThe authors proceed to plot the SQUID CPR and the CPRs of the individual junctions that correspond to the extracted model parameter. Finally, they report the results of the qubit characterization for SQUID flux Φ0/2 , extracting the characteristic decoherence times T1 and T2 *.\nThe work addresses one of the current big topics in the field of solid-state quantum information processing: the hunt for qubit systems with long coherence times and short gate operation times. The superconducting transmon qubit has the advantage of reduced charge sensitivity, but its small spectral anharmonicity increases the time-scale of gate manipulations. In this work, the authors make use of higher harmonics in the potential of a SQUID with Sm-JJs to engineer a qubit spectrum with larger anharmonicity. Whereas the idea of using higher harmonics is not entirely new, the authors present a neat demonstration of the concept that will be of interest to the wider community.\nI read the manuscript with great interest but think that the presentation and analysis lack depth. Additionally, important information is missing, which precludes the reader from reaching their own conclusions. In its current state, I cannot recommend the paper for publication in Nature Communications.\nLet me elaborate:\nThe data in Tab. 1 are not discussed.\nThe choice of parametrization of the junction potential U(φ,T), implies that the supercurrent transport can be modeled as identical channels in the short junction limit. Taking a look at the results of the fit (Tab. 1), I am wondering if this makes sense. I am particularly intrigued by the transparency values, T∼0.928-0.999. These are incredibly large values for semiconductor Josephson junctions, for which one expects a sizable mismatch between the Fermi surfaces of semiconductor and metal, consequently interface scattering and lower transparencies.\n- Have the authors critically accessed the interpretation of highly transmitting channels?\n- Has the interface quality been studied in similar JJs? (References?)\n- Are the Sm-JJs indeed in the short junction limit? I.e., L≪ξ≈ℏvF/2Δ? (L≡ electrode separation; vF≡ Fermi velocity in the semiconductor)\nI didn’t find the value of Δ in the text. Has it been measured or inferred from the critical temperature of the Al layer?\n- Do the values Ni match expectations based on lateral size quantization?\n- It is mentioned that all Sm-JJs are nominally identical. Why is there are large spread in the parameter values of Tab. 1? What are the values of the critical current for each junction?\n- What is the mean free path in the semiconductor material? Can the authors rule out the diffusive transport limit?\nI recommend the authors to provide this information in the revised manuscript and include references to available literature.\nThere are several aspects of the CPR discussion (page 12f and Fig.4) that I find strange:\nIn Fig. 4, the normalized CPRs of JJ2 of Device I and II match. They are calculated based on the fit parameters in Table 1. However, judging by the transparency values given there, I would expect JJ1 of Device II to match with JJ2 of Device I. Are the labels in Fig. 4 wrong, or am I missing something in the analysis? Is there a way to know which one of the two physical junctions has the parameters of JJ1 and JJ2 in this experiment? Or are the labels interchangeable?\nWhat do we learn from Fig. 4c? The authors provide a plot of the first 16 amplitudes of the Fourier series expansion of eq. 2 for the parameters Ti in Tab. 1. In other words, an (effectively) one-parameter expression (as Ni and Δ drop out by normalization) is recast into another expression with 16 parameters. It is stated that the magnitudes of the amplitudes strongly decrease with increasing index n. This is true but easily understood when we look at the Fourier series of a sawtooth waveform, |An|∝1/n, which represents the extreme limit of a skewed CPR. I don’t see the value of this analysis. On the other hand, in Fig. 3, the authors compare the single-transparency fit with a model for USQUID composed of eight cos⁡ φ-harmonics. How do the coefficients of the Fourier series expansion (An) in Fig. 4c compare to the eight fitting parameters of the second model? Can we learn something about the junction physics or the fitting procedure by comparing the numbers?\nI disagree with the statement “More notably, the differences in the CPR and harmonic contributions between individual JJs are minimal, enabling us to extract fine details in the current-phase relations, far surpassing the capabilities of any standard transport measurement.”\nThe first part is wrong. It is observed that the CPRs of two junctions are identical. The other two are substantially different. Therefore their harmonic content is different. The second part is not discussed in the manuscript, and I highly doubt it is true. Excluding other systematic errors for now, the accuracy of the CPR extraction depends on the choice of a suitable parametrization and the quality of the fit to the experimental data. The authors have omitted a good deal of error analysis here. What are the confidence intervals for the parameters in Tab. 1? How large are the error bars in Fig. 4a,b? In Fig. 4c? Can the authors provide examples by detailed comparison with transport experiments in the literature to support their statement?\nFinally, the authors should provide an analysis of systematic errors in their experiment. E.g., generally, the phase drop across the SQUID loop inductance must be taken into account when determining the CPR. If the inductance is large, the shape of the CPR has to be determined self-consistently. Have the authors calculated/estimated the loop inductance (geometric and kinetic components) and ascertained that it is safe to omit it from their analysis? How would it affect the precision of their CPR extraction and the fit values for transmission in Tab. 1?\nAdditional comments:\nThere is a reference missing on page 16, bottom line.\nI, personally, find expressions like “gateless” gatemon or “super-semi” Josephson junctions confusing and distracting. The oxymoron in the title made me skip the paper when it first appeared on the arXiv. I presume other researchers feel similarly about colloquialisms.\nThe authors state that their transmon qubit allows for faster gate operations than traditional transmon and gatemons. As someone who works in the periphery of the quantum information field, I would greatly appreciate if the authors gave more context and examples. It would benefit the general readership."}, "author_response": "Our response:\nWe thank the referee for accurately summarizing our work.\nReferee #1\nI find that there is a significant issue with a main claim of the paper. The authors state that the gate time they have achieved using their qubit (≤10 ns) is significantly faster than existing transmons, which is not true. The typical anharmonicity of a transmon is ~250 MHz, which sets the speed limit for when leakage to non-computational states will occur. While it is true that Rabi rates above 100 MHz in AlOx-based transmons will exhibit leakage, multiple groups have used the DRAG protocol to make sub-10 ns gates in standard AlOx-based transmons. I cannot support publication of this paper until the authors clarify their claim regarding their gate time relative to existing transmons.\nOur response:\nWe thank the referee for this helpful comment. We agree that the conventional transmons can achieve sub-10 ns single-qubit gates through precise calibration with the DRAG protocol [Motzoi et al., PRL 103, 110501 (2009); Lucero et al., Phys. Rev. A 82, 042339 (2010)]. We have revised our statement to avoid claiming a unique speed advantage, but rather emphasize that our devices exhibit large intrinsic anharmonicity which enables >100 MHz raw Rabi frequency without the need of any specialized DRAG-type pulse shaping and sequences. Such high Rabi frequency facilitates strong drives for fast gate operations, thereby simplifying control and calibration with fewer resources, while intrinsically reducing leakage.\nTo address the referee’s concern, we include the following changes (marked in red, in the revised manuscript):\n1. We now state that our devices enable “fast gates with raw Rabi frequencies exceeding 100 MHz, without any pulse shaping, enabled by the large intrinsic anharmonicity of our devices.”\n2. We explicitly note that fast gates are achievable in both our devices and standard AlOx-based transmons, our architecture naturally reduces leakage through spectrum engineering, thereby simplifying experimental implementation and control.\nReferee #1\nThe following comments should also be addressed in their manuscript:\n1. Since the authors do not use any electrostatic gating in their hybrid superconductor-semiconductor Josephson junctions, the term “gateless gatemon” appears somewhat redundant. Referring to the device simply as a flux-tunable transmon with hybrid junctions more accurately and clearly reflects the nature of the work.\nOur response:\nWe thank the referee for this comment. We have changed the term “gateless gatemon” to “flux-tunable transmon” in the revised manuscript.\nReferee #1\n2. In Figure 1(d), the measured anharmonicity for devices 1 and 2 is marked with dots. However, given the already dense presentation of the graph, these indicators are dieicult to understand as data and may be misinterpreted as markers.\nOur response:\nWe thank the referee for the suggestion to improve figure 1d. In the revised figure, we have made the correspondence between the plotted data and the left/right y-axis clearer. In particular, we have added a new legend and improved axis color-coding to unambiguously indicate which data belongs to a particular axis. The key device parameters, i.e. anharmonicity and charge dispersion linewidths which are suppressed below 1 Hz, are now directly reported in the figure caption for better comprehensibility.\nReferee #1\n3. In Figure 2(h), a transition is visible above the 0–1 transition and below the 0–2 transition, but the authors do not provide any explanation for this feature. Clarifying its origin would be helpful.\nOur response:\nWe thank the referee for pointing this out. The observed transition corresponds to a cavity mode, which arises from the resonant frequency of the sample holder cavity. This cavity mode strongly couples to all modes in our sample, inducing a measurable response in the qubit’s readout through dispersive coupling. We note here that this mode is specific to the cavity used for device 3, whereas devices 1 and 2 were measured in a dieerent, although similarly designed, cavity.\nReferee #1\n4. The authors model the Andreev bound state (ABS) energy using the short junction approximation. Given the junction length of 250 nm, which is relatively long, have the authors considered alternative models beyond this approximation? As highlighted in Fatemi et al. (SciPost Phys. 18, 091), such junctions may deviate significantly from the short junction regime.\nOur response:\nWe thank the referee for this important question, as the validity of the short-junction approximation is indeed central to our modeling of the Andreev bound states (ABS) and the current-phase relation (CPR). In a superconductor-semiconductor-superconductor (S-Sm-S) junction, the short-junction limit is defined by:\n𝐿 ≪ 𝜉 = ℏ𝑣𝐹/2∆\nwhere 𝐿 is the JJ length, 𝜉 is the superconducting coherence length, 𝑣𝐹 is the Fermi velocity in the semiconductor and ∆ is the induced superconducting gap. Using ∆ ≈ 230 𝜇eV determined from the Al critical temperature of 𝑇𝑐 ~ 1.5K [Yuan et al., J. Vac. Sci. Technol. A 39, 033407 (2021)] and 𝑣𝐹 ~ 10^6m/s corresponding to an electron density of 𝑛 ≈ 7.68 X 10^11cm-2, we estimate 𝜉~1 𝜇m. For our junction length 𝐿 = 250 nm, this gives 𝐿⁄𝜉~ 0.25, placing our devices in the crossover of the short and intermediate regime.\nTo explicitly determine the validity of our approximation, we compare the extracted CPRs to expectations from a finite-length scattering formalism as discussed in Fatemi et al. (SciPost Phys. 18, 091). In the long-junction limit, the CPR evolves toward a more linear, sawtooth-like form near Φ = 𝜋 with suppressed higher harmonics. In contrast, our CPR consistently displays strong forward skewness, indicative of high transparency channels due to substantial higher-harmonic content – clear hallmarks of the short-junction regime. To further investigate this, we also employ an empirical harmonic expansion (up to 4 harmonics per JJ) to extract a model-independent CPR alongside the short-junction ansatz, where we find that both approaches show excellent agreement to experimental data, confirming that the essential skewness and anharmonicity are robust. This behavior is consistent with prior works on accurately capturing CPR features in highly transparent, ballistic InAs-Al JJs [e.g., Mayer et al., Nat. Comm. 11, 212 (2020); Goeman et al., New J. Phys. 19, 092002 (2017)], and our simplified single-transparency model yields excellent fits across a wide range of flux and frequency.\nWhile more elaborate finite-length models could, in principle, be used, such models require detailed knowledge of the JJ potential and disorder profile, information inaccessible from our two-tone qubit spectroscopy alone and would introduce many unconstrained degrees of freedom with minimal additional constraint from our data. Therefore, the lack of any experimental signatures of long-junction behavior and the excellent agreement of our fits, we conclude that the short-junction approximation is justified and sueicient to describe the dominant physics of our devices.\nReferee #1\n5. The authors should also provide additional motivation for why they expect all of the channels in their junction to have the same transparency.\nOur response:\nWe thank the referee for this insightful question. In our short-junction single transparency model, we initially considered a distribution of channel transparencies by dividing them into bins of width 0.2 (e.g. 0-0.2, 0.2-0.4 etc.). For each bin, we then assigned a characteristic channel transparency, which is treated as a fit parameter. In practice, the fits consistently converge the population into a single high-transparency bin for each JJ, e.g. around 0.99 for the 0.8-1 bin. This motivated us to adopt a simplified single-characteristic transparency for each JJ. Such behavior is reasonable for our InAs-Al 2DEG junctions, where molecular beam epitaxy (MBE) growth yields highly uniform, atomically clean interfaces. The resultant suppression in the interface roughness and inhomogeneous disorder reduces channel-to-channel variations in the transmission probabilities, consistent with prior studies in similar systems [e.g., Mayer et al., Nat. Comm. 11, 212 (2020); Goeman et al., New J. Phys. 19, 092002 (2017)], which have reported junctions with a few dominant highly transparent channels.\nTo further critically assess our single-transparency assumption, we implemented a two-transparency model by introducing an additional characteristic transparency and its corresponding channel population as additional fit parameters for each JJ. The resulting fits show two closely spaced transparencies and only a marginal improvement compared to the single-characteristic transparency case, as shown by the normalized residual plots in supplementary figure 6. This suggests that an average single-transparency approximation already captures the dominant features of the qubit dispersion and extracted CPR. While it is likely that small variations may exist among individual channels, they are not resolvable within the sensitivity of our current spectroscopy and do not meaningfully improve the fits.\nReferee #1\n6. The authors could provide a discussion on the importance of the higher-order harmonic contributions to the current. I assume a numerical derivative is performed on the potentials that they found to get the current-phase relation. The extracted current-phase relation is skewed so the authors then expand it in a sine series and find the higher-order harmonic contributions. It would be useful to understand how these higher-order contributions could provide information about the junction or guidance for future designs that was not previously available.\nOur response:\nWe thank the referee for this valuable suggestion. Indeed, the higher-order harmonic content of the current-phase relation (CPR) provides rich information about the microscopic transport properties of the JJ, the shape of the Josephson potential and ultimately the qubit performance. We have included a discussion on these aspects in the revised manuscript.\nMore specifically, in transport measurements, the relative amplitude of the harmonic terms 𝐴𝑛 serve as direct evidence of the junction’s microscopic transport regime. A large 𝐴1 with negligible 𝐴𝑛 (𝑛>1) corresponds to a purely sinusoidal CPR, characteristic of tunnel junctions with low transparency and dieusive transport. In contrast, significant second and third harmonic terms reflect highly transparent, ballistic or quasi-ballistic modes, which skew the CPR forward. A slow decay of 𝐴𝑛 with increasing n indicates short-junction behavior with near-unity transmission channels. In our device, we find 𝐴2/𝐴1 ≈ 1/2 for the SQUID loop and 𝐴2/𝐴1 ≈ 1/6 for individual JJs, along with non-negligible higher harmonics (𝑛 ≥ 3), confirming the presence of a small number of highly transparent channels.\nIn addition, the Josephson potential landscape is determined by the relation:\n𝑈(ϕ) ∝ −∑𝐴𝑛/𝑛 𝑐𝑜𝑠(𝑛ϕ)\nHere, higher harmonics reshape the potential beyond the simple cosine form – dominant 𝐴𝑛 (𝑛>1) terms create skewed or multi-well potential landscape within a single 2𝜋 phase period, producing multiple local energy minima. This directly aeects the curvature of the potential and thus the qubit’s anharmonicity, charge dispersion and sensitivity to flux noise, as demonstrated in our work.\nFinally, for qubit design, controlling the harmonic content provides a versatile route to engineer novel qubit modalities. For example, 𝑐𝑜𝑠(2𝜙) qubits can be realized when the first harmonic is suppressed (𝐴1 = 0), achieved in symmetric SQUIDs [Larsen et al., PRL 125, 056801 (2020)] where the Josephson potential becomes 𝜋-periodic, i.e. a double well potential, with degenerate ground states, leading to an increase in 𝑇1 at the half-flux sweet spot. Similarly, “harmonium” qubits [Hays et al., arxiv:2502.15459 (2025)] exploit the regime 𝐴2 ≪ 𝐴1 with 𝐴2 < 0, leading to non-degenerate but noise resilient ground states.\nIn summary, the higher-order harmonics are not just mathematical artifacts but a powerful diagnostic and design tool that encode real information - they provide a direct probe of the microscopic transport regime, determine the Josephson potential landscape, and enable harmonic engineering for optimized qubit performance and novel qubit architectures.\nReferee #1\n7. Have the authors performed T2E (echo coherence time) versus flux measurements? Such data would provide further insight into dephasing mechanisms and flux noise sensitivity in these devices.\nOur response:\nWe thank the referee for pointing this out. Yes, we have performed T2E (Hahn echo coherence time) as a function of flux and qubit frequency around 0.5Φ0, which are shown below:\nWe observe that T2E is independent of the flux around 0.5Φ0, even though 𝜕𝑓𝑞⁄𝜕Φ increases away from the half-flux sweet spot, as shown in supplementary figure 3. This indicates that 1/f type flux noise is not the dominant dephasing mechanism in our qubit. Since 𝑇2 < 2𝑇1, we note that other dephasing mechanisms such as critical current fluctuations in the JJ, high frequency thermal photon noise might limit our coherence times.\nReferee #1\n8. The conclusion mentions suppressed charge matrix elements as a key result, but this point is only addressed in the supplementary material. Given its importance, incorporating a brief discussion in the main text would strengthen the narrative and highlight its relevance to qubit coherence. It would also be useful to plot the expected T1 times vs flux for the quality factor quoted in the text in the same plot as their measured T1 times.\nOur response:\nWe thank the referee for this constructive suggestion. We agree that the suppression of charge matrix elements is a central result of our work with direct implications for qubit relaxation. Specifically, we find that, in our model, the charged matrix elements are strongly reduced near the half-flux sweet spot. This suppression diminishes the dipole coupling of the qubit to charge noise and dielectric loss channels, thereby enhancing the relaxation time.\nTo explicitly demonstrate this, we compare our measured T1 values with the expected T1 values from a dielectric loss dominated model with\n1⁄𝑇1 ∝ 𝐸𝑐 |𝑛𝑔|2, tan𝛿\nwhere 𝐸𝑐 is the charging energy, 𝑛𝑔 is the charge matrix element and tan𝛿 = 1⁄𝑄 is the dielectric loss tangent experimentally extracted in supplementary figure 7b. The measured and expected T1 times coincide around the half-flux sweet spot, as shown below as a function of qubit frequency (also included in supplementary figure 7 in the revised manuscript). This good agreement with the dielectric loss model supports the suppression of charge matrix element at half flux and identifies dielectric loss as the dominant relaxation mechanism at 0.5Φ0.\nAway from the sweet spot, the increase in charge coupling results in lower measured T1 values than the dielectric-loss model values, suggesting that additional relaxation channels start contributing. In summary, we demonstrate that suppressed charge matrix elements play a protective role in qubit relaxation and presents an eeective design principle for engineering qubits with longer lifetimes.\nReferee #2 (Remarks to the Author):\nThe manuscript reports on a implementation of gatemon qubits using split-junction devices based on InAs/Al 2D heterostructures. The key advance is the demonstration of significantly enhanced anharmonicity (>100%) at the half-integer flux sweet spot, enabling fast and potentially more robust gate operations without requiring gate voltages. The authors also use high-resolution spectroscopy to extract detailed features of the current-phase relation, oeering insight beyond standard transport techniques.\nThe results are timely and relevant to the community working on hybrid superconducting-semiconducting qubits and may present a significant step toward leveraging transparent junctions in scalable quantum hardware.\nOur response:\nWe thank the referee for the positive assessment of the novelty and relevance of our results.\nReferee #2\nHowever, there are several points that should be addressed before the manuscript can be considered for publication.\nComments:\n- The phrase \"anharmonicity in excess of 100%\" in the abstract is confusing. Anharmonicity is defined formally in Eq. (4), but the term is used repeatedly before that point in the manuscript. The authors should clarify this early in the text, preferably when the concept is first introduced. For instance, is 100% referring to the qubit frequency itself, or relative to the fundamental harmonic level spacing?\nOur response:\nWe thank the referee for pointing this out. In the revised manuscript, we have introduced the definition of anharmonicity at the beginning of the manuscript (marked in red), where the concept is first introduced. In particular, we now explicitly state that an anharmonicity of 100% is defined relative to the qubit transition frequency 𝑓𝑞. For comparison, the qubit frequency in a standard AlOx transmon is given by 𝑓𝑞 ≈ 𝐸𝐽/ℎ−𝐸𝑐/ℎ, which typically amounts to an anharmonicity of only a few percent of 𝑓𝑞.\nReferee #2\n- In Fig. 1(d), the mapping between plotted lines and the left/right y-axes is unclear. The caption and figure labeling should be revised to indicate clearly which data correspond to which axis. As it stands, this figure is dieicult to interpret.\nOur response:\nWe thank the referee for the suggestion to improve figure 1d. In the revised figure, we have made the correspondence between the plotted data and the left/right y-axis clearer. In particular, we have added a new legend and improved axis color-coding to unambiguously indicate which data belongs to a particular axis. The key device parameters, i.e. anharmonicity and charge dispersion linewidths which are suppressed below 1 Hz, are now directly reported in the figure caption for better comprehensibility.\nReferee #2\n- The transition from Eq. (2) to Eq. (3) introduces the concept of multiple channels in the Josephson junction, but the text does not explain this development adequately. Please expand this section to clarify the physical picture, assumptions, and implications of including multiple conduction channels.\nOur response:\nWe thank the referee for pointing out the need for more explanation between Eq. (2) and Eq. (3). In the revised manuscript, we have included a sentence to clarify the inclusion of multiple conduction channels in each JJ. Our junctions have a finite lateral width of ~ 800 nm, which allows for multiple transverse modes/channels to contribute to the supercurrent across each JJ based on the lateral confinement. Each channel corresponds to an Andreev bound state (ABS) characterized by its transparency T. The total supercurrent is then the sum of the contribution from all these channels.\nHowever, only a subset of these channels is highly transparent and eeectively contributing to the supercurrent, thereby dominating Josephson transport across the junction (consistent with prior works in Mayer et al., Nat. Comm. 11, 212 (2020); Goeman et al., New J. Phys. 19, 092002 (2017)). In our single-transparency model, we categorize these dominant channels into an eeective single-transparency channel with N representing the corresponding number of such highly transparent modes (see response for point 5 of Referee #1 on page 3 for further details). This assumption captures the essential nuances of multichannel transport dominated by the highly transparent modes, while keeping the model simple and easily applicable to experimental data.\nReferee #2\n- The coupling strength gg is introduced in connection with the spectroscopy data, but its precise definition is missing. This is crucial, as multiple conventions exist in the literature. For example, whether the Hamiltonian term is written as hbar g(adagger sigma- + a sigma+) or with hbar g/2, or hg etc which all leads to numerical dieerences. Please state the exact convention used.\nOur response:\nWe thank the referee for highlighting the need for precise definition of the coupling strength g. In our work, we use the Jaynes-Cummings Hamiltonian denoted by:\n𝐻 = ℏω𝑟 𝑎†𝑎 + ℏω𝑞 σ+σ− +ℏ𝑔(𝑎†σ− +𝑎σ+)\nwhere 𝑎†,𝑎 are the photon creation and annihilation operators, 𝜎+,𝜎− are the Pauli and ladder operators for the qubit, 𝑔 is the vacuum coupling strength and 𝜔𝑟,𝜔𝑞 are the resonator and the qubit transition frequency respectively. The resultant shape of the avoided crossing is then given by:\n𝐸± = ℏ𝑛ω𝑟 ± 1/2√(4𝑛𝑔2+Δ2)\nwhere Δ = 𝜔𝑞 −𝜔𝑟 denotes the detuning of the qubit from the resonator frequency.\nReferee #2\n- Furthermore, how is g extracted? Is it taken directly from the vacuum Rabi splitting at resonance, or obtained via a full fit of the qubit-resonator avoided crossing as a function of flux? For devices with broad linewidths, a full fit generally provides a more reliable estimate, and this should be clarified.\nOur response:\nIn our devices, 𝑔 is extracted by performing a full fit of the above equation for the avoided crossing spectrum shape to the experimental data. We now show the fits for our 3 devices in supplementary figure 8. From the fits, we obtain 𝑔 = 122 MHz for device 1, 𝑔 = 101 MHz for device 2 and 𝑔 = 170 MHz for device 3, respectively.\nReferee #2\n- The extracted values of g dieer significantly between the three devices, despite nominally identical qubit designs and similar resonator frequencies. The authors should comment on this variation. Is it due to fabrication variation, junction asymmetry, or some other source? If it is due to the dieerent matrix elements from the various junction transparencies, then it should be mentioned clearly.\nOur response:\nWe thank the referee for highlighting the variations in the coupling strength g across devices. In general, the Josephson energy E depends on the JJ critical current 𝐼𝑐 (𝐸𝐽 = ℏ𝐼𝑐 /2𝑒), which is further determined by the junction dimensions and the eeective channel transparencies. Although the three devices are nominally identical in geometry and resonator frequency, mesoscopic variations in channel transparency due to microscopic variations in barrier thickness or disorder causes a substantial variation in 𝐼𝑐 among the dieerent devices. This changes the 𝐸𝐽/𝐸𝑐 ratio which directly aeects the charge matrix element n𝑔. Since the coupling strength is directly proportional to the charge matrix element, i.e. 𝑔 ∝ 𝑛𝑔, any variation in the channel transparency translates into an observed change in 𝑔.\nTo explicitly quantify our claim, we extract the critical current 𝐼𝑐 of the SQUID loop for device 1 and device 2 using the fit parameters in Table 1 of the main text. We obtain 𝐼𝑐 ~ 147 nA for device 1 and 𝐼𝑐 ~ 65 nA for device 2, resulting in 𝐼𝑐1⁄𝐼𝑐2 ≈ 2.26. In the transmon regime with 𝐸𝐽/𝐸𝑐 ≫ 1, the coupling strength 𝑔 is related to 𝐸𝐽/𝐸𝑐 [Koch et al., PRA 76, 042319 (2007)] by:\n𝑔 ∝ 𝑛𝑔 ≈ (𝐸𝐽/8𝐸𝑐)1/4\nwhich results in:\n𝑔1/𝑔2 ≈ (𝐼𝑐1/𝐼𝑐2)1/4\nFor 𝐼𝑐1⁄𝐼𝑐2 ≈ 2.26 in our work, we obtain 𝑔1/𝑔2 = 1.21, which is in agreement with the measured coupling strengths of 𝑔 = 122 MHz for device 1 and 𝑔 = 101 MHz for device 2, ultimately validating our argument that dieerent matrix elements due to variations in the JJ transparencies predominantly lead to the observed spread in 𝑔.\nReferee #2\n- There appears to be a missing T2 in the inline equation before Table 1, which makes the expression incomplete. Please correct this.\nOur response:\nWe thank the referee for pointing this out. We have included the missing T2 in the revised manuscript.\nReferee #2\n- The manuscript claims suppression of the charge matrix element near the sweet spot. However, this does not seem to be supported by experimental evidence in the current work. In the supplemental material, a theoretical calculation is shown, but the flatness of T1-times across the sweet spot implies no observable suppression. Do the authors see any change in Rabi frequency that might indicate such suppression? If so, they should quantify this, but note that such a measurement would require a careful calibration of the entire drive line including PCB and device response. Without this, the claim should be moderated.\nOur response:\nWe thank the referee for this very insightful comment. In our dielectric-loss dominated model, the 𝑇1 relaxation time is given by:\n1⁄𝑇1 ∝ 𝐸𝑐 |𝑛𝑔|2, tan𝛿\nwhere 𝐸𝑐 is the charging energy, 𝑛𝑔 is the charge matrix element and tan𝛿 = 1⁄𝑄 is the dielectric loss tangent. Typically, tan𝛿 does not have a strong frequency/flux dependence, therefore any variations of 𝑇1 with flux is primarily determined by the charge matrix element 𝑛𝑔.\nTo verify this relation, we compare the model-predicted charge matrix elements with values extracted from our measured 𝑇1 values. To eliminate the unknown prefactors, including the assumed constant loss tangent, we normalize both the theoretical and experimental values to their respective values at half-flux sweet spot. The resulting plot for 𝑛𝑔 ⁄𝑛𝑔,sweet vs flux is shown below:\nThe model predicts a suppression of 𝑛𝑔 at Φ = 0.5Φ0, corresponding to an enhancement in 𝑇1, and an increase in 𝑛𝑔 away from the sweet spot. In contrast, our measured 𝑇1 values exhibit significant fluctuations, which obscures this dependence. Nevertheless, the predicted variation of 𝑛𝑔 lies withing the envelope of the experimental fluctuations, signifying that the experimental data is consistent with – but do not directly resolve – the expected suppression.\nIn the revised manuscript, we now explicitly mention that the suppression of the charge matrix elements could not be experimentally resolved in our present data. As suggested by the referee, a fully calibrated Rabi frequency measurement, which accounts for entire the fridge drive lines and packaging, would be required for a quantitative validation.\nReferee #2\nThe work presents interesting and potentially impactful experimental results that merit publication in Nature Communications after the above issues have been addressed. I recommend revision to clarify the points above.\nOur response:\nWe thank the referee for their time and careful review of our work and finding our experimental results interesting and impactful.\nReviewer #3 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.\nOur response:\nWe thank the referee and their co-reviewer for their participation and time in carefully peer reviewing our manuscript.\nReferee #4 (Remarks to the Author):\nThe authors of the manuscript “Liu et al., Strongly-anharmonic gateless gatemon qubits based on InAs/Al 2D heterostructure,” study the properties of transmon devices in which the qubit Josephson junction has been replaced by an asymmetric DC SQUID with semiconductor Josephson junctions (Sm-JJs). The Sm-JJs are characterized by a current-phase relation (CPR) that contains higher harmonics in phase, φ. When the SQUID flux is fixed at half of a flux quantum (Φ0/2), the supercurrent interference partially cancels the cos ⁡φ-component of the SQUID energy-phase relation. This results in a modified transmon spectrum with increased anharmonicity while, at the same time, retaining an acceptable charge dispersion linewidth. The authors report measurements of the qubit dispersion as a function of SQUID flux on three dieerent devices. Next, they fit the transmon spectra, replacing the Josephson energy term by an ansatz for the flux/phase-dependent energy of a SQUID with two junctions. They compare two dieerent approaches. The first one uses a parametrization of the junction energy-phase relation that assumes N identical channels of transparency Ti and gap Δ in the short channel limit (following Beenakker, Ref 37). The other model constitutes of a series of cos ⁡φ-harmonics up to eighth order. Both models yield fits of roughly similar accuracy. The authors proceed to plot the SQUID CPR and the CPRs of the individual junctions that correspond to the extracted model parameter. Finally, they report the results of the qubit characterization for SQUID flux Φ0/2 , extracting the characteristic decoherence times T1 and T2*.\nOur response:\nWe thank referee for precisely summarizing our work.\nReferee #4\nThe work addresses one of the current big topics in the field of solid-state quantum information processing: the hunt for qubit systems with long coherence times and short gate operation times. The superconducting transmon qubit has the advantage of reduced charge sensitivity, but its small spectral anharmonicity increases the time-scale of gate manipulations. In this work, the authors make use of higher harmonics in the potential of a SQUID with Sm-JJs to engineer a qubit spectrum with larger anharmonicity. Whereas the idea of using higher harmonics is not entirely new, the authors present a neat demonstration of the concept that will be of interest to the wider community.\nOur response:\nWe thank the referee for the positive assessment on the quality of our results.\nReferee #4\nI read the manuscript with great interest but think that the presentation and analysis lack depth. Additionally, important information is missing, which precludes the reader from reaching their own conclusions. In its current state, I cannot recommend the paper for publication in Nature Communications.\nOur response:\nWe are very grateful to the referee for their time to review our work. We note that the listed points are important technical issues, the corrections of which have significantly improved our manuscript. We hope the new heavily presented information will convince the referee to recommend publication.\nReferee #4\nLet me elaborate:\nThe data in Tab. 1 are not discussed.\nThe choice of parametrization of the junction potential U(φ,T), implies that the supercurrent transport can be modeled as identical channels in the short junction limit. Taking a look at the results of the fit (Tab. 1), I am wondering if this makes sense. I am particularly intrigued by the transparency values, T∼0.928-0.999. These are incredibly large values for semiconductor Josephson junctions, for which one expects a sizable mismatch between the Fermi surfaces of semiconductor and metal, consequently interface scattering and lower transparencies.\n- Have the authors critically accessed the interpretation of highly transmitting channels?\n- Has the interface quality been studied in similar JJs? (References?)\nOur response:\nWe thank the referee for raising this important point. The transparencies extracted in table 1 are indeed very high (T ~ 0.93 - 0.999), and we have carefully examined their interpretation. We emphasize that these values do not represent the literal transmission probabilities of every microscopic channel, but rather as eeective parameters within our short-junction, single-transparency model. More importantly, when we allow for a distribution of channel transparencies in the fit, the population consistently converges into a single dominant, high-transparency value (please see response on point 5 of Referee #1 for additional details), demonstrating that the qubit spectra are well captured by few channels with an average eeective transparency close to unity.\nExperimentally, the qubit spectra reveal a strongly forward-skewed CPR. Such strong forward skewness arises only when higher harmonics are prominent in the CPR, possible only at high transparencies - signatures of highly transmitting channels in the short-junction limit. This close agreement between the observed CPR shape and our short-junction model provides robust independent evidence of near-unity channel transparencies in our devices.\nPhysically, such high eeective transparencies are consistent with the materials system we employ in our work. InAs-Al 2DEGs grown by in-situ molecular beam epitaxy (MBE) exhibit atomically abrupt, epitaxial interfaces with strong hybridization between the semiconductor and the Al layer. This minimizes Fermi-surface mismatch and interface scattering, enabling exceptionally transparent supercurrent transport. Prior works on similar epitaxial junctions have shown skewed CPRs and near-unity transmissions [Mayer et al., Nat. Comm. 11, 212 (2020); Goeman et al., New J. Phys. 19, 092002 (2017); Kjaergaard et al., Nat. Comm. 7, 12841 (2016)]. In addition, tunneling electron microscopy (TEM) and spectroscopy studies have further confirmed the atomic-scale uniformity due to lattice alignment and absence of any interfacial oxide in such heterostructures [Shabani et al., PRB 93, 155402 (2016)], providing direct evidence of the high interface quality needed for such near-unity transparency.\nIn summary, although microscopic variations in individual channel transmissions exists, the combination of skewed CPR in our measured data, fits that consistently converges to a single dominant transparency and prior material platform studies support the conclusion that supercurrent transport in these devices is dominated by a few nearly ideal channels.\nReferee #4\n- Are the Sm-JJs indeed in the short junction limit? I.e., L≪ξ≈ℏvF/2Δ? (L≡ electrode separation; vF≡ Fermi velocity in the semiconductor)\nOur response:\nWe thank the referee for this important question, as the validity of the short-junction approximation is indeed central to our modeling of the CPR. The junction length in our devices is 𝐿 = 250 nm. Using the extracted electron carrier density 𝑛 ≈ 7.68 X 10^11cm-2 of the InAs 2DEG, determined during the heterostructure growth, we obtain the Fermi velocity as 𝑣𝐹 ≈ 1.17 × 10^6m/s. With the induced Al superconducting gap (∆ ≈ 230 𝜇eV), we obtain a coherence length of:\n𝜉 = ℏ𝑣𝐹/2∆ ~ 1 𝜇m\nwhich gives 𝐿⁄𝜉~ 0.25, placing our devices in the crossover of the short and intermediate regime.\nThis is further supported by the observation of strongly forward-skewed CPR, indicative of high transparency channels with substantial higher-harmonic content – clear hallmarks of the short-junction limit (please see response to point 4 of Reviewer #1 for additional details). In addition, both an empirical harmonic expansion that allows us to extract a model-independent CPR alongside our short-junction single transparency model show excellent agreement with the experimental CPR, confirming that the short-junction approximation justifiably and sueiciently captures the dominant physics of our junctions.\nReferee #4\nI didn’t find the value of Δ in the text. Has it been measured or inferred from the critical temperature of the Al layer?\nOur response:\nWe thank the referee for pointing this out. The induced superconducting gap of the Al layer is ∆ ≈ 230 𝜇eV, determined from its critical temperature 𝑇𝑐 ~ 1.5K. This value was measured on a similar InAs-Al heterostructure grown in the same MBE chamber using an identical in-situ process [Yuan et al., J. Vac. Sci. Technol. A 39, 033407 (2021)]. We note here that in our model, the ∆ is absorbed into the product 𝑁×∆ as one single fit parameter in table 1, where N is the eeective number of channels.\nReferee #4\n- Do the values Ni match expectations based on lateral size quantization?\nOur response:\nWe thank the referee for this insightful question. For a junction width of 𝑊 = 800 nm and measured InAs 2DEG density of 𝑛 ≈ 7.68 X 10^11cm-2, the number of propagating transverse modes based on geometric lateral confinement is given by:\n𝑁 ≈ 2𝑘𝐹𝑊/𝜋, 𝑘𝐹 = √(2𝜋𝑛)\nwhere the factor of 2 arises due to spin degeneracy and 𝑘𝐹 is the 2D Fermi wavevector. This yields 𝑁~110 available modes, consistent with expectations for InAs-based heterostructures. More importantly, this value represents the maximum number of available transverse modes. In practice, only a small subset of these channels are strongly coupled to the superconductor, thereby making them highly transparent and contributing significantly to the Josephson current. Various factors such as disorder, electrostatic edge depletion while wet-etching, nonuniform lead coupling, strong mode-dependent transparencies etc. typically reduce the eeective number of contributing channels.\nSeveral prior works have emphasized this distinction. Mayer et al. [Nat. Commun. 11, 212 (2020)] and Costa et al. [Phys. Rev. B 108, 054522 (2023)] showed that the current–phase relation in InAs-Al JJs is governed primarily by a small number of high-transmission modes, despite the presence of many geometric channels. Similarly, Goeman et al. [New J. Phys. 19, 092002 (2017)] directly measured the channel distribution in InAs–Al nanowire JJs, finding that only a few channels dominate the supercurrent. Earlier theoretical work by Volkov et al. [Physica C: Superconductivity 210 (1993)] also predicted that proximity coupling to superconductors selectively enhances the contribution of certain modes.\nIn conclusion, our fitted values of 𝑁~ 10−12 corresponds to the eeective number of highly transmissive Andreev channels, consistent with the quasi-ballistic transport regime of our devices. This interpretation is in agreement with theoretical expectations and previous experimental reports, providing a coherent and physically grounded description of our JJs.\nReferee #4\n- It is mentioned that all Sm-JJs are nominally identical. Why is there are large spread in the parameter values of Tab. 1? What are the values of the critical current for each junction?\nOur response:\nWe thank the referee for highlighting this variation in the parameter values of table 1. Although the three devices are nominally identical in geometry and fabrication, mesoscopic variations in channel transparency due to device-to-device microscopic variations in material properties such as barrier thickness or disorder causes a substantial spread in the extracted parameters. Therefore, the reported values in table 1 represents junction-specific fit parameters within the same device, capturing this natural variation. From our fits, the critical current for each JJ is: 𝐼𝑐1 ≈ 178.9±19.7 nA for JJ1 and 𝐼𝑐2 ≈ 71.9±1.1 nA for JJ2 in device 1, and 𝐼𝑐1 ≈ 138±17.6 nA for JJ1 and 𝐼𝑐2 ≈ 100.8±1.2 nA for JJ2 for device 2. For the SQUID loop, we obtain 𝐼𝑐1 ~ 147.2±23.4 nA for device 1 and 𝐼𝑐2 ~ 65.3±14.8 nA for device 2, respectively.\nThese extracted values from our fits can be independently verified with the measured variations in the qubit-resonator coupling strength 𝑔, which depends on the Josephson energy 𝐸𝐽 and, in turn, the critical current 𝐼𝑐 via the relation: g ∝ I𝑐1/4 [Koch et al., PRA 76, 042319 (2007)] (see response to Referee #2 on page 10 for details). For 𝑔1/𝑔2 = 1.21 measured in our devices, as shown in supplementary figure 8, we obtain 𝐼𝑐1⁄𝐼𝑐2 ≈ 2.26, consistent with the extracted values of 𝐼𝑐 for the SQUID loops in device 1 and 2. Thus, the spread in 𝐼𝑐 and the corresponding fit parameters are real and physical, and not an error.\nReferee #4\n- What is the mean free path in the semiconductor material? Can the authors rule out the dieusive transport limit?\nOur response:\nWe thank the referee for this important question. The mean free path of the InAs 2DEG is given by:\n𝑙 = ℏ𝑘𝐹𝜇/𝑒\nwhere 𝑘𝐹 = √(2π𝑛) is the 2D Fermi wavevector, 𝑛 is the electron carrier density and 𝜇 is the electron mobility. Using the measured values of 𝑛 ≈ 7.68 X 10^11cm-2 and 𝜇 ≈ 20,000 cm2/Vs, we obtain 𝑙 ≈ 290 nm. Since the junction length is 𝐿 = 250 nm, the ratio 𝑙/𝐿 ≈ 1.2 places our device in the quasi-ballistic regime, but well away from the strongly dieusive limit (𝑙 ≪ 𝐿). In this regime, many electron trajectories can traverse the junction without any scattering, giving rise to ballistic contributions to transport, while a non-negligible fraction will scatter, resulting in only a few eeective high-transparency channels that dominate supercurrent transport. Furthermore, the Al coherence length is 𝜉~1 𝜇m, ensuring that the device also lies in the short-junction regime. In conclusion, although dieusive contributions to transport cannot be entirely excluded, the combination of length scales indicates that our junctions operate in the quasi-ballistic regime, consistent with transport picture presented in the manuscript. A more quantitative determination could be obtained by directly comparing the normal-state resistance to the Sharvin resistance, which we view as a natural extension beyond the scope of the present qubit spectroscopy devices.\nReferee #4\nI recommend the authors to provide this information in the revised manuscript and include references to available literature.\nOur response:\nWe thank the referee for the suggestion to include more information. We have added the above points along with the necessary references in the revised manuscript.\nReferee #4\nThere are several aspects of the CPR discussion (page 12f and Fig.4) that I find strange:\nIn Fig. 4, the normalized CPRs of JJ2 of Device I and II match. They are calculated based on the fit parameters in Table 1. However, judging by the transparency values given there, I would expect JJ1 of Device II to match with JJ2 of Device I. Are the labels in Fig. 4 wrong, or am I missing something in the analysis? Is there a way to know which one of the two physical junctions has the parameters of JJ1 and JJ2 in this experiment? Or are the labels interchangeable?\nOur response:\nWe thank the referee for raising this question. In our analysis, the labels JJ1 and JJ2 are assigned by comparing the extracted critical currents of the two junctions obtained from the CPR fits. Specifically, we define the junction with the larger critical current as JJ1. Beyond this convention, the junction properties are interchangeable, i.e. swapping the labels does not alter the simulated qubit spectrum or the interpretation of the results. Since the two junctions are fabricated to be nominally identical, their distinction arises only from the physical parameters extracted through our modeling. Therefore, the notation JJ1 and JJ2 can be used interchangeably without aeecting any conclusions of our work.\nReferee #4\nWhat do we learn from Fig. 4c? The authors provide a plot of the first 16 amplitudes of the Fourier series expansion of eq. 2 for the parameters Ti in Tab. 1. In other words, an (eeectively) one-parameter expression (as Ni and Δ drop out by normalization) is recast into another expression with 16 parameters. It is stated that the magnitudes of the amplitudes strongly decrease with increasing index n. This is true but easily understood when we look at the Fourier series of a sawtooth waveform, |An|∝1/n, which represents the extreme limit of a skewed CPR. I don’t see the value of this analysis.\nOur response:\nWe thank the referee for this valuable comment. Indeed, the overall transport regime can already be inferred from the skewness of the CPR: a sinusoidal CPR reflects dieusive transport, while forward skewness signals the presence of high-transparency modes. The value of the harmonic expansion, however, lies in going one step further - the relative ratios of the harmonic terms 𝐴𝑛/𝐴1 directly connect the microscopic transport properties of the JJ to the shape of the Josephson potential landscape. This information is crucial for qubit engineering, since the higher harmonics reshape the potential beyond the simple cosine form and thereby determine the anharmonicity, charge dispersion and flux-noise sensitivity of the qubit.\nIn our devices, we find 𝐴2/𝐴1 ≈ 1/2 for the SQUID loop and 𝐴2/𝐴1 ≈ 1/6 for the individual junctions, together with substantial higher harmonics. These values indicate that the Josephson potential substantially deviates from the standard 𝑐𝑜𝑠(ϕ) form and acquires additional 𝑐𝑜𝑠(2𝜙) and higher-order contributions. As shown in figure 1c, the Josephson potential now approaches a π-periodic landscape, which makes the curvature of the potential minima steeper and more asymmetric within each 2𝜋 phase period. This modified curvature naturally reduces the qubit’s charge dispersion, since the multi-well potential suppresses the sensitivity to oeset charge due to suppressed charge matrix elements, and explains the strongly anharmonic spectra observed in our devices.\nMore broadly, tuning the harmonic ratios 𝐴𝑛/𝐴1 provides a versatile design tool for engineering novel qubit designs. For example, complete suppression of the first harmonic, i.e. 𝐴1 = 0, yields a 𝑐𝑜𝑠(2𝜙) potential with exact 𝜋-periodicity, enabling a 𝑐𝑜𝑠(2𝜙) qubit with degenerate ground states and enhanced T1 relaxation time at the half-flux sweet spot [Larsen et al., PRL 125, 056801 (2020)]. Such qubits can be realized by naturally extending our design to symmetric SQUIDs where the first harmonics of the two JJs cancel each other. Similarly, devices with 𝐴2 ≪ 𝐴1 and 𝐴2 < 0 satisfy the conditions for novel “harmonium” qubits, which feature non-degenerate but noise-resilient ground states [Hays et al., arXiv:2502.15459 (2025)]. Our devices, with intermediate ratios such as 𝐴2/𝐴1 ≈ 1/2, showing partial suppression of the charge matrix elements at half-flux sweet spot, naturally fall within this broader design landscape, and illustrate how harmonic content extracted from transport directly connects to qubit performance and engineering of future qubit modalities.\nReferee #4\nOn the other hand, in Fig. 3, the authors compare the single-transparency fit with a model for USQUID composed of eight cos⁡ φ-harmonics. How do the coeeicients of the Fourier series expansion (An) in Fig. 4c compare to the eight fitting parameters of the second model? Can we learn something about the junction physics or the fitting procedure by comparing the numbers?\nOur response:\nWe thank the referee for this insightful question. The comparison between the Fourier coeeicients in figure 4c and the eight-harmonic model (4 harmonics per JJ) for each individual JJ in device 1 and 2 is shown below:\nHowever, we note that the two representations are not directly comparable term by term, as they are obtained using dieerent approaches and assumptions: The Fourier coeeicients result from a direct numerical decomposition of the extracted CPR, while the eight-harmonic model corresponds to a fit in which the amplitudes of the first four harmonics of each JJ are treated as free parameters. Nevertheless, comparing the two representations still provides valuable insight.\nIn particular, it shows that higher-order harmonics 𝐴𝑛 (𝑛>4), although much smaller in magnitude than the leading terms, still play a significant role in accurately describing the qubit spectrum and the resulting fits. Their inclusion systematically reduces the fitting residuals and improves agreement with the measured transition frequencies. This is evident in figure 4d, 4e and 4f, where the residuals of the eight-harmonic model fluctuate around zero (oscillatory behavior) in contrast to the single transparency model fit. Therefore, higher harmonics cannot be neglected in the modeling. From a junction-physics perspective, their inclusion and identification reinforce the interpretation of highly transparent Andreev channels, rather than behaving like tunnel junctions.\nReferee #4\nI disagree with the statement “More notably, the dieerences in the CPR and harmonic contributions between individual JJs are minimal, enabling us to extract fine details in the current-phase relations, far surpassing the capabilities of any standard transport measurement.”\nThe first part is wrong. It is observed that the CPRs of two junctions are identical. The other two are substantially dieerent. Therefore their harmonic content is dieerent. The second part is not discussed in the manuscript, and I highly doubt it is true.\nOur response:\nWe thank the reviewer for this helpful comment. We agree that our original statement was misleading. While two JJs exhibit nearly identical CPR, others show substantial dieerences, resulting in distinct harmonic content that reflects variations in microscopic junction parameters. We have revised the text to reflect this observation accurately, without implying uniformity across all measured junctions.\nRegarding the claim about surpassing transport measurements, we do not intend to claim that our approach universally provides more information than all standard transport techniques. Rather, the harmonic decomposition of the CPR oeers complementary insights into the microscopic transport regime, Josephson potential landscape and qubit properties, which ultimately aeect the qubit’s performance, that are not directly accessible from DC I-V characteristics alone. In the revised manuscript, we now include: “While two junctions exhibit similar CPRs, others show substantial deviations, leading to distinct harmonic content. This diversity reflects the sensitivity of the CPR to microscopic junction parameters. The harmonic analysis provides complementary information to standard transport measurements, allowing one to identify deviations from purely sinusoidal behavior and to engineer the Josephson potential landscape, thereby tailoring qubit properties for better performance.”\nReferee #4\nExcluding other systematic errors for now, the accuracy of the CPR extraction depends on the choice of a suitable parametrization and the quality of the fit to the experimental data. The authors have omitted a good deal of error analysis here. What are the confidence intervals for the parameters in Tab. 1?\nOur response:\nWe thank the referee for pointing out the need for confidence intervals and error analysis. The confidence intervals of the fitting parameters in table 1 are obtained from the covariance matrix of the nonlinear least-squares fit. We follow White’s heteroskedasticity-consistent method [H. White, Econometrica, vol. 48, no. 4, pp. 817–838 (1980)], where we first compute the Jacobian matrix 𝐽 of the fit numerically using a finite-dieerence scheme. The covariance matrix 𝐶 is then approximated as:\nC = ϵ2(𝐽𝑇𝐽)−1\nwhere 𝜖 is the mean standard error of the fit. The 95% confidence interval for each fitted parameter 𝛽 is then given by:\n𝛽 ±𝑍√𝐶𝑖𝑖\nwhere 𝑍 ≈ 1.96 is the Z-score that corresponds to 95% confidence interval. The resulting fit parameters with their 95% confidence intervals is listed below:\nTable of 95% confidence interval for Device I and II\nDevice 𝑁1 Δ1 (GHz) T1 𝑁2 Δ2 (GHz) T2 𝐸𝐽 (GHz)\nI 221.0±22.9 0.999±0.099 98.9±2.9 0.926±0.006 0.197±0.003\nII 138.0±3.0 0.928±0.006 467.0±88.5 0.542±0.095 0.163±0.002\nWe have now added the confidence intervals in table 1 of the revised manuscript.\nTo propagate these uncertainties to the extracted CPR and the corresponding Fourier harmonics, we employ a bootstrapping scheme with 500 Monte-Carlo samples. Each parameter 𝛽 is assumed to follow a normal distribution with standard deviation 𝜎 = √𝐶𝑖𝑖, which is justified by the large number of data points relative to the number of fit parameters. Unphysical parameters, such as transmission exceeding unity, are excluded by post-selection. For each valid sample, the CPR is then computed numerically on a uniform grid of 200 phase points, and the harmonic amplitudes are extracted using a Fast Fourier Transform (FFT). The reported error bars for the current and the harmonic amplitude, as shown in the responses below, correspond to the median 95% interval of the resulting distributions.\nReferee #4\nHow large are the error bars in Fig. 4a,b?\nOur response:\nUsing the procedure described above, we extracted the normalized CPR of the individual JJs in device 1 and the corresponding SQUID loop. The resulting plots, including error bars depicted by the shaded area, are shown below (also shown in supplementary figure 9 of the revised manuscript):\nSimilarly, for device 2, we obtain:\nReferee #4\nIn Fig. 4c?\nOur response:\nFigure 4c with error bars included for the individual JJs and the SQUID loops are shown below:\nReferee #4\nCan the authors provide examples by detailed comparison with transport experiments in the literature to support their statement?\nOur response:\nThe parameters extracted from our fits in table 1 yield transparencies and CPR skewness consistent with prior transport measurements. Specifically, we obtain a small number of highly transparent channels with transmission probabilities 𝑇 ≥ 0.92, leading to strongly forward-skewed CPRs with 𝐴2/𝐴1 ∼ 0.2−0.5. These values are in excellent agreement with previous transport studies in similar hybrid systems- for example, Kjaergaard et al. [Nat. Comm. 7, 12841 (2016)] reported forward-skewed CPRs in epitaxial InAs-Al JJs consistent with 𝑇 ≥ 0.9; Mayer et al. [Nat. Comm. 11, 212 (2020);] observed skewness corresponding to few-mode transport with 𝑇 ≈ 0.6− 0.9; and Goeman et al. [New J. Phys. 19, 092002 (2017)] demonstrated distributions with near-unity transparency channels in atomic-scale contacts. Our results therefore lie within the established transport regime of highly transparent, quasi-ballistic junctions, while harmonic analysis provides a complementary and independent confirmation of these properties.\nReferee #4\nFinally, the authors should provide an analysis of systematic errors in their experiment. E.g., generally, the phase drop across the SQUID loop inductance must be taken into account when determining the CPR. If the inductance is large, the shape of the CPR has to be determined self-consistently. Have the authors calculated/estimated the loop inductance (geometric and kinetic components) and ascertained that it is safe to omit it from their analysis? How would it aeect the precision of their CPR extraction and the fit values for transmission in Tab. 1?\nOur response:\nWe thank the referee for highlighting the role of the loop inductance and its potential influence on the CPR extraction. We first carry out a quantitative estimate of the geometric and kinetic inductance of the SQUID loop and its associated energy scale, followed by an explicit evaluation of the screening parameter 𝛽.\nTypically, the geometric loop inductance for a square loop with rectangular conductors is estimated by [Grover, Inductance Calculations: Working Formulas and Tables, Dover (1946)]\nL = µ0𝑎/2𝜋 [𝑙𝑛(𝑎/ω+𝑡)+0.5]\nwhere 𝑎 is the length of one side of the SQUID loop and ω,𝑡 are the width and thickness of the Al layer. For our geometry of 𝑎 = 50 𝜇𝑚,ω = 8 𝜇𝑚,𝑡 = 30 𝑛𝑚, we obtain 𝐿 ~207 𝑝𝐻.\nKinetic inductance for such structures with 30 nm Al is typically around 5% of 𝐿, resulting in a total inductance of 𝐿~220 𝑝𝐻. The corresponding loop inductive energy 𝐸𝐿 = Φ02/2𝐿 ⋙ 1000 GHz. For reference, the Josephson energy of the SQUID loops is 𝐸𝐽 ~75 GHz for device 1 and 𝐸𝐽 ~33 GHz for device 2, while the charging energy is 𝐸𝑐 ~200 MHz. Therefore, the loop inductance provides an energy scale several orders of magnitude larger than both the Josephson and charging energies, thereby:\n𝐸𝐿 ⋙ 𝐸𝐽 ≫ 𝐸𝑐\nThis firmly places our devices in the regime where the loop acts as a stie phase bias, and inductive corrections to the CPR are expected to be very small.\nIn addition, a complementary way to assess the relevance of the loop inductance is to estimate the screening parameter\n𝛽 = 2π𝐿𝐼𝑐/Φ0\nwhich results in an extra phase shift due to the current 𝐼 flowing in the SQUID loop, i.e.\nΔϕ = −2πΦ/Φ0 − 2π𝐿𝐼/Φ0\nUsing 𝐿 𝑎𝑛𝑑 𝐼 determined above, we obtain 𝛽 ≈ 0.09 for device 1 and 𝛽 ≈ 0.04 for device 2.\nBoth the values are well below unity, meaning the system operates deeply in the weak-screening regime. In this limit, the inductive term enters only as a small phase shift, i.e. Δϕ ~ 𝛽, and produces very minimal corrections to the CPR harmonics.\nIn summary, the inductive energy is far larger than both 𝐸𝐽 and 𝐸𝑐. The associated screening parameter 𝛽 ≪ 1, thereby ascertaining that the precision of our CPR extraction is robust.\nReferee #4\nAdditional comments:\nThere is a reference missing on page 16, bottom line.\nOur response:\nWe thank the referee for pointing this out. We have added the missing reference in the revised manuscript.\nReferee #4\nI, personally, find expressions like “gateless” gatemon or “super-semi” Josephson junctions confusing and distracting. The oxymoron in the title made me skip the paper when it first appeared on the arXiv. I presume other researchers feel similarly about colloquialisms.\nOur response:\nWe thank the referee for this comment. We have changed the term “gateless gatemon” to “flux-tunable transmon” and “super-semi” to “superconducting-semiconducting” in the revised manuscript.\nReferee #4\nThe authors state that their transmon qubit allows for faster gate operations than traditional transmon and gatemons. As someone who works in the periphery of the quantum information field, I would greatly appreciate if the authors gave more context and examples. It would benefit the general readership.\nOur response:\nWe thank the referee for the valuable suggestion to clarify our statement. We note that conventional AlOx-based transmons can achieve sub-10 ns gates using advanced pulse shaping protocols such as DRAG [Motzoi et al., PRL 103, 110501 (2009); Lucero et al., Phys. Rev. A 82, 042339 (2010)]. Our point here is not to emphasize that such gate times are unattainable in other platforms, rather that our device reaches comparable speeds intrinsically, without relying on advanced DRAG protocols.\nThe key advantage of our architecture is the strongly anharmonic qubit spectra engineered by the higher-harmonic contributions. This produces an energy level spectrum where the 𝑓𝑞 transition is well separated from higher levels, intrinsically suppressing leakage under strong drives. As a result, we can apply large drive amplitudes and achieve raw Rabi frequencies exceeding 100 MHz using simple Gaussian-like pulses, while maintaining low leakage. In contrast, standard transmons with ~250 MHz anharmonicity experience significant leakage at these drive rates, and therefore require careful DRAG calibration to compensate.\nThus, the contribution of our work is not that we surpass the absolute gate speed demonstrated in optimized AlOx devices, but that we oeer a simplified and robust route to fast gates. By embedding large anharmonicity directly into the device, our qubit design reduces the need for complex pulse shaping, minimizes calibration overhead, and naturally supports fast, high-fidelity gate operations."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have adequately addressed the concerns and comments from the previous round of review.\nI do have one significant issue with their new revision regarding the suppression of the charge matrix element from their dielectric loss model. Their procedure seems to be circular - they extract the dependence of the loss tangent at each frequency using their measured T1 and then claim good agreement between their model, which uses the extracted loss tangent, and their experimental T1. The typical method for comparing with a loss model is to consider the expected functional form of T1 for some loss model, which includes the matrix element and a well-motivated frequency-dependent form for the loss tangent, and check whether the data can be fit using that model. The authors need to either remove the claim that their model fits their data well or provide reasoning for why their procedure is physically sound.", "reviewer_2": "The authors have addressed the previosuly raised points from myself and the other refereees and I recommend publication", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "The authors have addressed all points that needed clarification in their rebuttal letter. I noticed, however, the sentence \"This approach is justified for atomically clean JJs, where channel-to-channel variations are minimal.\" has been added on page 11 of the revised manuscript. I think this statement is misleading. Firstly, in their rebuttal letter, the authors point out that the variations in the critical current of the individual junctions likely originate from disorder or some other mesoscopic imperfections. Secondly, only about 1/10 of ~110 modes are highly transmissive due to either selective coupling or some sort of disorder/mesoscopic imperfection. I recommend this information be stated clearly (including the references on selective coupling and mode transmissions provided in the rebuttal letter) for the benefit of the reader.\nI support the publication of the paper in Nature Communications."}, "author_response": "Our response:\nWe thank the referee for the positive assessment of the previous round of review.\nOur response:\nWe thank the referee for highlighting the circular nature in our earlier approach for determining the charge matrix element from the dielectric loss model. In the revised manuscript, we now adopt a weakly frequency dependent dielectric loss model based on previously reported measurements. We assume that the dielectric loss tangent follows the empirical relation:\ntan𝛿(𝜔) / tan𝛿(𝜔 = 2𝜋×2.63 GHz) = (𝜔 / 𝜔0)𝜀\nwith 𝜀 = 0.15 as reported by Nyugen et. al. [Phys. Rev. X 9, 041041 (2019)]. The reference loss tangent value of tan𝛿(𝜔 = 2𝜋×2.63 GHz) = 2.44×10^-6 corresponds to typical dielectric losses for InP substrates [Strickland et. al., Phys. Rev. Res. 6, 023094 (2024)].\nAt the finite operating temperature of our qubit system 𝑇 = 50 mK, the dielectric loss limited relaxation time is given by:\n1/𝑇1,diel(𝜔) = 1/16𝜋𝐸𝑐 tan𝛿(𝜔) 𝑛𝑔2 ⋅ [1+𝑐𝑜𝑡ℎ(ℏ𝜔/2𝑘𝐵𝑇)]-1\nThe resulting plot for the reference quality factor 𝑄ref = 1/𝑡𝑎𝑛𝛿(𝜔 = 2𝜋×2.63 GHz) = 4100 is plotted in the revised figure below (also included in supplementary figure 7c).\nAlthough our measured T1 times show no clear frequency dependence, indicating that the measured data is not suZiciently constraining, the results are consistent with a dielectric loss model at finite temperature of 𝑇 = 50 mK. This demonstrates that our analysis does not rely on circular fitting and that the observed relaxation times are physically consistent with realistic dielectric losses.\nOur response:\nWe thank the referee for their time and careful review of our manuscript and the recommendation for publication in Nature Communications.\nOur response:\nWe thank the referee and their co-reviewer for their participation and time in carefully peer reviewing our manuscript.\nOur response:\nWe thank the referee for the positive assessment of the rebuttal letter.\nOur response:\nWe thank the referee for this insightful comment. In the revised manuscript, we have now clarified that only a small fraction of the total transmitted channels, i.e. approximately 10%, are highly transmissive, while the majority exhibit reduced transparency due to mesoscopic imperfections such as disorder and selective mode coupling at the interface. We have also included the relevant references [34] and [39-41] in the revised manuscript to support this discussion.\nTherefore, the revised text now states: “This approach is justified for atomically clean JJs, where channel-to-channel variations are expected to be minimal. However, in practice, only a fraction of the total channels (~10% in our devices) is highly transmissive due to mesoscopic imperfections such as disorder and selective mode coupling.”\nOur response:\nWe thank the referee for the recommendation to publish the manuscript in Nature Communications."}, {"round": 3, "reviewer_comments": {"reviewer_1": "The authors have addressed my comments in the revised version of their manuscript.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Our response:\nWe thank the referee for their time and careful review of our manuscript.\nOur response:\nWe thank the referee and their co-reviewer for their participation and time in carefully peer reviewing our manuscript."}]

Decision letter