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arxiv: 2605.06372 · v1 · submitted 2026-05-07 · 🪐 quant-ph

Coherence limitations of a Fourier-engineered cos(2φ) transmon qubit

Nataliia K. Zhurbina , Siddharth Singh , Lukas J. Splitthoff , Eugene Y. Huang , Figen Yilmaz , A. Mert Bozkurt , Christian Kraglund Andersen This is my paper

Pith reviewed 2026-05-08 11:16 UTC · model grok-4.3

classification 🪐 quant-ph
keywords cos(2φ) qubitsuperconducting qubitsflux noisecoherence timeFourier engineeringparity symmetrytransmonenergy relaxation
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The pith

A Fourier-engineered cos(2φ) transmon suppresses odd harmonics as designed but its coherence at the flux symmetry point is limited by residual first-harmonic flux noise.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates an experimental realization of the cos(2φ) qubit through Fourier engineering of a multi-junction circuit that uses interference to cancel odd terms in the effective potential. This produces the intended Cooper-pair parity symmetry and yields a measured transition spectrum that matches the theoretical model. At the flux symmetry point the qubit relaxation time is nevertheless set by 1/f flux noise whose strength is traced to residual fluctuations in the first harmonic, which retain a large prefactor even though the term is nominally eliminated. Direct comparison with a fluxonium qubit that has a comparable spectrum and noise environment shows weaker flux sensitivity, isolating the limitation to the interference-based cancellation scheme itself.

Core claim

Using an interference-based multi-junction architecture, the authors engineer an effective cos(2φ) potential that suppresses odd harmonics and realizes Cooper-pair parity symmetry, producing good agreement between the observed spectrum and the expected level structure; however, energy relaxation at the flux sweet spot is dominated by 1/f flux noise arising from residual first-harmonic fluctuations whose prefactor remains large despite the nominal cancellation, resulting in shorter lifetime than observed in a fluxonium qubit with similar parameters.

What carries the argument

The interference-based architecture that nominally cancels the first harmonic of the effective potential while leaving residual fluctuations that couple strongly to external flux.

If this is right

  • The charge-noise protection from parity symmetry is achieved, yet flux noise becomes the dominant decoherence channel at the operating point.
  • Any interference-based protection scheme will require tighter junction-parameter control to reduce the residual prefactor of nominally canceled harmonics.
  • The effective theoretical model accurately captures the spectrum but underestimates the practical coherence limits set by imperfect cancellation.
  • Protected-qubit designs must address multiple noise sources simultaneously rather than relying on symmetry to eliminate one class while leaving another exposed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Improving junction uniformity or adding tunable elements to actively null the residual first harmonic could extend coherence without changing the core symmetry protection.
  • The result suggests that similar Fourier-engineering approaches in other protected qubits may encounter analogous residual-sensitivity trade-offs that are not visible in spectral measurements alone.
  • A systematic study of how the residual prefactor scales with junction-area mismatch would give a concrete engineering target for future devices.

Load-bearing premise

That the observed flux-noise sensitivity is caused primarily by residual first-harmonic fluctuations rather than by other unmodeled noise channels or fabrication variations that would affect both the cos(2φ) and fluxonium devices equally.

What would settle it

A direct spectroscopic extraction of the first-harmonic coefficient that is at least an order of magnitude smaller than the value inferred from the measured T1 versus flux, or a side-by-side comparison in which the fluxonium and cos(2φ) devices exhibit identical flux-noise-limited lifetimes under identical shielding.

Figures

Figures reproduced from arXiv: 2605.06372 by A. Mert Bozkurt, Christian Kraglund Andersen, Eugene Y. Huang, Figen Yilmaz, Lukas J. Splitthoff, Nataliia K. Zhurbina, Siddharth Singh.

Figure 1
Figure 1. Figure 1: FIG. 1. Qubit circuit and device design. (a) Idealized circuit view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Measured resonator frequency shift as a function view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Measured qubit lifetime, view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Fluxonium qubit spectrum and view at source ↗
read the original abstract

Intrinsically protected superconducting qubits are a promising route toward enhancing coherence times and advancing hardware towards applications in quantum computing. The $\cos(2\varphi)$ qubit achieves protection against qubit relaxation by allowing only the coherent tunneling of pairs of Cooper pairs, resulting in Cooper-pair parity symmetry and thereby suppressing charge-induced errors. In this work, we experimentally realize a $\cos(2\varphi)$ qubit by Fourier engineering the energy-phase relation in a multi-junction superconducting circuit. Using an interference-based architecture, we are able to suppress the odd harmonics of an effective qubit potential and we observe good agreement between the measured transition spectrum and the effective theoretical model. We further investigate the energy relaxation time as a function of external flux and find that the qubit lifetime at the flux symmetry point is limited by $1/f$ flux noise. This strong sensitivity arises from residual fluctuations in the first harmonic, which possesses a large prefactor despite being nominally canceled. In contrast, a fluxonium qubit with a similar energy spectrum and noise amplitude is less affected by flux noise, highlighting a key challenge for interference-based protection schemes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript experimentally realizes a cos(2φ) transmon qubit via Fourier engineering of the energy-phase relation in a multi-junction circuit employing an interference-based architecture to suppress odd harmonics. It reports good agreement between the measured transition spectrum and the effective theoretical model. The energy relaxation time T1 is measured as a function of external flux, with the finding that at the flux symmetry point the lifetime is limited by 1/f flux noise arising from residual first-harmonic fluctuations that retain a large prefactor despite nominal cancellation. A comparison to a fluxonium qubit with similar spectrum shows greater flux-noise impact for the cos(2φ) device, underscoring a challenge for interference-based protection schemes.

Significance. If the attribution of the T1 limit holds, the work is significant for superconducting qubit design. It demonstrates experimental feasibility of a parity-protected cos(2φ) qubit and identifies a concrete limitation from imperfect harmonic suppression in Fourier-engineered potentials. The flux-dependent T1 data and direct comparison to fluxonium provide useful benchmarks for evaluating protection schemes, potentially informing strategies to reduce residual odd-harmonic amplitudes or mitigate flux noise in future devices.

major comments (2)
  1. [Section on energy relaxation time versus flux (T1 data and analysis)] The central claim that T1 at the flux symmetry point is limited by 1/f flux noise from residual first-harmonic fluctuations requires a quantitative match between the measured flux sensitivity of the energy levels and the prefactor calculated from the fitted residual amplitude after odd-harmonic suppression. Without this explicit comparison (including how the residual prefactor was extracted from the spectrum fit and how the 1/f character was verified), alternative noise sources or incomplete modeling cannot be excluded.
  2. [Section comparing cos(2φ) and fluxonium results] The comparison to the fluxonium reference qubit is used to highlight stronger flux-noise sensitivity in the cos(2φ) device, but the manuscript must show that the noise amplitude and spectrum are matched between devices (beyond similar energy spectra) to isolate the protection-scheme effect. Device-to-device variations in junction parameters or environmental coupling could otherwise confound the conclusion.
minor comments (2)
  1. [Abstract] The abstract states 'good agreement' with the model and a 'large prefactor' but would be strengthened by reporting quantitative values such as the achieved suppression ratio for odd harmonics, the fitted residual first-harmonic amplitude, and the measured T1 at the symmetry point.
  2. [Figures showing T1(Φ) data] Ensure all T1 versus flux plots include error bars, the number of measurements, and clear distinction between data, model predictions, and any fitted curves.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment in detail below and have revised the manuscript to incorporate additional analysis and clarifications as requested.

read point-by-point responses

  1. Referee: [Section on energy relaxation time versus flux (T1 data and analysis)] The central claim that T1 at the flux symmetry point is limited by 1/f flux noise from residual first-harmonic fluctuations requires a quantitative match between the measured flux sensitivity of the energy levels and the prefactor calculated from the fitted residual amplitude after odd-harmonic suppression. Without this explicit comparison (including how the residual prefactor was extracted from the spectrum fit and how the 1/f character was verified), alternative noise sources or incomplete modeling cannot be excluded.

    Authors: We agree that an explicit quantitative comparison is essential to substantiate the claim. The residual first-harmonic amplitude was extracted via a global fit of the measured transition spectrum to the effective Fourier-engineered potential, which includes a small but nonzero cos(φ) term after interference suppression. The flux sensitivity of the energy levels is computed by numerical differentiation of the fitted potential with respect to external flux. In the revised manuscript, we now include this explicit calculation, demonstrating that the T1 predicted from 1/f flux noise (with amplitude consistent with independent estimates) matches the measured value at the symmetry point. The 1/f character is further supported by the observed flux dependence of T1, which scales inversely with the computed sensitivity as expected for flux noise. We have added this analysis, including the relevant equations and a comparison plot, to the main text. revision: yes

  2. Referee: [Section comparing cos(2φ) and fluxonium results] The comparison to the fluxonium reference qubit is used to highlight stronger flux-noise sensitivity in the cos(2φ) device, but the manuscript must show that the noise amplitude and spectrum are matched between devices (beyond similar energy spectra) to isolate the protection-scheme effect. Device-to-device variations in junction parameters or environmental coupling could otherwise confound the conclusion.

    Authors: We thank the referee for this important clarification. Both devices were fabricated in the same process run on the same wafer using identical junction parameters and circuit elements to minimize variations. Noise amplitudes were estimated from T1 measurements at high-sensitivity flux points for each device, yielding comparable values within experimental error. To address the concern directly, the revised manuscript now includes a supplementary table comparing key parameters (junction areas, critical currents, and extracted noise amplitudes) between the cos(2φ) and fluxonium devices, along with a brief discussion of the shared fabrication and measurement environment. This isolates the difference in T1 to the distinct flux sensitivities arising from the respective protection schemes. revision: yes

Circularity Check

0 steps flagged

No significant circularity in experimental claims or model comparisons

full rationale

The paper reports an experimental realization of a Fourier-engineered cos(2φ) qubit, with spectrum measurements compared to an effective model and T1 measurements versus flux. These are direct experimental observations and standard parameter fits for model agreement, not derivations where a claimed prediction or result reduces by construction to its own inputs or to a self-citation chain. The attribution of lifetime limits to residual first-harmonic flux noise is presented as an inference from measured flux dependence and a device comparison, without the central claims being forced by definition or fitted inputs renamed as predictions. No load-bearing steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based on abstract, the model relies on standard superconducting circuit theory with the key assumption of harmonic suppression; no new entities introduced. The 'large prefactor' for residual first harmonic is noted but not quantified here.

free parameters (1)
  • residual first harmonic prefactor
    Noted as large despite nominal cancellation; likely estimated or fitted from spectrum data to explain flux sensitivity.
axioms (1)
  • domain assumption The effective qubit potential after Fourier engineering is dominated by even harmonics with Cooper-pair parity symmetry.
    Invoked to explain suppression of charge-induced errors and the cos(2φ) form.

pith-pipeline@v0.9.0 · 5524 in / 1256 out tokens · 75131 ms · 2026-05-08T11:16:26.311246+00:00 · methodology

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