Error-detected coherence metrology of a dual-rail encoded fixed-frequency multimode superconducting qubit

Brian Vlastakis; James Wills; Mohammad Tasnimul Haque

REVIEW 3 major objections 5 minor 57 references

Error-detected coherence metrology of a dual-rail encoded fixed-frequency multimode superconducting qubit

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A fixed-frequency multimode superconducting qubit encodes a dual-rail logical qubit whose detected bit-flip and phase-flip rates are 48x and 11x below the physical modes.

desk verdict A plausible and well-executed fixed-frequency dual-rail erasure demo, but missing readout-confusion numbers leave the headline 48x/11x coherence improvements unproven. read the letter →

arxiv 2506.15420 v1 pith:IWLNQ3NB submitted 2025-06-18 quant-ph

James Wills , Mohammad Tasnimul Haque , Brian Vlastakis This is my paper · ORCID

classification quant-ph

keywords dual-railencodingerasureconversionmultimodetransmondimonsuperconductingqubitamplitudedampingerrordetectionlogicalcoherence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to establish that a single fixed-frequency multimode transmon—a three-island, two-junction device supporting two transmonlike microwave modes—can host a dual-rail logical qubit in which amplitude damping, normally the dominant error, becomes a detectable erasure. It encodes $|0\rangle_L=|10\rangle$ and $|1\rangle_L=|01\rangle$, so relaxation of either logical state leaks into $|00\rangle$; a three-state Gaussian-mixture readout flags that leakage and postselection (Eq. 3) removes it. On three devices measured over more than 50 hours, the error-detected logical coherence times are $T_1^L=3.12$ ms and $T_{2E}^L=0.76$ ms median, corresponding to 48x and 11x reductions in bit-flip and phase-flip rates relative to the physical modes. A sympathetic reader would care because erasure conversion is one of the more hardware-efficient routes to fault tolerance, and this implementation requires no flux tuning, no galvanic coupling, and no larger footprint than a conventional coaxial transmon.

What carries the argument

The central mechanism is a dual-rail encoding inside a dimon, a three-island, two-junction superconducting device with two transmonlike modes, $D$ and $Q$. The logical states $|0\rangle_L = |10\rangle$ and $|1\rangle_L = |01\rangle$ have different electric-field symmetries, so there is no direct single-photon transition between them; amplitude damping therefore exits the logical subspace to the ground state $|00\rangle$. A readout tone placed between the two dispersive resonances produces three separable Gaussian clouds, and a Gaussian mixture model assigns each shot to $|00\rangle$, $|01\rangle$, or $|10\rangle$. Eq. (3) renormalizes the surviving logical populations, converting a raw decay measurement into an error-detected coherence measurement. This combination—symmetry-forbidden logical leakage, three-state readout, and postselection—is what carries the reported $T_1^L$ and $T_{2E}^L$ improvements.

What would settle it

Retrain the GMM classifier on a different readout drive frequency or with a stricter $|00\rangle$ rejection threshold and remeasure the logical bit-flip decay: if $T_1^L$ shifts, readout misclassification is biasing the reported suppression. Independently, search spectroscopically for any transition between $|01\rangle$ and $|10\rangle$; observing direct coupling between the logical states would break the symmetry argument and invalidate the erasure interpretation.

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Extended reading notes

Core claim

The central discovery is that the two single-excitation states of the dimon form a dual-rail code with no direct single-photon decay channel between them, so energy relaxation from either logical state lands in $|00\rangle$ and is heralded by the end-of-line readout. After discarding those heralded events, the logical subspace exhibits median $T_1^L = 3.12$ ms and $T_{2E}^L = 0.76$ ms, i.e. bit-flip and phase-flip rates suppressed by factors of 48 and 11 relative to the constituent $D$ and $Q$ modes, with Ramsey coherence improved by about 4x. The erasure rate closely tracks the physical $T_1$ fluctuations, confirming that the detected leakage is ordinary amplitude damping. The same numbers reproduce on three devices with mode detunings from 0.76–1.05 GHz, showing the improvement is architectural rather than sample-specific.

Load-bearing premise

The protocol assumes that relaxation from either logical state always goes to the empty $|00\rangle$ state, never directly between $|01\rangle$ and $|10\rangle$, and that the three-state readout separates $|00\rangle$ from the logical states well enough that postselection removes essentially all amplitude-damping events.

Editorial extensions

If this is right

A fixed-frequency multimode transmon can act as an erasure-converted logical qubit with all-microwave control and no flux or galvanic-coupling overhead.
The error-detected suppression is stable over 50-hour runs on three separate devices, so it is not a one-sample coincidence.
Because the logical subspace cancels common-mode frequency noise, comparing logical and physical Ramsey or Hahn-echo decays gives a per-device estimate of the common versus differential noise budget.
The erasure rate follows the physical $T_1$ fluctuations, so raising physical mode relaxation times directly lowers the postselection overhead of this logical protocol.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper measures only end-of-line detection; a natural extension would be a mid-circuit erasure check, which would turn the postselected memory into a repeat-until-success primitive whose overhead is set by the measured erasure rate.
The observed Ramsey improvement on Q2 (~4.5x) exceeds the photon-shot-noise-only estimate from Eq. (7) (~1–2x), suggesting that other common-mode noise is dominant; sweeping the readout photon number while tracking $T_{2E}^L$ would test this separation directly.
If junction-asymmetry noise is the limiting differential source, then fabricating devices with $r = E_{J1}/E_{J2}$ closer to unity, or post-processing the junctions, should raise $T_{2E}^L$; the paper's tabulated $r$ values make this a quantitative prediction across Q1–Q3.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a real experimental step: dual-rail encoding in a fixed-frequency multimode transmon (dimon), all-microwave control, three devices, and 50 hours of data. The physical mode characterization is careful, the time series are extensive, and the noise analysis (Allan deviation, spectral density, TLS-like jumps) is thoughtful. The three-device reproducibility and bootstrap bounds are genuine strengths. The paper is also honest about what it has not identified, e.g., the precise noise source limiting the logical Ramsey coherence.

What is genuinely new is applying dual-rail erasure conversion to this specific device class in a coaxial architecture, with mode detunings (0.76–1.05 GHz) larger than prior tunable-transmon dual-rail work. That is a useful hardware-oriented contribution, not a conceptual breakthrough.

The load-bearing soft spot is readout. The postselection in Eq. (3) renormalizes over shots classified as |01> or |10>. If even a couple percent of true |00> erasure shots are misclassified as one of the logical states, those shots add an apparent logical bit-flip rate of roughly ε·Γ_erasure. With Γ_erasure ≈ 13–17 kHz and the claimed Γ_L1 ≈ 0.32 kHz, ε ≈ 2% is enough to produce the entire 48x improvement even if the true logical bit-flip rate is zero. The paper reports no numeric readout assignment fidelity or confusion matrix; Fig. 1(c) shows overlapping IQ distributions and Fig. 1(d) is described only qualitatively. This is not a minor omission—it bears directly on the central quantitative claim. A referee needs to see the confusion matrix for the GMM classifier, ideally for the same integration time and drive settings used in the coherence measurements.

Other soft spots are secondary: the 30 µs linear fit is short but defensible for short circuit depths, Eq. (7) is stated without derivation, the Q3 Ramsey data are absent, and raw data/code are not provided. None of these individually sinks the paper, but together they lower confidence that the reported ratios are robust.

The paper deserves a serious referee. The device work is real and the erasure-conversion angle is timely. My recommendation: send it to peer review with a request for major revision, and make the readout-fidelity quantification a condition for acceptance. If the confusion matrix shows ε well below the 2% threshold for all three devices, the central claim will stand; if not, the headline numbers need to be revised.

Referee Report

3 major / 5 minor

Summary. This manuscript reports a dual-rail logical qubit encoded in the single-excitation subspace of a fixed-frequency two-mode multimode transmon ('dimon'), with logical states |01> and |10> and an end-of-line readout that classifies the |00> state as an erasure. By postselecting on non-erasure shots, the authors extract effective logical bit-flip and phase-flip rates and report improvements over the physical mode coherence times: median T_L1 = 3.12 ms (48x), T_L2E = 0.76 ms (11x), and T_L2R = 0.07 ms (4x), based on roughly 50-hour interleaved measurements on three devices. Section II describes the device, Hamiltonian, and three-state GMM readout; Section III defines the coherence metrics and postselection; Section IV presents noise analysis via Allan deviation and spectral density, attributing the observed logical coherence improvement to suppression of common-mode dephasing sources while retaining sensitivity to differential noise.

Significance. If the reported rates are correct, this is a significant step for erasure-conversion hardware: a flux-free, fixed-frequency, three-island transmon provides all-microwave dual-rail encoding with detectable amplitude-damping errors in a compact coaxial architecture, with repeatability across three devices. The long time series, interleaved measurements, and bootstrap bounds are notable strengths, as is the explicit stability analysis over 50 hours. However, the headline improvements depend on the three-state GMM readout being nearly confusion-free and on the definition of the short-time logical error rates; without a confusion matrix and a clearer fitting model, the central claim is not yet fully established. The work is of high interest to the superconducting-qubit and quantum-error-correction communities, but requires additional readout and fit characterization.

major comments (3)

[Section III, Eq. (3) and Fig. 1(c)] The postselection in Eq. (3) is valid only if the GMM classifier separates |00> from the logical states with negligible error, but the manuscript reports no confusion matrix or readout fidelity for the three-state classifier. Because the erasure population grows approximately as Gamma_erasure * t with Gamma_erasure ~ 13-17 kHz, a fraction epsilon of |00> shots misclassified as |01> or |10> contributes an apparent bit-flip rate of order epsilon * Gamma_erasure. For the reported median T_L1 = 3.12 ms (Gamma_L1 ~ 0.32 kHz), epsilon of a few percent is sufficient to produce the claimed 48x bit-flip improvement even if the true logical bit-flip rate were zero, and the overlapping distributions in Fig. 1(c) make such confusion non-negligible. The authors should report the full confusion matrix for the EOL readout, including |00>->|01>, |00>->|10>, and |01><->|10> misclassification probabilities, and show that the extracted logical rates are stable under an upper bound on these probabilities.
[Section III and Fig. 2(d)-(f)] The logical T_L1 and T_L2E values are obtained by fitting a linear slope with constant offset to data over the first 30 microseconds, whereas the physical coherence times are extracted from exponential fits. The manuscript does not report fit residuals, uncertainties, or a model justifying the linear approximation over this window; if the true short-time logical bit-flip probability is quadratic in time (as expected when a bit flip requires relaxation followed by re-excitation), the fitted slope depends on the chosen window length. This makes the reported 'logical error rate' not directly comparable to the physical decay rates. Please provide the fits with residuals, a stability test of the extracted rates under varying the fit window (e.g., 10, 20, and 40 microseconds), and, if possible, fits to the expected short-time functional form.
[Section II, 'no direct single photon decay channel'] The statement that there is no direct single-photon decay channel from |01> to |10> due to the different symmetries of the modes is load-bearing for the logical bit-flip model, since any direct |01><->|10> matrix element would contribute a first-order bit-flip rate. The manuscript gives no derivation or reference for the vanishing of this matrix element. Please provide a quantitative symmetry argument (or a citation) showing that the relaxation coupling operator has zero matrix element between |01> and |10>, or bound the residual direct transfer rate experimentally, for example by preparing |01> and measuring the |10> population in the absence of re-excitation.

minor comments (5)

[Eq. (3)] There is a typographical error in the denominators of Eq. (3): 'P (|01))' should read 'P (|01>).'
[Section III, paragraph on erasure rate] The sentence 'The erasure rate is a important metric' should read 'an important metric.'
[Fig. 2 caption] The caption says 'Y-axis in (d) showing difference'; this should be 'shows the difference' for grammatical completeness.
[Section IV, Eq. (7)] Equation (7) is introduced without derivation or a reference; please provide a derivation sketch or citation so that the dephasing ratio can be checked.
[Section IV, text near Fig. 4(i)] The units of the noise amplitudes A and B should be stated explicitly when the fit model sigma_delta_f(tau) = (B/2)^{1/2} tau^{-1/2} + (2 ln 2 A)^{1/2} is introduced.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the dual-rail coherence improvements are measured against independent physical benchmarks, with no fitted parameter forced to produce the claimed ratios.

full rationale

The paper's central quantitative claims are experimental comparisons between postselected logical-subspace coherence times and independently measured physical-mode coherence times on the same devices. Equation (3) defines the error-detected logical populations by renormalizing over the |01> and |10> classifications; this is the erasure-detection protocol under test, not a hidden input. The reported T_L1=3.12 ms, T_L2E=0.76 ms, and T_L2R=0.07 ms are extracted from fits to time traces (linear slope over 30 microseconds for bit/phase flip, exponential for Ramsey), and the physical T1, T2E, and T2R are conventional measurements listed in Table I. None of these fits contains the 48x/11x/4x ratios as parameters; the ratios are computed afterwards from the medians. The theoretical photon-shot-noise dephasing estimate in Eq. (7) is an independent formula using measured chi and kappa parameters, and the paper explicitly notes that the measured Ramsey improvement (~4.5x) exceeds the Eq. (7) prediction (1-2x), which is inconsistent with fitting-to-conclusion. Self-citations to earlier work by the same group (refs. [29], [34], and [40]) support the device architecture and fabrication methods but are not used to justify the coherence-improvement claim. The readout-confusion scenario raised by the skeptic is a possible calibration or validity bias that could affect the absolute logical error rates, but it is not a circular reduction of the measured coherence to an input of the measurement, so it does not raise the circularity score.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The main coherence improvements are experimental measurements, not derived quantities, so the ledger is light. The entries above are the assumptions and fitted noise-analysis constants that support the secondary interpretations and the rate extraction. No invented physical entities are introduced; the dimon device and dual-rail encoding are known from prior work.

free parameters (5)

Logical coherence times T_L1, T_L2E, T_L2R from short-window fits = T_L1 median 3.12 ms; T_L2E median 0.76 ms; T_L2R median 0.07 ms
These are the reported results, extracted by fitting linear or exponential models to measured decays. They are not ad hoc constants, but they are fit-derived, so they are listed for transparency.
Junction energy ratio r = 0.963 (Q1), 0.959 (Q2), 0.931 (Q3)
Inferred from room-temperature resistance measurements and used in the claimed junction-asymmetry dephasing dependence (1-sqrt(r))/Delta; no uncertainty is given.
Noise model amplitude A = 5.9e5 Hz^2 (logical), 3.5e5 Hz^2 (D), 2.0e5 Hz^2 (Q)
Fitted to Welch spectral densities using A/f+B; supports the qualitative noise-source discussion, not the main coherence claim.
White noise amplitude B = 0.5e6 Hz^2/Hz (logical), 1.3e6 Hz^2/Hz (D and Q)
Fitted to the same spectral-density model; used to argue that the logical encoding lowers white-noise sensitivity.
Erasure time constant T_erasure = Not tabulated
Fit parameter in Eq. (6) for P(|00>) = 1 - exp(-Delta t / T_erasure) + D; used to report leakage rates, but values are not listed.

assumptions (6)

domain assumption The dimon's two modes have no direct single-photon transition between |01> and |10> because of their different symmetries.
Central to erasure conversion; stated in Section II after the energy level diagram.
domain assumption Every amplitude-damping event from the logical subspace lands in |00>.
Used by postselection Eq. (3); higher-lying states such as |11> or |02> are not treated in the error model.
domain assumption The GMM classifier trained on 10k shots correctly separates |00>, |01>, and |10> sufficiently for renormalized logical populations.
No assignment fidelity is reported, but all logical probabilities depend on this classification.
ad hoc to paper Short-time logical decay is linear up to 30 microseconds.
The bit-flip and phase-flip rates are defined by a linear fit over this window; this is the operational definition of the reported T_L1 and T_L2E.
domain assumption The dispersive approximation for resonator coupling holds, yielding Eq. (2) and the dephasing ratio Eq. (7).
Standard circuit QED assumption, but Eq. (7) is not derived in the text.
domain assumption Noise sources can be separated into common and differential classes for the two modes.
Organizes the interpretation of Ramsey frequency jumps and coherence improvements; validated only qualitatively.

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Cite this review

Pith. "Pith review of Error-detected coherence metrology of a dual-rail encoded fixed-frequency multimode superconducting qubit." pith.science (2026). https://pith.science/paper/IWLNQ3NB

@misc{pith2026250615420,
  author       = {Pith},
  title        = {Pith review of: Error-detected coherence metrology of a dual-rail encoded fixed-frequency multimode superconducting qubit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IWLNQ3NB}},
  note         = {Machine review of arXiv:2506.15420}
}

read the original abstract

Amplitude damping is a dominant source of error in high performance quantum processors. A promising approach in quantum error correction is erasure error conversion, where errors are converted into detectable leakage states. Dual-rail encoding has been shown as a candidate for the conversion of amplitude-damping errors; with unique sensitivities to noise and decoherence sources. Here we present a dual-rail encoding within a single fixed-frequency superconducting multimode transmon qubit. The three island, two junction device comprises two transmonlike modes with a detuning of 0.75-1 GHz, in a coaxial circuit QED architecture. We show the logical bit-flip and phase-flip error rates are more than one order of magnitude lower than the physical error rates, and demonstrate stability and repeatability of the architecture through an extended measurement of three such devices. Finally, we discuss how the error-detected subspace can be used for investigations into the fundamentals of noise and decoherence in fixed-frequency transmon qubits.

Figures

Figures reproduced from arXiv: 2506.15420 by the authors.

**Figure 2.** FIG. 2. Error-detected coherence metric traces, extracted from 200 repeated measurements of Q1. (a-c) Fraction of states [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Logical coherence metric distributions. Boxplots of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Ramsey interferometry measurements of logical encoded qubit, and physical modes of Q2. (a-b) Energy level diagrams [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. Time series traces of error-detected coherence metrics. (a-c) Extracted logical qubit [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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