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REVIEW 3 major objections 5 minor 57 references

High-fidelity measurement of qubits encoded in multilevel superconducting circuits

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Encoding a qubit across widely spaced multilevel states suppresses relaxation-induced measurement errors, giving logical assignment infidelity as low as $5.8\times10^{-5}$ and resolving transmon gate errors near $1.4\times10^{-3}$.

desk verdict A real experimental advance in multilevel readout, but the headline 5.8e-5 infidelity may be limited by unmodeled 0.2% stuck reset events rather than relaxation. read the letter →

arxiv 1908.01869 v2 pith:ECUSL6JB submitted 2019-08-05 quant-ph

Salvatore S. Elder , Christopher S. Wang , Philip Reinhold , Connor T. Hann , Kevin S. Chou , Brian J. Lester , Serge Rosenblum , Luigi Frunzio
show 2 more authors
Liang Jiang Robert J. Schoelkopf
This is my paper · ORCID
classification quant-ph
keywords multilevelencodingstate-preservingreadoutsuperconductingqubitsbosoniccodesbinomialshelvinggatecharacterizationassignmentinfidelity
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

Superconducting qubit readout is usually capped by a tradeoff: collect signal longer to beat noise, but risk the qubit relaxing during the measurement and being assigned the wrong bit. This paper tries to break that tradeoff by encoding the bit in states separated by several energy levels, so one relaxation event no longer flips the answer. It demonstrates the idea on a transmon with a shelving pulse, and on a bosonic storage cavity with repeated state-preserving readouts of Fock and binomial encodings. The result is a logical assignment infidelity of $5.8\times10^{-5}$ when distinguishing $\lvert 0\rangle$ from $\lvert 5\rangle$, and $4.2\times10^{-3}$ for a binomial error-correcting code, with transmon gate errors resolved directly at the $10^{-3}$ level and the Fock result surpassing previous circuit-QED readout fidelities.

What carries the argument

The carrying object is the code distance $L$ between the logical codewords with respect to the photon-loss channel: for Fock codes the codewords are $\lvert 0\rangle$ and $\lvert L\rangle$, and the measured subspace $S$ is chosen so that losing or gaining a small number of photons does not move a state across the $\{S,\bar S\}$ partition. The readout is formalized as a projective measurement of membership in $S$, with fidelity $F=1-P(S\mid \lvert 1_L\rangle)-P(\bar S\mid \lvert 0_L\rangle)$, and the repeated-readout protocol combines a dispersive storage–ancilla map that flips the ancilla only when the storage state lies in $S$, a real-time ancilla reset that makes the readout effectively repeatable, and a hidden Markov maximum-likelihood classifier that weighs all votes against the growing chance of relaxation. The theoretical infidelity, Eq. (4), splits into majority-vote terms that fall with the number of readouts and transition terms proportional to $(\kappa_\downarrow\tau)^{L-1}$ and $(\kappa_\uparrow\tau)^2$ that rise as the code distance grows.

What would settle it

Prepare the same storage codewords with an independent tomographic verification rather than the heralded number-selective checks, then measure the logical assignment infidelity while scanning the storage loss rate $\kappa_\downarrow$ or the readout drive power; if the error does not follow the predicted $(\kappa_\downarrow\tau)^L$ scaling, or if the inferred demolition probability rises above $0.02\%$ with drive power, the central claim is false.

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Extended reading notes

Core claim

The central discovery is that the leading relaxation error is promoted to a higher order when codewords are separated by $L$ photon-number steps: the probability that relaxation corrupts the measurement scales like $(\kappa_\downarrow \tau)^L$ rather than $\kappa_\downarrow \tau$, so a code of distance $L$ is robust until $L$ errors accumulate. On the transmon, this means reading out the $\lvert g\rangle$–$\lvert h\rangle$ manifold instead of $\lvert g\rangle$–$\lvert e\rangle$; applying an $e$–$f$ shelving pulse turns the usual qubit measurement into a higher-distance measurement and exposes gate errors of order $10^{-3}$ that ordinary readout hides. In the storage cavity, a number-selective pulse maps the encoded bit onto a transmon ancilla, the ancilla is read out and reset by real-time feedback, and the sequence of outcomes is classified by a maximum-likelihood hidden Markov model. The reported best logical assignment infidelities are $5.8\times10^{-5}$ for the Fock encoding $\lvert 0\rangle$ versus $\lvert 5\rangle$ and $4.2\times10^{-3}$ for the $S=2,N=1$ binomial code.

Load-bearing premise

The entire result rests on the repeated readout being essentially state-preserving and on independent photon loss and gain being the only significant error processes; if a readout or reset event occasionally shuffles the storage state in a way not captured by Eq. (4), then the quoted infidelities no longer describe errors about the true initial state.

Editorial extensions

If this is right

  • If the central claim is right, relaxation no longer sets the practical floor for superconducting-qubit readout; the floor moves to state preparation, ancilla reset, and the state-preserving quality of each readout.
  • Gate and state-preparation characterization can be made an order of magnitude more precise, because shelved readout resolves transmon gate errors directly at the $1.4\times10^{-3}$ level instead of hiding them behind larger measurement errors.
  • For bosonic encodings, logical assignment infidelity should continue to improve roughly exponentially with $L$, so larger-distance Fock codes or higher-order binomial codes should push below $10^{-5}$ on the same hardware.
  • Repeated state-preserving readout with real-time reset is directly applicable to error-syndrome extraction, where the same ancilla must be reused many times without destroying the logical information.
  • Because the mechanism is generic to multilevel systems with a one-photon loss channel, the same encoding-and-voting strategy should transfer to other qubit platforms with relaxation-limited readout.

Reading between the lines

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

  • If the per-readout demolition probability could be pushed below the calibrated $0.02\%$, the same protocol should reach assignment errors set by preparation and reset rather than by relaxation; this can be tested by measuring the $\lvert 0\rangle$–$\lvert L\rangle$ infidelity at still larger $L$.
  • The gap between the Fock result ($5.8\times10^{-5}$) and the binomial result ($4.2\times10^{-3}$) suggests that photon gain and single-round readout errors, not code distance, are the next limit; reducing the storage thermal excitation should narrow that gap.
  • A direct test of the model would vary $\kappa_\downarrow\tau$ (by changing readout duration or cavity lifetime) and check that the optimal number of votes and the infidelity scale as Eq. (4) predicts.
  • Codes whose logical states share photon-number support would require a phase-respecting map, because the photon-number-selective flip used here only measures membership in a number subspace; extending the scheme to such codes is a nontrivial next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper demonstrates that encoding a qubit in multilevel states suppresses relaxation-induced measurement errors in superconducting circuits. Two experimental settings are studied: (i) a transmon whose higher excited states |f> and |h> are used to improve readout contrast, including a shelving technique that directly resolves g–e gate errors at the level of about 1.4e-3; and (ii) a bosonic storage mode measured through a transmon ancilla with repeated map–measure–reset cycles. For the bosonic encodings, the authors report logical assignment infidelities of 5.8e-5 for the Fock encoding |0> vs |5> and 4.2e-3 for the S=2,N=1 binomial code, obtained with a hidden-Markov-model/MLE classifier. The paper also proposes Eq. (4) to describe the tradeoff between repeated readouts and state relaxation and compares Fig. 5 data to this model. The central claim is that multilevel encoding plus repeated readout yields record-level measurement fidelities in circuit QED.

Significance. If the reported numbers are robust, this is an important experimental advance: the 5.8e-5 assignment infidelity and the 1.4e-3 direct resolution of gate errors are substantially better than standard transmon readout, and the multilevel-encoding principle is general and applicable to other platforms. The strength of the paper is that the headline infidelities are direct experimental assignments with error bars, not theory extrapolations; the QND-ness of the readout is separately calibrated (PD = 0.02%); and the shelving demonstration cleanly separates measurement error from gate error in a way that is useful for SPAM characterization. The main caveats concern the treatment of rare reset failures and the circularity of the model comparison, as detailed below.

major comments (3)
  1. [Supplemental Material, 'Reset errors'; main text §IV, Fig. 6 and Table I] The paper explicitly states that 0.2% of 1.8 million repeated-measurement experiments contain a 'stuck' ancilla reset requiring five or more feedforward iterations, and that the mechanism of these high-level excitations is not apparent from the measurements. These events are not modeled in the hidden-Markov-model transition matrix, whose emission matrix is calibrated only for g/e/f/h readouts. Since the results in Fig. 6 and Table I are not postselected, the reported Fock-code infidelity of 5.8e-5 is about 34 times smaller than the 2e-3 stuck-event rate. If even a few percent of stuck events corrupt the MLE assignment, these rare correlated events would account for essentially the entire reported error budget. The authors should provide a conditional breakdown of misassignments by reset history (e.g., with and without stuck events), or otherwise demonstrate that the headline infidelities are not dominated by this unmodeled process. As written, the central quantitative claim is not yet supported.
  2. [§IV, Eq. (4); Supplemental Material, 'System parameters'] The theory curves in Fig. 5 are not an independent test of the photon-loss model: delta0 and delta1 are taken as the first points of the same curves being compared, and kappa_up tau is fitted to the 1-photon curve. The abstract's statement that the tradeoff is 'shown to be consistent with the photon-loss model' should therefore be qualified. The measured infidelities themselves are direct assignments and are not invalidated by this circularity, but the model agreement in Fig. 5 does not add independent confirmation. The authors should either refit the model to independent calibration measurements or explicitly state which parameters are fixed a priori.
  3. [Supplemental Material, 'State preparation'] The argument that preparation errors can be neglected relies on the assumption that the only significant residual error is photon loss during the final check measurement, which the protocol is robust to. This is plausible for the Fock and binomial codes studied, where a single loss event from |L> or |2>/|3> does not cross the S/ar S boundary. However, the argument is not fully quantitative: the paper does not provide an upper bound on other preparation errors (e.g., residual thermal population or mapping-pulse miscalibration) that could enter the reported infidelity. Adding an explicit error budget for state preparation would strengthen the claim that the quoted infidelities are measurement infidelities.
minor comments (5)
  1. [Fig. 2 caption] Typo: 'beloning' should be 'belonging'.
  2. [Supplemental Material, 'Simultaneous number-selective pulses'] Typo: 'simulataneous' should be 'simultaneous'.
  3. [§IV, Fig. 5] The caption mentions dashed and dash-dotted lines but does not define which is which in terms of the two contributions (majority-vote vs. state-transition). Please add a legend or textual description.
  4. [Table I] The notation 'S=2,N=1 binomial code' is used without a definition in the main text; a brief sentence citing Ref. [25] and explaining S and N would help readers.
  5. [§III] The phrase 'the measurement is only corrupted when multiple errors occur' is slightly imprecise for the transmon case, where a single relaxation event from |f> still leaves the state in the excited manifold; consider rewording to 'a single relaxation event does not corrupt the encoded bit' for clarity.

Circularity Check

1 steps flagged · score 6.0 of 10

Theory-curve agreement in Fig. 5 is partly a fit because δ0, δ1, and κ↑τ are taken from the same curves, but the headline Table I infidelities are measured assignments and not circular.

  1. fitted input called prediction [Section IV, Fig. 5 discussion; Supplemental Material, 'System parameters']
    "These trends are captured by their theoretical description [26] and follow the predictions of Eq. (4), shown in dashed and dash-dotted lines. ... The parameters used in the theory curves in Fig. 5 of the main text were determined as follows: δ0 = 5.2×10−2 and δ1 = 1.5×10−3 were taken as the first point in the 0-photon and 5-photon curve, respectively. κ↑τ = 2.7×10−4 was obtained by fitting to the 1-photon curve."

    In Eq. (4), the terms (N choose ceil(N/2)) δ0^{ceil(N/2)} and (N choose ceil(N/2)) δ1^{ceil(N/2)} reduce at N=1 to just δ0 and δ1. Setting δ0 and δ1 to the first data points of the 0-photon and 5-photon curves forces the N=1 theoretical points to coincide with those measurements, and fitting κ↑τ to the 1-photon curve forces the shape of that curve. The resulting dashed and dash-dotted lines are therefore partly a re-plotting of the data used to set the parameters, not an independent test of the photon-loss model. The abstract's statement that the tradeoff is 'shown to be consistent with the photon-loss model' is thus partially circular, although the measured infidelities in Table I and Fig. 6 do not themselves derive from Eq. (4).

full rationale

The main circularity is confined to the paper's theory-comparison claim. The supplemental material explicitly discloses that δ0 and δ1 are taken as the first points of the very curves being compared and that κ↑τ is fitted to the 1-photon curve; since Eq. (4) contains δ0 and δ1 linearly at N=1, the agreement with Eq. (4) is partly forced by construction. This is a real, but partial, circularity: it affects the 'consistency with the photon-loss model' statement in the abstract, not the central measured infidelities. The Table I numbers (5.8×10−5 for Fock |0> vs |5> and 4.2×10−3 for the S=2,N=1 binomial code) are obtained from actual MLE assignments of the recorded readout sequences and do not reduce to the fitted parameters of Eq. (4). The multilevel-suppression argument is also supported by a simple heuristic (robustness to L relaxation events) that does not depend on the self-cited proposal. Separately, the paper itself flags a non-circular limitation: 0.2% of reset attempts are 'stuck' due to high ancilla levels whose 'detailed mechanism ... is not apparent from our measurements,' and these events are postselected away only for the theory comparison. That is a validity concern for the headline number, but it is a correctness risk rather than a circularity. Overall, one secondary prediction is partially circular, while the primary experimental claims retain independent content, giving a score of 6.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The experiment is evaluated as a self-contained demonstration. The measured infidelities are direct outcomes, but the theoretical comparison (Eq. 4, Fig. 5) depends on three parameters that are taken or fitted from the same dataset, and the analysis assumes a bare photon-loss/gain model with 99.98% QND readout. No new entities are introduced.

free parameters (3)
  • delta0 (single-round readout error for |0> state) = 5.2e-2
    Taken as the first data point of the 0-photon curve in Fig. 5, then used in Eq. (4) to draw the theory curves. This makes the prediction partially a fit to the data it is compared with (Supplemental System parameters).
  • delta1 (single-round readout error for |1> state) = 1.5e-3
    Taken as the first data point of the 5-photon curve in Fig. 5; same circularity as delta0.
  • kappa_up tau (photon gain per readout cycle) = 2.7e-4
    Obtained by fitting to the 1-photon curve in Fig. 5; used in the theory model for state transitions during repeated measurement.
assumptions (4)
  • domain assumption The storage mode evolution between readouts is governed by independent photon loss and gain with rates kappa_down and kappa_up, with no other decoherence.
    This underlies Eq. (4) and the hidden Markov model used for the Bayesian classifier (Supplemental Maximum likelihood estimator).
  • domain assumption The ancilla readout is quantum nondemolition with respect to the storage mode, with a per-readout demolition probability PD=0.02% that is negligible.
    Calibrated in Supplemental Fig. S1; if the readout were more demolishing, repeated readouts would corrupt the stored state.
  • domain assumption Number-selective pulses can implement the conditional map U_map (Eq. S4) with negligible leakage outside the desired photon-number subspace.
    The protocol relies on the dispersive shift being large compared to the pulse bandwidth (Supplemental Simultaneous number-selective pulses).
  • domain assumption Heralded state preparation leaves residual errors only of the form of photon loss during the final check, which the encoded measurement is robust to.
    Argued in Supplemental State preparation; if preparation could produce errors outside this class, the quoted infidelities would be contaminated by preparation error.

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Pith. "Pith review of High-fidelity measurement of qubits encoded in multilevel superconducting circuits." pith.science (2026). https://pith.science/paper/ECUSL6JB

@misc{pith2026190801869,
  author       = {Pith},
  title        = {Pith review of: High-fidelity measurement of qubits encoded in multilevel superconducting circuits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ECUSL6JB}},
  note         = {Machine review of arXiv:1908.01869}
}
abstract

Qubit measurements are central to quantum information processing. In the field of superconducting qubits, standard readout techniques are not only limited by the signal-to-noise ratio, but also by state relaxation during the measurement. In this work, we demonstrate that the limitation due to relaxation can be suppressed by using the many-level Hilbert space of superconducting circuits: in a multilevel encoding, the measurement is only corrupted when multiple errors occur. Employing this technique, we show that we can directly resolve transmon gate errors at the level of one part in $10^3.$ Extending this idea, we apply the same principles to the measurement of a logical qubit encoded in a bosonic mode and detected with a transmon ancilla, implementing a proposal by Hann et al. [Phys. Rev. A \textbf{98} 022305 (2018)]. Qubit state assignments are made based on a sequence of repeated readouts, further reducing the overall infidelity. This approach is quite general and several encodings are studied; the codewords are more distinguishable when the distance between them is increased with respect to photon loss. The tradeoff between multiple readouts and state relaxation is explored and shown to be consistent with the photon-loss model. We report a logical assignment infidelity of $5.8\times 10^{-5}$ for a Fock-based encoding and $4.2\times 10^{-3}$ for a QEC code (the $S=2,N=1$ binomial code). Our results will not only improve the fidelity of quantum information applications, but also enable more precise characterization of process or gate errors.

Figures

Figures reproduced from arXiv: 1908.01869 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. A high-level description is given in 4(a), with [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (4 more)
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png]
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.