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Mitigating errors in state preparation and measurement with noncomputational states

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Measuring the $|1\rangle$–$|2\rangle$ Rabi oscillation of a transmon pins down the state-preparation error, fully constraining measurement noise models and separating preparation, gate, and readout mitigation.

desk verdict RabiEF-anchored SPAM splitting is a real advance for dynamic-circuit error mitigation; the protocol holds up in simulation, and the hardware evidence is suggestive but not complete. read the letter →

arxiv 2506.09145 v2 pith:VZPLYBBU submitted 2025-06-10 quant-ph

Conrad J. Haupt , Almudena Carrera Vazquez , Laurin E. Fischer , Stefan Woerner , Daniel J. Egger This is my paper
classification quant-ph PACS 03.67.Lx
keywords errormitigationstate-preparationnon-computationalstatesRabiEFmid-circuitmeasurementnoisemodelPaulilearningtransmonqubit
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

Error-mitigation methods that rely on learned noise models cannot separate state-preparation errors from measurement errors when only the two computational states are used; there is a fundamental gauge ambiguity. This paper shows that the extra $|1\rangle$ and $|2\rangle$ levels of a superconducting transmon provide the missing information. A RabiEF experiment, which drives oscillations between $|1\rangle$ and $|2\rangle$ and compares signals with and without an initial pi-pulse, directly measures the thermal population $p_{\rm sp}$ that defines the state-preparation error. Knowing $p_{\rm sp}$ anchors the measurement cycle-benchmarking noise model, so the assignment-error fidelity $f_a$, the state-error fidelity $f_s$, the correlated-error fidelity $f_c$, and the state-preparation fidelity $f_{\rm sp}$ can be learned separately. The result is that state-preparation, gate, and readout errors can be mitigated independently, including for circuits with mid-circuit measurements, preventing the unphysical over-corrections that joint state-preparation-and-measurement mitigation can produce.

What carries the argument

The load-bearing mechanism is the RabiEF experiment on the non-computational $|1\rangle$–$|2\rangle$ subspace. A thermal state with population $p_{\rm sp}$ in $|1\rangle$ is driven by an $R_{12}(\theta)$ rotation between $|1\rangle$ and $|2\rangle$; executing the sequence with and without an initial pi-pulse yields two oscillating signals whose fitted amplitudes give $p_{\rm sp} = a_{\rm A}/(a_{\rm A}+a_{\rm B})$, where A and B label the no-pi and pi signals. Because readout assignment errors affect both amplitudes in the same way, the ratio is first-order insensitive to measurement noise, giving a direct estimate of the state-preparation error. This estimate enters the measurement-cycle-benchmarking fit as the known fidelity $f_{\rm sp}$, splitting the product $f_{\rm sp}f_s f_a$ in Eq. (10) into individually learned factors. The twirled measurement model, with the classical bit represented as a second qubit in the Pauli transfer matrix, together with the RabiEF anchor, converts an underdetermined noise model into a fully specified one.

What would settle it

Prepare a qubit state with an independently calibrated state-preparation error, for example by a verified coherent excitation pulse, run RabiEF on it, and check that the estimated $p_{\rm sp}$ matches the calibrated value within the reported uncertainty; a mismatch, or a systematic drift of the estimate when the readout assignment error is deliberately varied, would show the anchor is biased.

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Extended reading notes

Core claim

The paper's central claim is that non-computational states break the no-go result for noise learning in the computational subspace: when the qubit is allowed to visit its $|1\rangle$ and $|2\rangle$ levels, the state-preparation error $p_{\rm sp}$ becomes directly measurable instead of being entangled with readout noise. Concretely, the measurement noise model written as a Pauli transfer matrix product $\Gamma_m = \Gamma_{\rm CX}\Gamma_{\rm s}\Gamma_{\rm a}\Gamma_{\rm c}$, and the full measurement cycle benchmarking expression of Eq. (10), contain only products of fidelities such as $f_a f_s$ and $f_a f_s f_{\rm sp}$; cycle benchmarking alone can learn the products but not the individual factors. The authors show that a RabiEF experiment supplies $p_{\rm sp}$, and therefore $f_{\rm sp}=1-2p_{\rm sp}$, as an independent anchor. With that anchor, the remaining fidelities $f_a$, $f_s$, and $f_c$ are separately determined, the combined state-preparation and measurement (SPAM) error is split in mitigation, and the same noise model applies to mid-circuit measurements. Hardware data on a GHZ-type stabilizer show that joint SPAM mitigation over-corrects the global observable $X^{\otimes n}$ because state-preparation errors affect only a subset of qubits, while the split mitigation restores physical expectation values consistent with the learned gate noise.

Load-bearing premise

The load-bearing premise is that the RabiEF amplitude ratio estimates $p_{\rm sp}$ without significant bias from readout assignment errors, gate errors, or shot-to-shot state fluctuations, and that the same $p_{\rm sp}$ describes the initial state in the RabiEF, measurement cycle benchmarking, and mitigation circuits; the paper supports this with numerical simulations and calls for an analytical error analysis.

Editorial extensions

If this is right

  • Joint SPAM mitigation no longer over-corrects global observables: the split mitigator restores expectation values consistent with the learned gate noise in circuits where joint TREX mitigation produces unphysical values.
  • Measurement noise models for both final and mid-circuit measurements become fully specified, so probabilistic error cancellation can mitigate dynamic circuits with classical feedforward, as demonstrated on a noisy teleportation circuit.
  • Fast active qutrit resets are sufficient in place of slow thermal resets; simulations show the RabiEF estimate of $p_{\rm sp}$ stays within about five percent relative error when the $|2\rangle$ population is actively reset.
  • The approach generalizes to other hardware platforms that have accessible non-computational states, such as spin qutrits and trapped ions.
  • The paper's random-X amplification procedure keeps the RabiEF anchor usable as state-preparation fidelities improve and $p_{\rm sp}$ becomes very small.

Reading between the lines

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

  • If the RabiEF ratio carries a small bias from gate errors or state fluctuations, the split noise model inherits that bias; a full analytical error-propagation treatment, which the authors call for, would turn the current numerical support into a bound on the final mitigation error.
  • The same anchoring idea could be extended to a full qutrit readout model, making leakage and $|2\rangle$ populations learnable rather than assumed negligible.
  • Combining this anchor with self-consistent gate-set Pauli-noise learning, the paper's stated outlook, would let one calibrated noise model predict errors for circuits whose gate structures were never individually benchmarked.
  • The amplification trick from Appendix G could double as an independent consistency check: comparing the amplified and unamplified RabiEF estimates would expose state-preparation instabilities between shots.
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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

2 major / 5 minor

Summary. The paper proposes using non-computational states of superconducting transmons (the |1⟩ and |2⟩ levels) to measure the state-preparation error p_sp with a RabiEF experiment, then uses this value to split the SPAM fidelity products that arise in a twirled measurement cycle benchmarking (MCB) protocol. Section II motivates the problem by showing that TREX over-corrects a global X⊗n observable when state-preparation and measurement errors are jointly mitigated. Section III C derives a Pauli-transfer-matrix model of noisy (mid-circuit) measurements with state, assignment, and correlated errors, and shows that with an external anchor f_sp the model is fully specified. Section IV reports simulations: RabiEF with fast qutrit resets, MCB noise learning with relative errors below 0.1%, SPAM mitigation of a stabilizer circuit, and PEC mitigation of a three-qubit teleportation circuit. The hardware demonstration in Section II is presented as motivation; the four-qubit point lies outside the predicted CNOT-noise range. The central claim is that non-computational states overcome no-go theorems for SPAM noise learning.

Significance. If the RabiEF anchor can be rigorously established, this is a valuable contribution: the PTM model in Sec. III C is explicit and self-consistent, and the simulated MCB experiments recover f_a, f_s, f_c, and f_sp with relative errors below 0.1% (Table II), including a second MCB pass that separates a fast-reset fidelity. The teleportation simulation in Sec. IV D demonstrates a concrete use of the split SPAM model with PEC, and the paper is honest about the n=4 hardware outlier and the non-convexity of the CNOT-noise bound. The main weakness is the numerical support for the RabiEF anchor, which is load-bearing for the central claim.

major comments (2)
  1. [Appendix A2, Eq. (A1)] The simulation does not test the claimed insensitivity of RabiEF to assignment errors. Because the |1⟩ and |2⟩ states are mapped to the same Gaussian N(μ=1, σ_m), the no-pi signal s_notpi(θ) is independent of θ; the R12(θ) gate only redistributes population between states that are detected identically, so the fitted amplitude a_notpi is sampling noise rather than a thermal-population signal. The observed centering of the estimates in Fig. 9 therefore does not support the ratio in Eq. (A1) as an unbiased estimator of α. This matters because Eq. (10) uses p_sp from RabiEF to split the products f_s f_c and f_s f_sp, so any bias in \hat p_sp propagates directly into the mitigated observables. Please provide either a corrected simulation with distinct |1⟩ and |2⟩ readout responses, or the analytical treatment called for in Sec. V, before the ground-truth claim is taken as established.
  2. [Sec. II and Fig. 1; Appendix C] The hardware demonstration is incomplete because the four-qubit point in the SPAM-mitigation plot lies outside the shaded CNOT-noise range, and that range is computed by solving non-convex optimization problems whose global optimality is not certified (Appendix C). The authors' attribution to out-of-model errors is plausible, but as presented the quantitative support for the claim that the split-mitigated values are consistent with the CNOT noise levels is weaker for n=4 than for n=6,8. Please quantify the systematic error using the framework of Ref. [38], or report the spread of the shaded region over many optimization restarts, and state explicitly whether the four-qubit outlier is compatible with that uncertainty.
minor comments (5)
  1. [Secs. II and III B] The TREX and split-SPAM results are referenced as Fig. 1(e), but the caption identifies the SPAM error mitigation plot as Fig. 1(c). Please correct the cross-references.
  2. [Sec. IV A] The phrase "true p_sp of 0.01206%" should read "true p_sp of 0.01206" (i.e., 1.206%), since p_sp is a probability rather than a percentage.
  3. [Sec. III C] The sentence introducing the learned products uses the same symbol f for two different quantities, namely f = f_a sqrt(f_c f_s) and f = f_c f_s. Please use distinct symbols to avoid ambiguity.
  4. [Appendix C] The constraint that Pauli fidelities remain physical should be stated as −1 ≤ λ ≤ 1, not only λ ≤ 1.
  5. [Appendix A2 and Eq. (A1)] The fit function is written as a sin²(θ+c)+b in Eq. (A1) but as a sin²(bθ)+c in the Fig. 1 caption; the offset and phase notation should be harmonized. Also, the text contains a doubled word in "The the normalization constraint" after Eq. (11).

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the RabiEF f_sp anchor is an independent external measurement, and the MCB fidelity-splitting in Eq. (10) is algebraic with that anchor supplied from outside the fit.

full rationale

Walking the derivation chain, the central move is to measure f_sp = 1 - 2 p_sp with a RabiEF experiment and use it as an external anchor to split the products f_a f_s f_sp, f_a sqrt(f_c f_s), and f_c f_s learned from measurement cycle benchmarking (Eqs. 7-10). This splitting is algebraically underdetermined without f_sp, and the paper supplies f_sp from an independent experiment (Sec. III A and Appendix A), not from the MCB decay fits. The mitigation formula Eq. (1) likewise uses independently measured f_sp values, so the corrected observable is not an output reinserted as an input. The no-go/gauge discussion cites external learnability results [29, 31], and those are used as background facts rather than as self-referential justification. There are minor self-citations ([13], [32], [43]) for background context and simulation methodology, but none carries the load-bearing step: the f_sp anchor is established by RabiEF with external references [35, 36] and by the hardware data in the paper. One non-circular caveat is that Appendix A2's numerical support maps |1> and |2> to a single readout Gaussian, which would suppress the RabiEF oscillation, so the claimed insensitivity to assignment errors is not as strongly tested as stated; the authors themselves call for an analytical treatment in Sec. V. This is a correctness or evidence concern, not a circularity, because the estimator is neither defined in terms of the MCB model nor fitted to the mitigated observables. Overall the derivation is self-contained against external benchmarks, and no specific circular reduction can be exhibited.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central claim rests on a small number of noise-model assumptions (bit-flip noise after twirling, noise-before-measurement convention, first-three-level confinement) and on the RabiEF measurement being an unbiased external estimate of psp. No new physical entities are introduced. The protocol's own fitted output is psp, which is then used to split learned fidelity products.

free parameters (2)
  • psp_i (state-preparation error probability per qubit) = Hardware RabiEF: 2.79% to 8.75% (Table III); simulation: 1.946% slow reset, 1.206% fast qutrit reset
    Obtained by fitting the RabiEF amplitude ratio a_no_pi/(a_no_pi+a_pi). It anchors the split of SPAM fidelities; its accuracy is validated only numerically in Appendix A2.
  • f_c (correlated-error fidelity in simulations) = 0.995
    Hand-set in the simulations because the readout assignment matrix R does not constrain the correlated error; used as ground truth for testing, not fitted to the central claim.
assumptions (7)
  • domain assumption State-preparation errors are bit flips (X errors) on the qubit.
    Assumption (i) in Sec. III C; standard after twirling for the error model.
  • domain assumption Measurement errors after twirling are bit flips, i.e., Pauli strings with only I or X.
    Assumption (iii) in Sec. III C, based on measurement twirling from Ref. [39].
  • domain assumption Noise occurs before measurements without loss of generality.
    Assumption (ii) in Sec. III C; standard convention for modeling readout.
  • domain assumption The transmon stays within its first three levels, and the |2>-state population is negligible for slow passive resets.
    Sec. III A and Appendix A1; hardware table shows beta < 0.9%, but Sec. V acknowledges escape and confined states are not captured.
  • domain assumption The RabiEF amplitude ratio yields psp with first-order insensitivity to readout assignment and gate errors.
    Sec. III A and Appendix A2 give numerical evidence only; the authors state an analytical treatment would be useful (Sec. V).
  • domain assumption The prepared initial state is the same diagonal thermal state across RabiEF, MCB, and mitigation circuits.
    Sec. III B requires slow passive reset or stable qutrit reset; Table I shows fast qubit reset fails this assumption.
  • domain assumption Measurement noise is Markovian and described by the three channels Lambda_s, Lambda_a, Lambda_c.
    Sec. III C; the authors point to out-of-model errors as a possible cause of the n=4 hardware discrepancy.

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Cite this review

Pith. "Pith review of Mitigating errors in state preparation and measurement with noncomputational states." pith.science (2026). https://pith.science/paper/VZPLYBBU

@misc{pith2026250609145,
  author       = {Pith},
  title        = {Pith review of: Mitigating errors in state preparation and measurement with noncomputational states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VZPLYBBU}},
  note         = {Machine review of arXiv:2506.09145}
}
read the original abstract

Error mitigation has enabled quantum computing applications with over one hundred qubits and deep circuits. Many error mitigation methods are noise-aware, relying on a faithful characterization of the noise channels of the hardware. However, fundamental limitations lead to unlearnable degrees of freedom of the underlying noise models when considering qubits. Here, we show how to leverage non-computational states as an additional resource to learn state-preparation errors in superconducting qubits. This allows one to fully constrain the noise models. We can thus independently and accurately mitigate state-preparation errors, gate errors and measurement errors. Our proposed method is also applicable to dynamic circuits with mid-circuit measurements. This work opens the door to improved error mitigation for measurements, both at the end of the circuit and mid-circuit.

Figures

Figures reproduced from arXiv: 2506.09145 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (8 more)
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: The expected result with TREX is f 1−n sp , shown by the dashed purple line in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: Here, the distribution of measurements is again centered around the ideal value. Importantly, increasing the shots reduces the variance of the distribution. These results are expected given assignment errors, and potentially other imperfections, affect both aπ and a ✁…
Figure 11
Figure 11. Figure 11: FIG. 11. In the CNOT picture, the final measurement corre [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]

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Cited by 1 Pith paper

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