Heisenberg-limited calibration of entangling gates with robust phase estimation

Akel Hashim; David I. Santiago; Irfan Siddiqi; J. P. Marceaux; Kenneth Rudinger; Kevin C. Young

REVIEW 3 major objections 4 minor 40 references

Heisenberg-limited calibration of entangling gates with robust phase estimation

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Robust phase estimation, extended to multi-qubit gates, provides Heisenberg-limited estimates of coherent errors in a controlled-Z gate, and using these estimates in a classical optimization loop improves the gate's diamond distance by…

desk verdict A solid, honest experimental demonstration of RPE-based calibration for a two-qubit gate; the Heisenberg-limited claim is real but conditional on a commutative error model that the paper itself shows is only approximately valid. read the letter →

arxiv 2502.06698 v2 pith:GKERAC7C submitted 2025-02-10 quant-ph cs.SYeess.SY

Kenneth Rudinger , J. P. Marceaux , Akel Hashim , David I. Santiago , Irfan Siddiqi , Kevin C. Young This is my paper · ORCID

classification quant-phcs.SYeess.SY MSC 81P68 PACS 03.67.-a

keywords quantumcomputingrobustphaseestimationgatecalibrationcoherenterrorsHeisenberglimitcontrolled-Zsettomographysuperconductingqubits

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

Robust phase estimation, a protocol originally designed for single-qubit gates, can be extended to two-qubit entangling gates and yields estimates of coherent errors whose uncertainty shrinks as O(1/2^k) — the Heisenberg limit, where each doubling of sequence length halves the error. The authors show that for a controlled-Z gate, three RPE experiments measuring linear combinations of the ZI, IZ, and ZZ phase parameters suffice to recover all three parameters, and that a classical optimizer can then adjust drive amplitude and frequency to minimize the estimated ZZ error while virtual Z rotations fix the local phases. On a superconducting transmon processor, this procedure reduced the estimated ZZ angle error by more than three orders of magnitude and decreased the diamond distance to the ideal gate by about a factor of two, while infidelity remained essentially unchanged because it is dominated by stochastic noise. The method is designed to transfer to other entangling gates such as CNOT or iSWAP and to other hardware platforms such as ion traps and neutral atoms.

What carries the argument

The central object is the robust phase estimation protocol acting on the spectral decomposition of a multi-qubit unitary, U = Σ_a $e^{{iφ_a}}$ |E_a⟩⟨E_a|. RPE prepares a superposition of two eigenstates, applies the unitary 2^k times, and measures in-phase and quadrature observables to estimate the relative phase φ_{a,b} with an error bounded by π/$2^{{k+1}}$ as long as out-of-model errors stay below a threshold. Branch-unwinding integers n_k and an angular-historical consistency check make the estimate robust to SPAM and stochastic errors. For the controlled-Z gate, the authors assume a commutative model exp(-i/2(θ_ZI ZI + θ_IZ IZ + θ_ZZ ZZ)), whose eigenstates are independent of the parameter values; three RPE experiments measure the relative phases φ_{00,01}, φ_{10,11}, and φ_{01,11}, which are linear combinations of the three parameters, and inverting that linear system yields Heisenberg-limited estimates of the parameters themselves.

What would settle it

Run the same RPE-based calibration on a two-qubit gate for which gate set tomography shows non-commuting error rates (e.g., ZX or YI) comparable to the ZZ error rate, and check whether the diamond distance after calibration tracks the RPE cost function. If a full tomography-based optimization finds a substantially different amplitude-frequency operating point, or if the diamond distance does not improve when the RPE phases are minimized, the commutative-model assumption is violated and the central claim fails.

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Extended reading notes

Core claim

The central claim is that robust phase estimation, extended to multi-qubit gates, provides Heisenberg-limited estimates of coherent errors, and that these estimates are sufficient objective functions for calibrating an entangling gate. Working with a commutative error model in which the controlled-Z gate is generated only by ZI, IZ, and ZZ Pauli terms, the paper constructs three RPE experiments whose measured relative phases are linear combinations of the three model parameters; inverting the linear system gives estimates of θ_ZI, θ_IZ, and θ_ZZ with uncertainty decaying as O(1/2^k). Experimentally, the authors use a CMA-ES optimizer to minimize the absolute error in θ_ZZ over the drive amplitude and frequency, then apply virtual Z rotations to correct the local phases. Gate set tomography confirms the improvement: the diamond distance to the target operation drops by roughly a factor of two, the RPE-measured angle error improves by over three orders of magnitude, and the infidelity does not change, consistent with the claim that infidelity is only quadratically sensitive to coherent errors. The paper also shows that the remaining diamond-distance error is dominated by small non-commuting Hamiltonian terms, such as ZX and YI interactions, that the commutative model does not measure.

Load-bearing premise

The load-bearing assumption is that the effective unitary of the CZ pulse is well described by a commuting set of Pauli terms, ZI, IZ, and ZZ, whose eigenstates do not depend on the parameter values; if non-commuting errors such as ZX or YI are significant, the RPE-estimated phases do not capture all the dominant coherent errors and the optimization can miss the true operating point.

Editorial extensions

If this is right

Calibration of a CZ gate reduces to a classical optimization loop: three RPE experiments supply the phase parameters, and the optimizer adjusts controls to minimize a linearly sensitive cost function, eliminating the need for full process tomography during calibration.
Because RPE is Heisenberg-limited, each additional depth doubling roughly halves the parameter uncertainty, so reaching a given coherent-error precision requires dramatically fewer shots than randomized benchmarking or standard process tomography.
The same three-experiment RPE design transfers to CNOT, iSWAP, and other entangling gates in ion-trap or neutral-atom platforms, provided the dominant errors fit a commuting Pauli model.
Infidelity and randomized benchmarking are weak calibration targets for coherent errors: in this experiment the ZZ angle error dropped by over 1000x while infidelity stayed flat, confirming that RB-style metrics would not have detected the improvement.
After calibration, the residual diamond-distance error is set by small non-commuting ZX and YI terms, which become the next limiting factor and a target for future estimation and correction.

Reading between the lines

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

A natural next step is to design RPE experiments sensitive to the non-commuting terms that GST reveals after calibration; the single-qubit axis-angle estimator suggests a route, though the multi-qubit Pauli structure makes closed-form estimators difficult.
The total calibration time of about 3.75 hours, of which only 8 minutes was classical optimization, is dominated by waveform recompilation; switching to FPGA-based control or a better-suited optimizer could cut this substantially.
Combining RPE with an interleaved benchmarking or amplified-error protocol could separate coherent from stochastic errors in situ, allowing calibration to target improvements in worst-case diamond distance rather than average infidelity.
A testable prediction of the commutative model is that the RPE-optimal operating point coincides with the maximum of the conditionality landscape only along a one-dimensional ridge; the optimizer's observed trajectory, which explores a single dominant component, is consistent with this and could be verified on other devices.

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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 practical result is real: this paper closes the loop between RPE estimates and a classical optimizer to calibrate a CZ gate, and shows a factor-of-two improvement in diamond distance, validated independently with GST. The angle-error improvement is dramatic on paper, though the authors are right to attach a caveat about the large decoherence-limited uncertainty region. This is a useful engineering contribution, not a conceptual breakthrough: the multi-qubit RPE estimator is from Russo et al. 2021, and the novelty here is the workflow and the hardware demonstration.

The paper earns credit for honesty. It states plainly in Sec. II.1 that the method assumes a commutative model with only ZI, IZ, and ZZ terms, and the Conclusions call that assumption critical. The GST data in Fig. 3(a) show non-commuting terms like ZX and YI that persist and become relatively more important after calibration. The stress-test concern lands: if those terms are significant, the prepared states are not eigenstates of the true unitary, the measured cos/sin signals are biased, and the linear inversion in Eq. 16 does not return the true Hamiltonian coefficients. The consistency check in Eq. 13 may not catch a bias below its detection threshold. So the headline "Heisenberg-limited estimates of coherent errors" is conditional on a model that this paper shows is only approximately true. That does not kill the paper, but it should temper the claim.

The Heisenberg-scaling proof in Appendix C is the weakest section. It asserts that because the Hilbert-space dimension does not enter the single-qubit theorem, the multi-qubit case follows by direct application. That is too quick, especially since the eigenstates themselves become uncertain when the model is wrong. A rigorous bound accounting for model mismatch, or at least a numerical study, would strengthen the claim.

Minor issues: the experimental data are not public, and the linked code has no commit hash, so exact reproducibility is limited. The paper does not hide its flaws, though, and the central conclusion — that RPE can serve as a practical, high-precision calibration tool for two-qubit gates — holds up despite the model caveat.

This paper deserves a serious referee. The demonstration is concrete, the validation is independent, and the limitations are acknowledged. I would not desk-reject it. With requested revisions on the Heisenberg-scaling proof and the model-mismatch discussion, it could be a useful reference for anyone building calibration routines.

Referee Report

3 major / 4 minor

Summary. The paper extends robust phase estimation (RPE) from single-qubit gates to multi-qubit gates, specializing to a controlled-Z gate whose coherent errors are modeled by a commuting Hamiltonian with only ZI, IZ, and ZZ terms. It specifies three RPE experiments whose measured relative phases are linearly inverted (Eq. 16) to estimate theta_IZ, theta_ZI, and theta_ZZ, then uses a CMA-ES classical optimizer to minimize |theta_ZZ + pi/2| while correcting the local phases with virtual Z rotations. On a superconducting transmon processor, the authors report an improvement of more than a factor of 1000 in the RPE-estimated angle error and about a factor of 2 in the GST-estimated diamond distance, with no significant infidelity change. The paper claims Heisenberg-limited calibration and states that the methods apply to other entangling gates and hardware platforms.

Significance. If the central claim holds, the work is significant because it offers an efficient calibration primitive that directly targets coherent errors, which are only quadratically visible to randomized-benchmarking-style metrics. The paper's strengths include the independent GST validation, the use of open-source analysis software (pyRPE and PRAQTICE), and the honest discussion of the limitations of the commutative model and of RB as a calibration metric. The experimental demonstration is credible, but the headline theoretical claim is narrower than stated: the Heisenberg-limited estimator is defined and proven only within the commuting model, and the GST data show that non-commuting coherent errors remain and become relatively more important after calibration. The manuscript therefore needs to qualify its central claims and quantify the model-mismatch bias before the advertised scope is justified.

major comments (3)

[II.1, Eq. (15) and Fig. 3(a)] The central estimator is defined under the commutative model U~CZ = exp[-(i/2)(theta_ZI ZI + theta_IZ IZ + theta_ZZ ZZ)], whose eigenstates are fixed computational-basis states independent of the parameter values. Equation (16) inverts the measured relative phases into theta coefficients only under this model. If non-commuting terms such as ZX or YI are present, the prepared superposition states in Table II are not equal superpositions of true eigenstates, the cos/sin signals used in Eq. (9) are biased, and the inverted parameters are not exactly the Hamiltonian coefficients. The paper's own GST data in Fig. 3(a) show such non-commuting terms persisting and becoming relatively more important after calibration, and Sec. IV explicitly calls the commutative assumption "critical." The abstract and Sec. I claim estimates of coherent errors in multi-qubit gates without this restriction. Please either bound the bias using the GST Hamiltonian-error rates or qualify all central claims as applying only to the commuting model.
[III.C] The reported improvement of "a factor of over 1000" in the angle error is measured with the same RPE estimator used in the cost function J(theta_ZZ) of Eq. (18), so this number is partly self-referential; the independent GST metric improves only by about a factor of 2. The text notes that the decoherence-limited uncertainty region is large, but the manuscript does not state how much of the 1000x factor survives when that uncertainty and the non-commuting residuals of Fig. 3(a) are accounted for. Please report the RPE angle-error estimate with its operational uncertainty range from Eq. (12) rather than presenting a single improvement factor, and explain explicitly the distinction between the RPE-measured angle and the independent GST diamond-distance improvement.
[Appendix C] The claimed Heisenberg scaling for multi-qubit RPE is asserted by direct application of Theorem 2 of [1], with the statement that the dimensions of the Hilbert space are not used in the proof. This is not a derivation of the multi-qubit estimator's behavior as defined in Sec. II, and the linear-combination argument in Eq. (C1) assumes the phase estimates are unbiased and statistically independent, which again relies on the commutative model and on the post-selection procedure of Eq. (8). Please provide a self-contained proof or a precise citation to a result that covers the multi-qubit RPE protocol presented here, including the effect of post-selection and the multi-experiment linear inversion.

minor comments (4)

[Fig. 1 caption] The caption contains a typo: "controsl" should be "controls."
[III] The sentence "on the Advanced Quantum Testbed" appears as "at the the Advanced Quantum Testbed"; the duplicated article should be removed.
[III.B] The statement that parameter sweeps "scale very poorly as the number of interdependent parameters increases" would benefit from a citation or a brief quantitative illustration; as written it is an unsupported assertion.
[II.1] The phrase "GST results (see Fig. 3(a)) verify this expectation" appears shortly before results showing that non-commuting terms become significant after calibration; consider rephrasing to distinguish leading-order validity from the non-negligible residuals that the paper itself reports.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-referential metric in reporting the RPE-optimized angle error; the central calibration claim is independently validated by GST.

fitted input called prediction [Sec. III.B, Eq. (18), and Sec. III.C, first Results paragraph]
"We use a cost function that is the absolute difference between the target θZZ phase of −π/2 and the RPE estimate θZZ: J(θZZ) = |θZZ + π/2|. ... Our most significant improvement is in the angle error as measured by RPE, where we found that the calibrated gate improved by a factor of over 1000"

The classical optimizer selects control parameters by minimizing J(θZZ), which is exactly the RPE-estimated θZZ error. Reporting the before/after RPE-measured angle error as the headline improvement is therefore reporting the decrease of the optimized objective, not an independent confirmation of the estimator. That decrease is forced by construction once the optimizer converges. The paper's external validation is the GST diamond-distance reduction, which is independent of the RPE cost function and keeps the central claim non-circular.

full rationale

The central claim—that RPE provides Heisenberg-limited estimates of coherent CZ errors and that using these estimates in a classical optimization loop improves the gate—is not circular. The estimator theory is inherited from published RPE results [1,16,18], and Appendix C applies Theorem 2 of [1] to the multi-qubit setting; the linear inversion in Eq. (16) is an explicit model, not a hidden reuse of the target. The experimental validation is primarily gate set tomography: diamond distance improves by roughly a factor of 2 while infidelity is unchanged, which is external to the RPE cost function. The commutative-model assumption of Eq. (15) is explicitly acknowledged as critical and is checked against GST data, making it a validity condition rather than a circularity. The one self-referential element is that the reported RPE angle-error improvement is the very quantity minimized by Eq. (18); this is a fitted-metric presentation, not an independent confirmation. Score 2 reflects this minor self-reference while the central claim remains independently grounded.

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

The central claim rests on the commutative error-model assumption (Eq. 15), the asserted transfer of the single-qubit Heisenberg-scaling theorem to multi-qubit RPE (Appendix C), and the standard RPE post-selection robustness. No free parameters are fit to data; the model parameters are estimated by RPE, not fitted. No new entities are introduced.

assumptions (3)

domain assumption The effective unitary of the CZ gate is generated only by the commuting Pauli terms ZI, IZ, and ZZ, so its eigenvectors are independent of the parameter values.
This is the core modeling assumption of Eq. 15 in Sec. II.1. The paper itself labels it 'critical' in the Conclusions and notes GST reveals other terms that become significant after calibration.
domain assumption Theorem 2 of Kimmel et al. [1] on Heisenberg-limited single-qubit RPE applies unchanged to multi-qubit RPE because Hilbert-space dimension is not used in its proof.
Appendix C asserts this transfer without a full derivation. It underpins the headline claim of Heisenberg-limited estimation for the multi-qubit phases.
domain assumption Post-selecting measured outcomes on the two eigenstates of interest (Eq. 8) yields unbiased estimates of the in-phase and quadrature expectation values despite leakage out of that subspace.
Used in Sec. II to define the empirical expectation values. This is a standard RPE robustness property inherited from prior literature.

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Cite this review

Pith. "Pith review of Heisenberg-limited calibration of entangling gates with robust phase estimation." pith.science (2026). https://pith.science/paper/GKERAC7C

@misc{pith2026250206698,
  author       = {Pith},
  title        = {Pith review of: Heisenberg-limited calibration of entangling gates with robust phase estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GKERAC7C}},
  note         = {Machine review of arXiv:2502.06698}
}

read the original abstract

The calibration of high-quality two-qubit entangling gates is an essential component in engineering large-scale, fault-tolerant quantum computers. However, many standard calibration techniques are based on randomized circuits that are only quadratically sensitive to calibration errors. As a result, these approaches are inefficient, requiring many experimental shots to achieve acceptable performance. In this work, we demonstrate that robust phase estimation can enable high-precision, Heisenberg-limited estimates of coherent errors in multi-qubit gates. Equipped with an efficient estimator, the calibration problem may be reduced to a simple optimization loop that minimizes the estimated coherent error. We experimentally demonstrate our calibration protocols by improving the operation of a two-qubit controlled-Z gate on a superconducting processor, and we validate the improved performance with gate set tomography. Our methods are applicable to gates in other quantum hardware platforms such as ion traps and neutral atoms, and on other multi-qubit gates, such as CNOT or iSWAP.

Figures

Figures reproduced from arXiv: 2502.06698 by the authors.

**Figure 1.** Components of RPE-based calibration. Panel (a) shows the layout of the superconducting chip used in our experiment. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Cost function landscape exploration and optimizer trajectories. Panel (a) depicts a coarse-grained sweep of the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Metrics before and after calibration. Panel (a) shows the during-gate Hamiltonian error rates and panel (b) shows [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Diamond distance and infidelity before and after [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

angular histor- ical

as a enhancement of the Heisenberg-limited phase es- timation protocol of [15] that added robustness both to incoherent noise in the logical gates and to errors in state preparation and measurement. Its original formulation estimatedtheuniquerelativephaseofasingle-qubitgate, but has since been expanded to measure relative phases between arbitrary eigensta...

work page

[2] [2]

However, GST can require hundreds to thousands of unique circuits to achieve high- precision estimates of the target coherent errors

overcomes this by running deep circuits consisting of many repeated gate applications to amplify errors and achieve Heisenberg scaling; a GST experiment with max- imum circuit depth O(L) can yield an estimate whose uncertainty scales asO(1/L). However, GST can require hundreds to thousands of unique circuits to achieve high- precision estimates of the tar...

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[3] [3]

The CZ gate has a target unitary operation UCZ = I ⊗ |0⟩ ⟨0| + Z ⊗ |1⟩ ⟨1|, or written in an equivalent exponential notation UCZ = exp − i 2 − π 2 II + π 2 ZI+ π 2 IZ− π 2 ZZ

Controlled-Z RPE We now specialize our discussion to the specific controlled-Z (CZ) RPE experiment design employed in this work. The CZ gate has a target unitary operation UCZ = I ⊗ |0⟩ ⟨0| + Z ⊗ |1⟩ ⟨1|, or written in an equivalent exponential notation UCZ = exp − i 2 − π 2 II + π 2 ZI+ π 2 IZ− π 2 ZZ . (14) An experimental implementation of the CZ unita...

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[4] [4]

Helsen, I

J. Helsen, I. Roth, E. Onorati, A. Werner, and J. Eisert, PRX Quantum 3, 020357 (2022)

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[5] [5]

Ry(−π 2) |0⟩ U2k CZ |0⟩ Ry(π

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[6] [6]

Rx(−π 2) ϕ10,11 = θIZ − θZZ |0⟩ X U2k CZ X |0⟩ Ry(π

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[7] [7]

Ry(−π 2) |0⟩ X U2k CZ X |0⟩ Ry(π

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[8] [8]

Details of the six classes of circuits used to characterize theZ-type phases of the controlled-Z gate

Rx(−π 2) ϕ01,11 = θZI − θZZ |0⟩ Ry(π 2) U2k CZ Ry(−π 2) |0⟩ X X |0⟩ Ry(π 2) U2k CZ Rx(−π 2) |0⟩ X X Table A1. Details of the six classes of circuits used to characterize theZ-type phases of the controlled-Z gate. Each pair of circuits in a row implements a particular state preparation and two measurements to estimate the phase in the first column. See als...

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