Easy better quantum process tomography

Kenneth Rudinger; Robin Blume-Kohout; Timothy Proctor

REVIEW 4 major objections 4 minor 1 cited by

Easy better quantum process tomography

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read SPAM-calibrated QPT removes most bias with only double the data.

desk verdict A clean, practical recipe for SPAM-corrected QPT with a correct gauge-invariant core; the headline fidelity claim is only empirically supported, and the novelty over the authors' earlier work should be stated. read the letter →

arxiv 2412.16293 v2 pith:ZMZ4XNPZ submitted 2024-12-20 quant-ph

Robin Blume-Kohout , Kenneth Rudinger , Timothy Proctor This is my paper · ORCID

classification quant-ph

keywords quantumprocesstomographySPAMerrorgaugefreedomlinearinversionmaximumlikelihoodestimationgateeigenvaluescalibration

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

Quantum process tomography (QPT) normally assumes the states used for preparation and the measurements used to probe a gate are exactly known, which is never true and biases the resulting gate estimate. This paper shows that running one extra "process tomography on nothing" experiment, measuring the SPAM operations on their own, gives enough information to correct that bias. The correction is explicit and simple: estimate the SPAM error superoperator from the calibration data and conjugate the standard linear-inversion estimate by its square root. The result is a gate estimate whose gauge-invariant properties, such as the eigenvalues of the process matrix, are exactly immune to SPAM error, at the cost of only twice as much data.

What carries the argument

The central object is the SPAM error superoperator $E = M_0^{-1} I S_0^{-1}$, estimated from calibration data as $\hat{E} = M_0^{-1} \hat{I} S_0^{-1}$. It captures how the true state preparations and measurement effects deviate from the a priori matrices $M_0$ and $S_0$. The load-bearing identity is the correction $\hat{G} = \hat{E}^{-1/2} \hat{G}_0 \hat{E}^{-1/2}$, which follows from choosing the factorization $\hat{E} = \alpha \beta$ with $\alpha = \beta = \hat{E}^{1/2}$ so that the SPAM error is split equally between states and effects; for overcomplete data the same logic uses Moore-Penrose pseudoinverses and rank-$d^2$ truncation of $\hat{I}$.

What would settle it

Run the protocol on a known gate with SPAM that drifts between the calibration and process circuits, then check whether the corrected gate eigenvalues remain accurate; if they drift with the delay time, the claimed SPAM immunity fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that standard linear-inversion QPT can be made SPAM-robust by measuring the SPAM calibration matrix $I = MS$, estimating the SPAM error superoperator $E = M_0^{-1} I S_0^{-1}$, and replacing the naive estimate $\hat{G}_0 = M_0^{-1} \hat{P} S_0^{-1}$ with $\hat{G} = \hat{E}^{-1/2} \hat{G}_0 \hat{E}^{-1/2}$. This symmetric square-root split assigns half the observed SPAM error to state preparation and half to measurement, which is the gauge choice that respects the a priori SPAM model as much as possible. The same procedure extends to overcomplete data by using pseudoinverses and truncating the calibration matrix to rank $d^2$, and to principled estimators like maximum likelihood by treating the SPAM operations as nuisance parameters. The paper demonstrates the correction on simulated single-qubit X$\pi/2$ gates with three SPAM error models, showing that the corrected estimates of gate fidelity are more accurate than standard QPT in every case, and that gate eigenvalues are completely unaffected by SPAM error.

Load-bearing premise

The correction breaks down if the state preparation and measurement operations change between the calibration experiment and the process experiment, or if the estimated SPAM error superoperator is singular so its square root is not well defined.

Editorial extensions

If this is right

Anyone already performing standard QPT can obtain a more accurate gate estimate simply by adding one calibration experiment and applying Eq. (3), with no new hardware or complex analysis.
Gauge-invariant quantities extracted from the process estimate, such as the eigenvalues of the transfer matrix, become exactly immune to SPAM error, so they can be used as reliable diagnostic figures even when SPAM is unknown.
For overcomplete data sets, the correction is still available through pseudoinverses and low-rank truncation, making the method applicable to typical experimental data with more circuits than the minimal informationally complete set.
The same calibration strategy can be incorporated into maximum-likelihood estimation, either independently, sequentially, or jointly, so statistically optimal estimates can also be freed from SPAM bias.

Reading between the lines

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

A direct testable consequence not pursued in the paper: the size of the SPAM correction can be used to flag unstable experiments, because if the calibration data and process data do not share the same SPAM, the corrected eigenvalues will drift with the time between the two data sets.
The gauge choice $p=0.5$ is a prior, not a measurement; users who have independent estimates of which of state preparation or measurement is noisier can set $p$ accordingly, and the paper's simulations suggest the residual error penalty for a wrong guess is small for small SPAM error.
The method could be combined with randomized benchmarking or other reference-free protocols to provide a low-cost SPAM-robust fidelity estimate, though the paper does not develop this connection.

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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

Short version: this is a well-written, accessible recipe for removing SPAM bias from linear-inversion QPT by adding one calibration experiment. The math is correct under the stated assumptions, and the eigenvalue immunity is a genuine formal result. But the advertised "more accurate" claim is not proven—it's an empirical observation from three simulations—and the core trick is essentially the same gauge-freedom idea the authors already used in their earlier closed-form self-consistent tomography (Ref. [8]). Worth taking seriously, but the framing needs care.

The paper's main contribution is practical clarity. It spells out the correction Ghat = E^{-1/2} G0 E^{-1/2} (and its p-weighted generalization) in a way that any QPT user can apply, with a clear derivation. The overcomplete pseudoinverse version in Section 5 is a useful extension. The simulations in Figure 1 cover three distinct SPAM-error models and show consistent improvement in estimated fidelity, and Figure 2 correctly demonstrates that the process eigenvalues are exactly immune to SPAM error—that part is a straightforward consequence of the gauge-orbit argument, and it's a nice point to make explicitly.

The main soft spot is the headline claim about accuracy. As your stress-test note shows, the correction places the estimate on the gauge orbit of the true process, which fixes the gauge-invariant content but not the gauge-variant fidelity. Whether the corrected fidelity is closer to the true value depends on the actual SPAM asymmetry and the user's choice of p. The paper's own footnote 1 admits that large coherent SPAM can produce unphysical fidelities. So "more accurate" is a statement about the simulated models, not a theorem. That's a legitimate limitation, but not a fatal one—the recipe delivers exactly what it says in the gauge-invariant sense, and the simulations do show the advertised behavior in realistic regimes.

A second soft spot is that the stationarity assumption (same M and S for calibration and gate data) is only mentioned in passing. If SPAM drifts, the correction is simply not applicable, and the paper should say so more prominently. Also, the novelty relative to Ref. [8] should be stated explicitly; the closed-form tomography paper already used the same gauge correction idea. That doesn't make this paper valueless—it's a simpler and more accessible presentation—but the authors should be clear about what's new.

Minor: no code or data are provided, which would help others apply the recipe.

Who is this for? Experimentalists and labs that use QPT and want a low-cost fix. It's a practical methods paper, not a deep theoretical advance. The algebra is sound and the recipe is likely useful, so it deserves peer review, but the final version should either temper the accuracy claim or add a condition under which it holds.

Referee Report

4 major / 4 minor

Summary. The paper proposes a modification of standard linear-inversion QPT that adds a SPAM-calibration experiment measuring the Gram matrix I = MS. From the two datasets, I = MS and P = MGS, the authors construct an estimate of the SPAM error superoperator E = M0^{-1} M S S0^{-1} and correct the naive process estimate via Ghat = Ehat^{-1/2} G0 Ehat^{-1/2} (or a generalization with a gauge parameter p). They claim this yields more accurate and less biased gate estimates, show simulations on three single-qubit SPAM models, and extend the idea to overcomplete data and to maximum-likelihood estimation. The core algebraic identity is correct: the corrected estimate lies on the gauge orbit of the true process, so gauge-invariant quantities such as process eigenvalues are exactly immune to SPAM error.

Significance. If the practical claims hold, the protocol gives a substantial accuracy improvement over standard QPT at the cost of only one additional calibration experiment, and the paper's closed-form correction is simple enough for immediate adoption. The simulations are clearly described and show consistent improvement on three SPAM models, and the overcomplete extension is a useful contribution. The paper also correctly identifies the gauge freedom and explains why gauge-invariant quantities are robust. However, the mathematical guarantee is limited to gauge-invariant properties, and the paper's stronger claim that the corrected estimate is 'more accurate' in terms of gate fidelity is not established by the algebra and is only empirically supported by the simulations.

major comments (4)

[Section 3, Eq. (3) and abstract] The paper's headline claim that SPAM-corrected QPT is 'more accurate' is not a consequence of Eq. (3). The derivation shows only that the corrected estimate lies on the gauge orbit of the true process G; specifically, Ghat(p) = C_p G C_p^{-1} with C_p = E^p B^{-1} and B = S S0^{-1}. Consequently, any gauge-dependent figure of merit, such as the process fidelity used in Figure 1, changes by an amount controlled by the unobserved state-preparation error B and the user's choice of p. The paper provides no bound or criterion for when the corrected fidelity estimate is closer to the truth than the uncorrected estimate. The fidelity improvement is an empirical observation for the three simulated SPAM models, and the paper should state this limitation explicitly or supply an analysis of the conditions under which improvement is guaranteed.
[Footnote 1] The admission that 'the estimated gate fidelity can be greater than 1 for large coherent SPAM errors' directly contradicts the introduction's claim that the corrected estimate is 'truthfully higher-fidelity.' As presented, Eq. (3) can return unphysical estimates. The paper should either restrict the fidelity claim to sufficiently small SPAM errors or incorporate the numerical gauge constraint mentioned in the footnote into the protocol itself. Without this, the central 'easy better' message is overstated.
[Section 3 and Section 5] The derivation assumes that the same SPAM operations M and S appear in both the calibration experiment (I) and the process experiment (P). This assumption is not stated as a formal condition. If SPAM drifts between the two data acquisitions, Ehat no longer describes the SPAM affecting the process data, and Eq. (3) can in principle move the estimate to a point on the gauge orbit that is farther from the truth. The paper only discusses a related failure in Section 5 for non-Markovian errors that enlarge the active state space; drift is a more general and equally realistic limitation that should be acknowledged.
[Section 3, Eq. (3)] The corrected estimate requires computing Ehat^{-1/2}, which presumes that Ehat is invertible and has a well-defined square root. The paper does not discuss conditions on E (e.g., no zero eigenvalues, or for real square roots, no negative real eigenvalues), nor does it specify which square-root branch is intended. For SPAM error superoperators that are not diagonalizable or not positive, Ehat^{-1/2} may be undefined or complex. This is a technical gap in the closed-form correction that should be addressed, even if the simulations happen to avoid it.

minor comments (4)

[Section 6] The phrase 'relatively näive' contains a typo; it should be 'relatively naive.'
[Section 3, Eq. (3)] The notation E^{1/2} is used without specifying that it is a matrix square root. For clarity, state that E is a d^2 x d^2 matrix in the Liouville representation and that a principal square root is intended when E is Hermitian positive semidefinite.
[Figure 1 caption] The caption describes the shot count and circuit count in the main text, but including the circuit counts (12 for standard QPT, 24 for SPAM-corrected) directly in the caption would improve readability.
[Section 5, Eq. (11)] The 'suggestive approximation' after Eq. (11) is presented without stating that it is not used in the simulations and is only heuristic; please label it as such.

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation is self-contained; the SPAM correction is an algebraic estimator from independent calibration data, not a fit of the target gate.

full rationale

Section 3 derives the corrected estimate from the stated model I=MS and P=MGS, with Ehat estimated only from the separate calibration data Ihat. No gate fidelity, eigenvalue, or other target quantity is used to fit Ehat, and the gauge parameter p is an explicitly acknowledged user choice rather than a fitted parameter. Equation (3) follows by the explicit choice beta = E^{1/2}; the eigenvalue immunity is a mathematical consequence of the resulting similarity transformation, not a quantity defined in terms of the estimate. The paper's Section 2 reference to circularity concerns the general state/measurement tomography problem that motivates the work, not a step in its own derivation. GST citations provide background context and are not load-bearing: gauge freedom is also supported by independent references [13-15]. The simulated fidelity comparisons in Figure 1 are empirical benchmarks against standard QPT using generated data, so no fitted input is renamed as a prediction. No self-definitional, fitted-input, self-citation, uniqueness-importation, ansatz-smuggling, or renaming pattern is present.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

No invented physical entities. The central derivation rests on a stationarity assumption, invertibility and square-root existence for the SPAM error matrix, and a truncation ansatz for overcomplete data. The gauge parameter p is a user choice, not a fitted parameter.

free parameters (1)

p (gauge division parameter) = p = 0, 0.5, 1 in simulations; arbitrary in [0,1]
User-selected split of SPAM error between state preparation and measurement. It does not affect gauge-invariant quantities, but the paper's fidelity-accuracy claim depends on it, and no rule is given for choosing p.

assumptions (3)

domain assumption The SPAM operations M and S are the same in the calibration experiment (I = MS) and process experiment (P = MGS), with no drift or non-Markovianity between the two data-taking stages.
Invoked in Section 3 when I and P are combined through E = M0^{-1} M S S0^{-1}; if SPAM drifts, Eq. (3) does not correct the actual process estimate. Non-Markovian failure is only discussed in Section 5.
domain assumption The a priori matrices M0 and S0 are invertible, and the estimated SPAM error E = M0^{-1} Ihat S0^{-1} has a square root, or a well-defined pseudoinverse analogue in the overcomplete case.
Used in Section 3 Eq. (3) and Section 5 Eqs. (8)-(11); if E is singular or the a priori SPAM matrices are rank-deficient, the symmetric split is not defined.
ad hoc to paper For overcomplete data, deviations of Ihat from rank d^2 are sampling noise, so truncating the smallest singular values improves accuracy.
Section 5 states this ansatz explicitly; if the deviations come from a larger state space, the paper declares the entire tomographic problem ill-posed.

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Cite this review

Pith. "Pith review of Easy better quantum process tomography." pith.science (2026). https://pith.science/paper/ZMZ4XNPZ

@misc{pith2026241216293,
  author       = {Pith},
  title        = {Pith review of: Easy better quantum process tomography},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZMZ4XNPZ}},
  note         = {Machine review of arXiv:2412.16293}
}

read the original abstract

Quantum process tomography (QPT), used to estimate the linear map that best describes a quantum operation, is usually performed using a priori assumptions about state preparation and measurement (SPAM), which yield a biased and inconsistent estimator. This estimate can be made more accurate and less biased by incorporating SPAM-calibration data. Unobservable properties of the SPAM operations introduce a small gauge freedom that can be regularized using the a priori SPAM. We give an explicit correction procedure for standard linear-inversion QPT and overcomplete linear-inversion QPT, and describe how to extend it to statistically principled estimators like maximum likelihood estimation.

Figures

Figures reproduced from arXiv: 2412.16293 by the authors.

**Figure 1.** Comparative accuracy of standard and SPAMcorrected QPT: We compared the accuracy of standard QPT (blue) and SPAM-adjusted QPT (orange, green, red) by simulating both protocols on an Xπ/2 gate with 1% depolarizing noise, with 5000 shots per circuit (see main text for details). To quantify accuracy, we used the process matrix estimated by each protocol to estimate the noisy gate’s fidelity with a noiseless Xπ/2 unit… view at source ↗

**Figure 2.** Error in estimated gate eigenvalues: Gaugeinvariant quantities like the eigenvalues of the estimated superoperator (transfer matrix) should be independent of how SPAM error is divided. We analyzed the results of simulated standard and SPAM-corrected QPT to determine how inaccurately they estimated gate eigenvalues, using the figure of merit δ = 1 4 P i |λi − λ˜i|, where λi are the true eigenvalues and λ˜i the est… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

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Nicolás Quesada, Agata M Brańczyk, and Daniel F V James. “Self-calibrating tomography for non- unitary processes”. In The Rochester Conferences on Coherence and Quantum Optics and the Quan- tum Information and Measurement meeting. Page W6.38. Optica Publishing Group (2013)

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Seth T Merkel, Jay M Gambetta, John A Smolin, Stefano Poletto, Antonio D Córcoles, Blake R Johnson, Colm A Ryan, and Matthias Stef- fen. “Self-consistentquantumprocesstomography”. Phys. Rev. A87, 062119 (2013)

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