Personalized Improvement of Standard Readout Error Mitigation using Low-Depth Circuits and Machine Learning

Melody Lee

REVIEW 4 major objections 6 minor 33 references

Personalized Improvement of Standard Readout Error Mitigation using Low-Depth Circuits and Machine Learning

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proposes a machine-learning update to the standard readout error confusion matrix, trained on low-depth random circuits, and reports median gains of 6.6% in fidelity, 29.9% in mean-squared error, and 10.3% in Hellinger distance…

desk verdict The reader's rank-1 objection to Eq. (22) is mathematically wrong; the paper's real weakness is that it omits shot counts, error bars, and code, making the modest reported gains impossible to evaluate. read the letter →

arxiv 2506.03920 v1 pith:UZDA2BKE submitted 2025-06-04 quant-ph

Melody Lee This is my paper

classification quant-ph

keywords readouterrormitigationquantumconfusionmatrixordinaryleastsquaresregressionmachinelearninglow-depthcircuitsNISQHellingerdistance

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

This paper proposes a personalized readout error mitigation (PEM) method that tunes the standard confusion-matrix readout error model using measured probability distributions from a collection of low-depth, randomly generated circuits. On a simulated seven-qubit superconducting backend with circuit depth four, it reports median improvements of 6.6% in state fidelity, 29.9% in mean-squared error, and 10.3% in Hellinger distance over the standard error mitigation approach. The intent is to make the readout error model reflect the actual gate set and calibration state of the device rather than a fixed predefined model. If the method holds, readout error mitigation could be recalibrated cheaply and periodically from ordinary low-depth circuit runs.

What carries the argument

The central object is the per-qubit readout error matrix $Q_i$, a $2\times 2$ confusion matrix whose entry $Q_{i,(j,k)}$ is the probability that qubit $i$ is measured as $j$ when prepared as $k$. The mechanism that carries the argument is an ordinary least squares update: for each qubit $i$ and each measured outcome $m \in \{0,1\}$, the design matrix $X_i$ stacks the ideal per-qubit probability vectors $q_{i,k} = [P(|0\rangle), P(|1\rangle)]^T$ across the training circuits, and the target $\bar{y}_{i,m}$ stacks the corresponding noisy probabilities; the regression weights $w_{i,m} = (X_i^T X_i)^{-1} X_i^T \bar{y}_{i,m}$ are then combined with the original $Q_i$ as $Q_i^* = c((1-\eta)Q_i + \eta [w_{i,0}, w_{i,1}])$, normalized by $c$, and used to assemble an updated full error matrix $E^*$ whose inverse is the personalized mitigation matrix. This update is what lets the readout model track the device's current behavior, and the matrix inversion step then applies the same standard mitigation protocol to the personalized model.

What would settle it

Compute $X_i^T X_i$ for any qubit's training data: because each row sums to one, the matrix has rank one and determinant zero, so the closed-form inverse in Eq. (22) does not exist. Running the training procedure with two different pseudo-inverse conventions on the same data would produce different $w_{i,m}$, different $Q_i^*$, and different mitigated distributions, showing that the reported improvements are implementation-dependent.

Watch

Extended reading notes

Core claim

Standard readout error mitigation constructs a per-qubit confusion matrix $Q_i$ from calibration circuits that prepare all-$|0\cdots 0\rangle$ and all-$|1\cdots 1\rangle$ states, builds the full $2^n \times 2^n$ error matrix $E$ under an independence assumption, and mitigates by inverting $E$. The paper's claim is that this matrix can be sharpened by collecting noisy measurement distributions from many random low-depth circuits and, for each qubit, regressing the measured probabilities $\bar{q}_{i,k}$ on the ideal probabilities $q_{i,k}$; the resulting regression weights $w_{i,m}$ are blended with the original $Q_i$ through a learning rate $\eta$, giving an updated $Q_i^*$, an updated $E^*$, and a personalized mitigation matrix $(E^*)^{-1}$. Tested on a simulated seven-qubit backend with depth-four training circuits, the personalized matrix outperforms standard mitigation on all three reported metrics, with 78%, 85%, and 91% of the 100 test circuits improving in fidelity, mean-squared error, and Hellinger distance, respectively.

Load-bearing premise

The load-bearing premise is that each qubit's least-squares regression update has a unique closed-form solution, but the two regressors — the probabilities of measuring $|0\rangle$ and $|1\rangle$ — always sum to one, so the matrix $(X_i^T X_i)$ is singular and the solution depends on an unspecified pseudo-inverse or regularization.

Editorial extensions

If this is right

The standard readout error matrix can be re-trained on low-depth circuits, so mitigation can track gate-set and calibration drift without repeating full $2^n$-state detector tomography.
On the tested backend, the improvement over standard mitigation persists for circuit depths from one to ten, so the personalization is not tied to a single training depth.
The method is limited to circuits whose useful output is captured by measurement-outcome magnitudes with positive, real coefficients; states with sign or phase information would require exponentially costly reconstruction.
Because small negative entries are dropped and the remaining vector renormalized, the post-processing assumes sparse output distributions; dense distributions would require exponentially more samples.

Reading between the lines

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

The regression's normal matrix $X_i^T X_i$ is singular for any training set, so the reported PEM matrix depends on the regression implementation's implicit pseudo-inverse; the improvement is not uniquely defined until that choice is specified.
A regularized or otherwise constrained update to $Q_i$ would make the personalization well-posed and could be benchmarked against the paper's numbers.
The same per-qubit regression update could in principle tune Bayesian unfolding or other mitigation strategies, not just matrix inversion.
The sparse-distribution assumption suggests the method's advantage may shrink on circuits that produce dense output distributions, a testable extension the paper leaves open.

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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 singularity objection in the reader's report doesn't survive contact with the paper. The columns of X_i are P(|0>) and P(|1>) across training circuits, and since P(|1>) = 1 - P(|0>), the column space is spanned by P(|0>) and the all-ones vector. For random low-depth circuits, P(|0>) is not constant across the 1,000 training samples, so the two columns are generically independent and X_i^T X_i is invertible. Eq. (22) is fine. Don't reject this paper on that ground.

What the paper actually does is sensible: it takes the standard per-qubit readout confusion matrix, then uses OLS on random low-depth circuits to update each row, with a learning rate that blends the old and new estimates. The idea is a natural extension of calibration and of prior ML-based readout mitigation (Kim, Liao), and the paper is clearly written and honest about its limitations, including the sparse-distribution assumption and exponential scaling.

The soft spots are empirical, not mathematical. The manuscript never states the shot count s used for the calibration circuits in Sec. 2.3, nor the per-circuit shot count for the training and test probabilities. All probabilities are frequencies, so every entry of Q_i and every regression target carries sampling variance. Without s, error bars, or significance tests, the median improvements in Table 2 (6.6% fidelity, 29.9% MSE, 10.3% Hellinger) could easily be finite-sampling artifacts. The algorithm is also under-specified: Eq. (23) says "some normalization constant c" and the negative-entry clipping threshold is undefined. No code or data are released. Everything runs on the Fake Perth simulator, which is fine as a proof of concept but does not test the calibration-drift motivation.

There is also no direct comparison against the prior ML-based methods the paper cites, so the incremental benefit over those approaches is unclear.

All that said, the core method is plausible and the math checks out. The missing details are exactly the kind of thing a competent referee can demand in revision. I would send it to peer review, with the expectation of a major revision that adds shot counts, error bars, a specified algorithm, and ideally real-hardware data.

Referee Report

4 major / 6 minor

Summary. The paper proposes 'personalized error mitigation' (PEM), a modification of standard readout error mitigation. Standard EM estimates a per-qubit readout error matrix Q_i from circuits preparing all |0...0> and |1...1>, constructs a full confusion matrix E, and inverts it. PEM additionally generates N random low-depth training circuits, uses ordinary least squares to regress measured outcome probabilities for each qubit on the ideal probabilities, and blends the resulting weights with the original Q_i through a learning rate eta, yielding an updated matrix E* that is then inverted. The authors evaluate PEM on the Qiskit Fake Perth simulator for n = 7 qubits and report median improvements over standard EM of 6.6% in fidelity, 29.9% in MSE, and 10.3% in Hellinger distance on 100 held-out depth-4 circuits, plus a depth-scaling study. The central regression step is, for generic random training circuits, mathematically well-posed, although the manuscript's justification is incomplete.

Significance. If the reported gains are real and reproducible, PEM would be a simple, low-overhead improvement to a standard readout mitigation pipeline, and the idea of using low-depth circuit statistics to refine a confusion matrix is worth exploring. The method's strength is its simplicity: the OLS update is standard and the training circuits are easy to generate. However, the paper currently lacks the experimental details needed to assess or reproduce the claim: the number of shots is never stated, no code is provided, and the reported medians come without error bars or significance tests. These omissions are load-bearing because the measured probabilities are frequency estimates with sampling noise, and the improvement could in principle be an artifact of that noise. The manuscript is also honest about known limitations (sparse distributions, exponential scaling), which is to its credit.

major comments (4)

[Sec. 2.3, Sec. 3.3, Tables 1-2] The manuscript never states the number of shots s used to estimate Q_i (Eqs. (11)-(14)) or the per-circuit shot count used to obtain the training targets ybar_i,m and the test probability vectors. All of these quantities are frequencies, so every entry of Q_i, every row of X_i, and every test vector carries sampling variance. Without s, the claimed median improvements in Table 2 and the depth scaling in Fig. 7 cannot be reproduced, and they cannot be distinguished from the effect of PEM smoothing sampling noise rather than improving the error model. This is the most important omission and should be fixed by reporting s and, ideally, the noise model's parameters.
[Sec. 3.2, Sec. 3.3, Fig. 6, Fig. 7, Table 2] The headline results are point estimates of median improvement over 100 test circuits, with no confidence intervals, error bars, or significance tests. Fig. 6 shows that a substantial fraction of circuits are better under standard EM (22% for fidelity, 15% for MSE, 9% for Hellinger), so the median improvement could be consistent with sampling fluctuations. The eta-selection plot in Fig. 3 and the quadratic fit in Eq. (27) also lack error bars, even though the underlying MSE values are estimated from 50 circuits. Please provide bootstrap or confidence intervals and a paired significance test (e.g., Wilcoxon signed-rank) for the three metrics.
[Sec. 2.5, Eq. (23), Sec. 2.6] The PEM algorithm is not fully specified. In Eq. (23), c is described only as 'some normalization constant,' and the post-processing step that 'drops small-magnitude, negative entries' does not define the threshold. The paper says code was written in Python but gives no repository or release. Because the method is defined by these implementation details, a reader cannot reconstruct E* from the text. Please specify c and the clipping rule, and make the code (or at least a pseudocode listing) available.
[Sec. 2.6, Sec. 3.3, Fig. 7] The training and test circuits are sampled from the same random distribution (n = 7, depth 4, and varying depths in Fig. 7), so the holdout evaluation shows generalization to new circuits of the same family, not personalization to a distinct gate set or to a different calibration period. The introduction motivates PEM as accounting for 'the entire gate set' and calibration drift, but no experiment varies the gate set or the time since calibration. Please either temper the claim or add an evaluation on circuits with a different gate distribution and on noise-model drift.

minor comments (6)

[Sec. 2.1, Eq. (5)] The equation P(|j>) = <j|psi> = |alpha_j|^2 should read P(|j>) = |<j|psi>|^2 = |alpha_j|^2; the inner product <j|psi> is alpha_j, not a probability.
[Sec. 2.4] The reference to 'Equation 2.2' should be to Eq. (9), and the statement about (E E_m)^T is confusing; clarifying that E_m = E^{-1} gives E_m^T E^T = I would make the mitigation step unambiguous.
[Sec. 2.5, Eq. (22)] The justification for invertibility is incomplete: the conditions 'N > 2' and 'X_i nonzero' do not guarantee that X_i^T X_i is invertible. The correct condition is that the training probabilities P(|0>) are not constant across circuits, since det(X_i^T X_i) = N * sum(p^2) - (sum(p))^2, which is positive for nonconstant p. This condition is presumably satisfied by the random training circuits, but it should be stated.
[Sec. 2.3 vs. Sec. 2.6] Section 2.3 says circuits are run 'on real hardware,' while Section 2.6 states that the Fake Perth simulator is used; please make the hardware/simulator distinction consistent throughout.
[Table 1] The MSE entry '0.005.32 x 10^-3' appears to be a formatting error; it should be a single number.
[Sec. 2.4] The phrase 'randomized lower-upper (LU) decomposition' is unusual because LU decomposition is normally deterministic; if a randomized solver was used, it should be described explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PEM matrix is fit on training circuits and evaluated on separate held-out-style test circuits, with no self-citation or definitional reduction.

full rationale

The paper's derivation chain is empirical and not circular. Section 2.5 fits the per-qubit error rows w_i,m by ordinary least squares to a training set T of N random low-depth circuits, using features X_i = ideal probabilities and targets ybar_i,m = noisy measured probabilities (Eqs. 16-22); the reported improvements in Tables 1-2 and Figure 7 are computed on test circuits described as 'randomly selected' after training, so the comparison is a generalization evaluation rather than a restatement of the training objective. Eq. (23) updates Q_i with a convex combination of the initial two-circuit estimate and the regression weights, but the update is not defined in terms of the evaluation metrics (fidelity, MSE over the full 2^n distribution, Hellinger distance) nor in terms of the test-set outcomes. The paper contains no self-citations, no imported uniqueness theorem, and no ansatz smuggled in via prior work: the linear readout model is stated in Eqs. (8)-(10) and (17) from measurement probability definitions, and the ML-QEM-related prior work [15] is cited as background rather than as justification for the personalization step. An initial concern about the singularity of X_i^T X_i is unfounded because the row dependence (each row sums to one) does not make the 2x2 Gram matrix singular for non-constant training features. The lack of stated shot counts and code availability is a reproducibility or statistical-validity issue, not a circularity issue; likewise the reported advantage possibly reflecting distribution matching is a possible bias in generalization claims but not a definitional reduction. Therefore no circular step can be exhibited, and the score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim depends on four fitted or unspecified numerical parameters and on the stated assumptions about readout error structure. The load-bearing mathematical premise that the least-squares normal matrix is invertible is false. No new physical entities are introduced.

free parameters (4)

learning rate eta = 0.23
Chosen by quadratic fit to validation MSE in Eq. (27); used in all main results.
regression weights w_i,m = not reported
Two weights per qubit per outcome, fit by OLS on 1,000 training circuits (Section 2.5).
normalization constant c = not reported
Used in Eq. (23) to renormalize the updated per-qubit matrix.
number of shots s = not reported
Shot count per circuit is never specified, but it controls the sampling noise in every measured probability.

assumptions (5)

domain assumption Assumption 1: measured and prepared states are random variables
Stated in Section 2.2; underlies the probabilistic confusion-matrix model.
domain assumption Assumption 2: readout errors are independent across qubits
Stated in Section 2.2; used to construct E as a product of per-qubit matrices. The authors acknowledge this is violated by crosstalk.
domain assumption Assumption 3: E is diagonally dominant
Stated in Section 2.2; used to justify invertibility of E.
ad hoc to paper The normal matrix X_i^T X_i is invertible
Implicit in Eq. (22); false because rows of X_i lie on the simplex p0+p1=1, making X_i rank 1.
domain assumption Fake Perth simulator faithfully represents IBM Perth noise
All experiments use the Qiskit Fake Perth backend (Section 2.6); real-device behavior is not tested.

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Cite this review

Pith. "Pith review of Personalized Improvement of Standard Readout Error Mitigation using Low-Depth Circuits and Machine Learning." pith.science (2026). https://pith.science/paper/UZDA2BKE

@misc{pith2026250603920,
  author       = {Pith},
  title        = {Pith review of: Personalized Improvement of Standard Readout Error Mitigation using Low-Depth Circuits and Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UZDA2BKE}},
  note         = {Machine review of arXiv:2506.03920}
}

read the original abstract

Quantum computers have shown promise in improving algorithms in a variety of fields. The realization of these advancements is limited by the presence of noise and high error rates, which become prominent especially with increasing system size. Mitigation techniques using matrix inversions, unfolding, and deep learning, among others, have been leveraged to reduce this error. However, these methods are not reflective of the entire gate set of the quantum device and may need further tuning depending on the distance from the most recent calibration time. This paper proposes a method of improvement to numerical readout error techniques, where the readout error model is further refined using measured probability distributions from a collection of low-depth circuits. We use machine learning to improve the readout error model for the quantum system, testing the circuits on the simulated IBM Perth backend using Qiskit. We demonstrate a median 6.6% improvement in fidelity, 29.9% improvement for mean-squared error, and 10.3% improvement in Hellinger distance over the standard error mitigation approach for a seven-qubit system with a circuit depth of four. With further focus directed towards such improvement of these error mitigation techniques, we are one step closer to the fault-tolerant quantum computing era.

Figures

Figures reproduced from arXiv: 2506.03920 by the authors.

**Figure 2.** Circuit 1, which generates a quantum state that, when measured, has an expected value of |11 . . . 1⟩. Used in initial readout error matrix construction. 2.4 Simple error mitigation Post-measurement error mitigation aims to approximate p using the measured p¯. We seek to compute some matrix Em where (EEm) T = E T mE ≈ I, such that when applied to both sides of Equation 2.2 we have E T mp¯ ≈ ET mE T p ⇒ p ≈ ET mp. ¯ … view at source ↗

**Figure 3.** Identifying the learning rate η by testing values from 0 to 0.5. The results have a quadratic best fit indicated in Equation 27 and an optimal η = 0.23. qubit 0 I 0 1DCX U(2.9905, 4.5383, 4.9233) qubit 1 P(0.51444) Y RZ(0.11833) qubit 2 √ X P(4.2593) S qubit 3 SDG qubit 4 S H qubit 5 qubit 6 T DG SDG classical [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Example of a randomly generated circuit on [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: Comparison of the mitigated probabilities of measuring various quantum states, arranged in increasing order [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: The difference in evaluated error metric between the standard error mitigation and proposed PEM methods. [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: Performance of the PEM method as the depth of the circuit scales. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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