{"id":"bd3df30d-5897-456d-af6d-205d24ccdfc2","arxiv_id":"2506.03920","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"A proposed machine-learning update to quantum readout error matrices claims 6.6-29.9% metric improvements in simulation, but its least-squares core is degenerate because each qubit's probability vector sums to one.","lead":"This paper proposes to improve standard quantum readout error mitigation by fitting each qubit's error model to a collection of random low-depth circuits with ordinary least squares. The authors report modest gains on a simulated seven-qubit IBM backend, but the central regression step is mathematically ill-posed as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reader's singularity objection is mathematically wrong; the real issue is that the experiments omit shot counts and code, so the claimed improvement is not reproducible.","rationale":"The paper's core idea — regressing measured single-qubit outcome frequencies on ideal marginal probabilities to adjust a per-qubit confusion matrix — is coherent. I verified the algebra of Eq. 17/20: for nonconstant ideal marginals, X_i has full column rank (det = N Σp^2 - (Σp)^2), so the reader's claimed singularity is a misreading of the row-sum constraint. The OLS solution is not the fatal flaw the reader identifies. The method's actual vulnerability is that all empirical quantities are frequency estimates and the manuscript does not report the shot counts on which they are based. The standard EM confusion matrix is estimated from two calibration circuits; PEM uses a large training set. If the standard matrix is noisy, PEM may improve simply by regression averaging, and the reported 6.6%/29.9%/10.3% improvements could shrink or vanish at higher shot counts. The lack of code, data, and significance testing makes this impossible to check from the paper. Eq. 23's undefined c and the ad hoc clipping add to the underspecification. I therefore recommend CONDITIONAL rather than REJECT: the mathematical objection should be retracted, but the authors need to provide exact shot counts, artifacts, and a statistical analysis before the empirical claim can be accepted.","tokens_in":10276,"tokens_out":17055,"duration_ms":160467,"concrete_test":"Request the authors' code and exact shot counts. Re-run the full pipeline for s ∈ {10^3, 10^4, 10^5, 10^6} shots, both for the initial Q_i calibration and for each training/test circuit, and recompute the median improvements in Table 2. If the PEM advantage over EM does not persist at high s, or if no s is supplied such that the advantage holds with a paired bootstrap 95% CI excluding zero, the central claim is an artifact of sampling noise rather than a genuine improvement.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The reader's singularity objection is incorrect. Rows of X_i sum to one, so X_i[1,1]^T = 1_N; that is a dependence among rows, not columns. Column dependence would require X_i v = 0, i.e., v0 p + v1(1-p)=0 for every training row; unless P(|0>) is identical across all 1000 training circuits, this forces v=0. Indeed det(X_i^T X_i) = N Σp^2 - (Σp)^2 > 0 for nonconstant p. So Eq. (22) is generically well-defined and the central derivation is not invalid for the stated random training set. The load-bearing problem is empirical: the paper never states the shot count s used to estimate Q_i (Sec. 2.3) or the per-circuit shot count used for training/test probabilities. All reported probabilities are frequencies, so every entry of Q_i and every training target has sampling variance. Standard EM estimates Q_i from only two circuits (all 0 and all 1), so its entries are noisy; PEM regresses on 1000 circuits and may simply be averaging away that sampling noise rather than 'personalizing' the error model. Without s, error bars, or significance tests, Table 2's median improvements and Figure 7's depth scaling cannot be distinguished from finite-sampling artifacts. In addition, Eq. (23) defines only 'some normalization constant' c and the negative-entry clipping threshold is unspecified, so the algorithm is not fully specified.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10515,"tokens_out":7941,"duration_ms":72949,"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":[{"comment":"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.","section":"Sec. 2.3, Sec. 3.3, Tables 1-2"},{"comment":"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.","section":"Sec. 3.2, Sec. 3.3, Fig. 6, Fig. 7, Table 2"},{"comment":"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.","section":"Sec. 2.5, Eq. (23), Sec. 2.6"},{"comment":"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.","section":"Sec. 2.6, Sec. 3.3, Fig. 7"}],"minor_comments":[{"comment":"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.","section":"Sec. 2.1, Eq. (5)"},{"comment":"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.","section":"Sec. 2.4"},{"comment":"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.","section":"Sec. 2.5, Eq. (22)"},{"comment":"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.","section":"Sec. 2.3 vs. Sec. 2.6"},{"comment":"The MSE entry '0.005.32 x 10^-3' appears to be a formatting error; it should be a single number.","section":"Table 1"},{"comment":"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.","section":"Sec. 2.4"}],"recommendation":"major_revision","confidential_remarks":"The reviewer's singularity objection in the earlier report is incorrect: the rows of X_i sum to one, which is a row dependence, and X_i^T X_i is generically invertible (det = N*sum(p^2) - (sum(p))^2). The paper's real weaknesses are empirical reproducibility and statistical support. The work is a concise methods paper that could be suitable after a major revision adding shot counts, code, error bars, and significance tests, and clarifying the scope of the 'personalization' claim. I would not reject on the basis of Eq. (22)."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nThere is also no direct comparison against the prior ML-based methods the paper cites, so the incremental benefit over those approaches is unclear.\n\nAll 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.","headline":"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.","tokens_in":11089,"tokens_out":2272,"would_cite":false,"duration_ms":23030,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["readout error mitigation","quantum error mitigation","confusion matrix","ordinary least squares regression","machine learning","low-depth circuits","NISQ","Hellinger distance"],"falsifier":"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.","tokens_in":9999,"feed_emoji":"⚛️","tokens_out":10755,"duration_ms":97203,"temperature":0.7,"pith_summary":"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.","feed_headline":"Per-qubit retraining cuts quantum readout error by 29.9%","feed_subtitle":"Training on low-depth random circuits adapts the error model to a device's current gates and calibration.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the detector-tomography based construction of readout noise confusion matrices that standard EM uses.","marker":"[18]"},{"why":"Supplies the scalable readout error mitigation approach whose inversion is the baseline PEM improves on.","marker":"[23]"},{"why":"Provides the Bayesian unfolding alternative that motivates statistical readout error correction.","marker":"[22]"},{"why":"Provides the machine-learning quantum error mitigation method whose fixed gate-set limitation motivates personalization.","marker":"[15]"},{"why":"Supplies a deep-learning readout error mitigation baseline that uses learned error models.","marker":"[14]"},{"why":"Supplies the sparse measurement outcome assumption used for post-processing and scalability discussion.","marker":"[33]"},{"why":"Supplies the software stack used to generate and run circuits on the simulated backend.","marker":"[13]"}],"fun_headline_variants":["Personalized readout error mitigation improves MSE by 29.9%","Machine learning refines quantum error models for better readout","Low-depth circuits tune error mitigation for qubit-specific readout","Retrain on random circuits to cut readout error by 29.9%","Per-qubit error model updates boost readout accuracy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Personalized readout error mitigation improves MSE by 29.9%","Machine learning refines quantum error models for better readout","Low-depth circuits tune error mitigation for qubit-specific readout","Retrain on random circuits to cut readout error by 29.9%","Per-qubit error model updates boost readout accuracy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001121,"raw_usage":{"total_tokens":4691,"prompt_tokens":1001,"completion_tokens":3690,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":617,"completion_tokens_details":{"reasoning_tokens":3601}},"tokens_in":617,"tokens_out":3690,"duration_ms":30086,"temperature":1.0,"reasoning_tokens":3601,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:53:18.734715+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":", Zimbor \\'a s, Z","cited_arxiv_id":null,"evidence_quote":"Supplies the detector-tomography based construction of readout noise confusion matrices that standard EM uses."},{"cited_title":", Kang, H","cited_arxiv_id":null,"evidence_quote":"Supplies the scalable readout error mitigation approach whose inversion is the baseline PEM improves on."},{"cited_title":", Urbanek, M","cited_arxiv_id":null,"evidence_quote":"Provides the Bayesian unfolding alternative that motivates statistical readout error correction."},{"cited_title":", Wang, D S","cited_arxiv_id":null,"evidence_quote":"Provides the machine-learning quantum error mitigation method whose fixed gate-set limitation motivates personalization."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies a deep-learning readout error mitigation baseline that uses learned error models."},{"cited_title":", Raymond, R","cited_arxiv_id":null,"evidence_quote":"Supplies the sparse measurement outcome assumption used for post-processing and scalability discussion."}],"review_version":1}