{"id":"cf15f236-e440-47c8-990d-80d53790b49b","arxiv_id":"2607.07988","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Procrustes tomography reconstructs quantum process matrices as the least-squares map between input and output density matrices and matches or exceeds standard methods under sampling and preparation noise.","lead":"The paper introduces Procrustes tomography, a method that reconstructs noisy quantum channels by solving a matrix Procrustes problem on reconstructed density matrices. It is simpler to implement than standard process tomography and often matches or beats it under realistic sampling and preparation errors.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the reader's already-noted scope limits.","rationale":"The paper's strongest claim is a controlled numerical comparison under a shared linear-inversion state-tomography front-end plus optional Procrustes-specific extensions (weighting, input tomography, unitary projection). Those extensions are correctly formulated (Eqs. 10-13) and the figures demonstrate the expected improvements. Because the mathematical equivalence to linear inversion is acknowledged and the only free parameters are the deliberately added extensions, there is no hidden assumption that would invalidate the reported trends. The reader's weakest assumption correctly identifies the missing comparison against stronger state estimators; that is a genuine but already-recognized scope limit, not a flaw that collapses the central claim. Consequently the CONDITIONAL verdict (pending code/real-device data and clearer superiority statements) remains appropriate; no adjustment is required.","tokens_in":11504,"tokens_out":489,"duration_ms":5320,"concrete_test":"Re-run the two-qubit Lindblad simulations of Figs. 1-3, replacing the linear-inversion state-tomography front-end with maximum-likelihood estimation (or projected least-squares) for every method; if Procrustes remains competitive or superior under identical MLE estimates, the claimed advantage is robust; if the gap closes or reverses, the advantage is an artifact of the linear-inversion front-end.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Procrustes Lambda = B A^+ matches or exceeds Choi and linear-inversion fidelity under the simulated conditions of Figs. 1-4) is internally consistent and correctly supported by the reported numerics. When linear-inversion state tomography is used for both Procrustes and the linear-inversion baseline, the two methods are mathematically equivalent (as the paper itself notes after Eq. 10); any residual advantage comes only from the optional PSD truncation, weighting matrix W, or input-state tomography steps that the baselines do not receive. Those extensions are correctly derived and the figures show the expected gains. The reader's weakest assumption (dependence on linear-inversion state tomography) is real but already correctly flagged as a scope limitation rather than an internal inconsistency; it does not undermine the claim as stated. No deeper load-bearing flaw (hidden non-physicality, incorrect projection, or mis-specified noise model) appears in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript introduces Procrustes tomography for reconstructing noisy quantum channels: after preparing a set of input states, performing state tomography on the outputs, and assembling the vectorized density matrices into matrices A (inputs) and B (outputs), the process matrix is obtained by the ordinary Procrustes solution Λ = B A⁺ (Eq. 10), optionally with a diagonal weighting matrix W that accounts for uneven shot counts (Eq. 11). The method is compared with the textbook Choi-matrix reconstruction and with linear-inversion process tomography under Lindblad noise (amplitude damping + dephasing) for two-qubit gates. Numerical experiments (Figs. 1–4) examine even and uneven sampling, imperfect state preparation, and optional projection onto the nearest unitary channel; they show that the Procrustes estimator matches or exceeds the fidelity of the two baselines, especially once input-state tomography or unitary projection is included. Finite-sample non-CPTP maps are corrected by the Knee et al. projection for all three methods.","tokens_in":11680,"tokens_out":1232,"duration_ms":12036,"significance":"If the reported performance advantage holds under realistic device conditions, the work supplies a lightweight, pedagogically transparent alternative to standard process tomography that is immediately usable by NISQ-cloud users who lack specialized tomography expertise. Strengths include an explicit, parameter-free least-squares estimator, a clean treatment of state-preparation errors (simply double the tomography), a natural weighting scheme for uneven sampling, and a unitary-projection step based on the Kronecker-product SVD. The numerical comparisons under controlled Lindblad dynamics are clear and support the claim within the simulated regime. The paper does not claim asymptotic optimality or experimental validation, so its contribution is best viewed as a practical, extensible reconstruction recipe rather than a fundamental advance in tomography theory.","major_comments":[{"comment":"Numerical simulations section (paragraph beginning “For the following examples…”) and the discussion after Eq. (10): all Procrustes results are generated with linear-inversion state tomography (plus optional negative-eigenvalue truncation). The paper itself notes that, under pure linear inversion, Procrustes and the linear-inversion baseline become mathematically identical. Consequently the observed advantage is attributable only to the optional PSD truncation, the weighting matrix W, or the input-state tomography step. Without at least one comparison against maximum-likelihood or Bayesian state tomography, it remains unclear whether the claimed superiority survives once a higher-quality state estimator is used for every method. A short additional panel or appendix addressing this point would substantially strengthen the central claim.","section":null},{"comment":"Figs. 1–4 and the associated text: all numerical evidence is restricted to two-qubit channels generated by a single Lindblad model (T1/T2 ratios and absolute T2 values listed in the figure captions). While the algebraic construction is dimension-independent, the performance ranking versus Choi and linear inversion could change for larger systems or for noise that includes coherent control errors, leakage, or non-Markovian effects. The manuscript should either (i) state explicitly that the superiority claim is limited to the two-qubit Lindblad setting examined, or (ii) supply at least one higher-dimensional or differently noised example.","section":null}],"minor_comments":[{"comment":"Abstract and opening paragraph: the rhetorical question “What is the most expensive part…?” and the unqualified claim that Procrustes “outperforms established methods in a number of aspects” should be tempered to match the more careful language used in the concluding remarks.","section":null},{"comment":"Eq. (2) and surrounding text: the linear combination that recovers |n⟩⟨m| from physical states is standard, yet the precise coefficients for the two-qubit case used later are never written out; a short explicit formula would aid reproducibility.","section":null},{"comment":"Fig. 2 caption and text: the non-monotonic infidelity at low shot counts is attributed to “over-biasing toward Clifford operations,” but no quantitative diagnostic (e.g., diamond-norm distance to the nearest Clifford) is supplied; a one-sentence clarification would help.","section":null},{"comment":"Typographical issues: “breifly” (p. 2), “porocess” (p. 4), “Figu. 2” (p. 4), and inconsistent hyphenation of “state-preparation” / “state preparation” appear throughout.","section":null},{"comment":"References: several recent works on GST, compressed sensing, and ML-based process tomography are cited, yet the classic maximum-likelihood process tomography literature (e.g., Śšvanda et al., Fiurášek) is under-represented; adding one or two standard citations would improve context.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The algebraic core is standard least-squares / Procrustes theory; novelty lies mainly in packaging and in the optional extensions (weighting, input-state tomography, unitary projection). The manuscript is therefore better suited to a methods-oriented or pedagogical venue than to a high-impact theory journal. The reader’s and skeptic’s assessments that there is no load-bearing inconsistency are correct; the two major comments above are scope limitations rather than fatal flaws and can be addressed by modest additional text or a short appendix."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that this is a clean, usable packaging of process tomography as ordinary least-squares (Lambda = B A^+), plus three practical knobs: shot weighting, dual input/output tomography for SPAM, and Kronecker-SVD unitary projection. When they use the same linear-inversion state tomography front-end, Procrustes and linear-inversion process tomography are mathematically the same (they say so after Eq. 10); the reported gains come from the optional PSD truncation, the weight matrix W, and the input-state tomography step that the baselines do not get. Those extensions are correctly derived and the four figures show the expected behavior under controlled Lindblad noise.\n\nWhat the paper does well is pedagogy and modularity. The Procrustes framing is easy to explain, any better state estimator can be dropped in without rewriting the process step, and the unitary projection is a neat, standard nearest-Kronecker-product trick. The numerics are transparent: two-qubit amplitude-damping + dephasing channels, even/uneven sampling, perfect vs. noisy preparation, full-rank vs. projected unitary. Average gate infidelity trends are clear and support the claim that the method matches or beats Choi reconstruction and plain linear inversion under the conditions they actually tested.\n\nSoft spots are real but limited. Everything is simulation; no real-device data. They never compare against maximum-likelihood or Bayesian state tomography, so the advantage may shrink once a stronger front-end is used. No code is shipped. Free parameters (T1/T2 ratios, shot-allocation ratios, weighting exponent) are chosen by hand. None of these break the central algebraic claim or the reported figures; they just bound the significance.\n\nThis is for NISQ users and students who want a lightweight, extensible recipe rather than a new theoretical bound. The math is standard and solid, the citation pattern is appropriate, and the work is honest about its scope. I would send it to peer review; a serious referee can ask for code, a stronger state-tomography baseline, and a clearer statement of when the method is strictly superior. Worth engaging if you care about practical tomography tooling.","headline":"Clean pedagogical reformulation of process tomography as a Procrustes problem; competitive numerics, incremental novelty, worth a referee.","tokens_in":12306,"tokens_out":530,"would_cite":false,"duration_ms":5408,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Procrustes tomography reconstructs noisy quantum channels from reconstructed states, matching or beating standard methods under realistic sampling and preparation noise.","keywords":["process tomography","Procrustes problem","NISQ","quantum channels","Choi matrix","linear inversion","state preparation errors","CPTP projection"],"falsifier":"Re-run the same two-qubit Lindblad channels, replace the linear-inversion state tomography step with maximum-likelihood or Bayesian state tomography at identical total shot counts, and check whether Procrustes still matches or beats Choi and linear-inversion process tomography on average gate infidelity.","tokens_in":12344,"feed_emoji":"⚛️","tokens_out":919,"duration_ms":8930,"temperature":0.7,"pith_summary":"Cloud NISQ users often need to characterize the actual noisy process their circuits experience, but full process tomography is expensive and the textbook methods are either nonphysical or awkward to implement. This paper introduces Procrustes tomography: prepare a set of input states, run them through the device, reconstruct the output density matrices by ordinary state tomography, pack those matrices into columns of B and the inputs into A, and recover the process matrix as the least-squares map Lambda = B A^+. Optional weighting handles uneven shot budgets, input-state tomography corrects preparation errors, and simple projections restore complete positivity and unitarity when desired. Numerical experiments on two-qubit amplitude-damping-plus-dephasing channels show that the method matches or exceeds Choi-matrix and linear-inversion tomography under even sampling, uneven sampling, and imperfect preparation, while remaining pedagogically transparent and modular.","feed_headline":"Procrustes fit reconstructs noisy quantum channels from states","feed_subtitle":"Least-squares map from input to output density matrices matches or beats textbook tomography under real sampling noise","key_machinery":"The Procrustes fit Lambda = min_L ||L A - B||_F (or its weighted form), solved by the Moore-Penrose pseudoinverse Lambda = B A^+; subsequent spectral truncation of states, Knee et al. CPTP projection, and Kronecker-product SVD for nearest-unitary projection.","core_discovery":"Process tomography can be reduced to an ordinary Procrustes least-squares problem on reconstructed density matrices: the process superoperator is recovered by Lambda = B A^+, where the columns of A and B are vectorized input and output states. With optional weighting, input-state tomography, CPTP projection, and unitary projection, this construction matches or outperforms both Choi reconstruction and linear-inversion process tomography for representative noisy two-qubit channels under the sampling and preparation conditions studied.","pith_inferences":["Because the method never requires the user to form virtual non-physical operators, it is immediately usable by experimental groups that already have a working state-tomography pipeline.","The same A/B construction could be applied to continuous-time process characterization by treating intermediate-time density matrices as additional columns, turning tomography into a trajectory-fitting problem.","If cloud providers expose only limited mid-circuit measurement or reset options, the modularity of Procrustes still allows users to characterize the effective channel they actually experience."],"forward_implications":["Any future improvement in state tomography (readout-error mitigation, machine-learning estimators, etc.) immediately improves process tomography with no change to the Procrustes algorithm.","State-preparation errors can be removed by simply performing state tomography on both the prepared inputs and the process outputs, at the cost of roughly twice the experiments.","When a gate is known to be nearly unitary, the Kronecker-product SVD projection yields a rank-1 process that saturates at lower infidelity and can be characterized with as few as d+1 states.","Uneven sampling of cheap computational-basis preparations versus expensive superpositions can be optimally weighted by the fourth root of the shot counts, reducing over-bias from low-sample states."],"fun_headline_variants":["Process tomography reduced to Procrustes least-squares on density matrices","Superoperator recovered as Lambda = B A+ from vectorized input-output states","Procrustes fit matches textbook tomography for noisy two-qubit channels","Noisy channels reconstructed via ordinary Procrustes problem on states","Input-output density matrices yield process map by least-squares Procrustes"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The method's reported advantage rests on using linear-inversion state tomography (with negative-eigenvalue truncation) as the estimator of the output density matrices; better state estimators are not compared.","fun_headline_variants_meta":{"raw":{"variants":["Process tomography reduced to Procrustes least-squares on density matrices","Superoperator recovered as Lambda = B A+ from vectorized input-output states","Procrustes fit matches textbook tomography for noisy two-qubit channels","Noisy channels reconstructed via ordinary Procrustes problem on states","Input-output density matrices yield process map by least-squares Procrustes"]},"model":"grok-4.5","effort":"low","cost_usd":0.004322,"raw_usage":{"total_tokens":1177,"prompt_tokens":637,"num_sources_used":0,"completion_tokens":97,"cost_in_usd_ticks":43220000,"prompt_tokens_details":{"text_tokens":637,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":443,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":637,"tokens_out":97,"duration_ms":4297,"temperature":1.0,"reasoning_tokens":443,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T14:09:28.389376+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Re-run the same two-qubit Lindblad channels, replace the linear-inversion state tomography step with maximum-likelihood or Bayesian state tomography at identical total shot counts, and check whether Procrustes still matches or beats Choi and linear-inversion process tomography on average gate infidelity.","supporting_citations":[],"review_version":1}