{"id":"bf25c1fa-30de-4cb5-9667-5a7c5a8b24eb","arxiv_id":"2607.26756","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.5,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"Coherent-error Fisher information discarded by standard randomized compiling is fully preserved in twirl labels and can be recovered unbiasedly at the quantum limit with zero extra circuit cost.","lead":"Randomized compiling does not erase coherent gate errors; it moves that information into the random twirl labels that labs usually throw away. Keeping those labels recovers the coherent phases at the quantum information limit on existing hardware with no extra circuits.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the strongest claim (conservation + label recovery at QFI, Theorems 1–2 and Eq. 5) and the weakest modeling assumptions (\theta-independent \theta and self-consistent v under standard incoherent noise). Those assumptions are load-bearing for guaranteed unbiasedness outside the controlled setting, yet they are standard, explicitly scoped, and not violated by the proofs or the injected-phase experiments. Simulation to machine precision and hardware recovery of known \theta∗ supply independent support. No stronger concern (e.g., a flaw in the chain-rule step, a missing cross-term, or circular use of the marginal) appears on close reading. Therefore the ACCEPT verdict and low correctness_risk stand; the concrete test above is a useful robustness check rather than a necessary condition for the claim as stated.","tokens_in":5922,"tokens_out":584,"duration_ms":11531,"concrete_test":"Re-run the statevector oracle suite (or the ibm_marrakesh protocol) with a controlled non-unital non-Pauli channel (e.g., a small coherent ZZ term plus amplitude damping) whose PTM is not diagonal in the Pauli basis; extract v from the marginal as prescribed and check whether |ˆ\theta-\theta∗| remains inside the shot-noise envelope. If it does, the practical domain is larger than the theorem statement; if it systematically exceeds ~0.01 rad, the unbiasedness claim needs an explicit noise-class caveat.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—an exact Fisher-information conservation law relocating coherent-error information into twirl labels, with a noise-aware unbiased estimator recovering \theta at the QFI limit—holds under the paper’s stated conditions. Theorem 1’s score identity follows immediately from \theta-independence of \theta (standard for classical RC twirls chosen by the experimenter). Theorem 2’s PTM decomposition and the estimator ˆ\theta=arcsin(E[s·b]/v) are standard for Pauli-twirled channels; the self-consistent extraction of v=\theta_Y from the same-run marginal is consistent with the model class of “standard incoherent channels.” Hardware recovers known injected phases to within shot noise while the marginal stays near zero, and simulation bias is reported at 2e-8 across 12 families. The reader’s weakest assumption (possible bias in v under unmodeled non-Pauli/non-unital noise, and the \theta-independence premise) is real but does not undermine the theorems as proved or the controlled experiments as reported; it only bounds the domain of guaranteed unbiasedness, which the paper already states. No internal inconsistency or hidden circularity is evident.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript argues that randomized compiling does not destroy coherent-error information but relocates it into the discarded twirl labels. Theorem 1 establishes an exact Fisher-information conservation law: F_cond = F_marg + Δ with Δ ⪰ 0 equal to the label-recoverable information, reducing at full twirl to F_marg = 0 and Δ = F_0. A complementarity relation (Eq. 5) makes this explicit in the tailoring parameter T. Theorem 2 gives a PTM-based noise-aware estimator θ̂ = arcsin(E[s·b]/v) that is unbiased once the visibility v = λ_Y is read from the same-run marginal. The conservation identity is checked to machine precision (bias ≤ 2×10^{-8}) across 12 circuit families, and a controlled injection experiment on ibm_marrakesh recovers θ* to within 0.0063 rad while the marginal estimator remains near zero at all depths.","tokens_in":6138,"tokens_out":1347,"duration_ms":51817,"significance":"If correct, the result reframes a standard premise of RC and yields a zero-extra-circuit coherent-error map from data already collected in any RC run. The derivation uses only the θ-independence of the twirl and the chain rule, so it is not model-fragile within that setting. Strengths that raise confidence include: (i) machine-checked numerical conservation across diverse circuits with reported bias six orders below shot noise; (ii) an explicit unbiased estimator under the stated PTM noise class with self-consistent visibility; (iii) a falsifiable hardware test in which the marginal null and labeled recovery are shown side-by-side; and (iv) public simulation code and oracles. The work is a natural coherent-sector complement to Pauli-noise learnability and is of immediate practical interest to groups already running RC.","major_comments":[{"comment":"Abstract and Main Result claim recovery 'at the quantum Fisher-information limit,' and Fig. 1(b) plots E_s[F_Q]. Theorem 1 rigorously conserves the classical joint/conditional Fisher information of (b,s). Equality with the quantum FI is justified in the pure/ideal generator calculation (Depth Scaling, Eq. 9) but is not proved for the noisy CPTP setting of Theorem 2; the Outlook itself lists a Cramér–Rao converse for Eq. (8) as future work. Please restrict the QFI-limit language to the regime where it is shown (ideal or pure-state generators), or supply the missing argument that the labeled classical FI saturates the QFI under the noise model of Theorem 2. This is a wording/scope fix, not a challenge to conservation or unbiasedness.","section":"Abstract; Theorem 1; Depth Scaling; Outlook"},{"comment":"Theorem 2 and consequence C assert an exactly unbiased estimator under 'all standard incoherent channels' via v = λ_Y. The derivation uses the Pauli-twirled adjoint form E^†_tw(Y) = t_Y I + λ_Y Y. Unmodeled coherent leakage outside the twirled Pauli model, strong non-unitality beyond the offset cancellation, or SPAM that correlates with labels could bias v or E[s·b]. The hardware run reports v ≈ 0.993 and sub-shot-noise recovery for injected R_z errors, which supports the claim in that instance, but the manuscript should state the precise noise class (Pauli-twirled CPTP with label-independent SPAM) under which unbiasedness is guaranteed, rather than the broader phrase in the abstract.","section":"Theorem 2; Eq. (8); Abstract"}],"minor_comments":[{"comment":"Fig. 1(a) caption and Eq. (5) give leading-order (in θ) complementarity; the exact conservation F_cond = F_marg + Δ holds beyond that order. A sentence distinguishing exact conservation from the first-order split in T would avoid confusion.","section":"Fig. 1(a); Eq. (5)"},{"comment":"Identifiability matrix F ∘ Σ (Eq. 6) is important for multi-parameter or correlated twirls but is not exercised in the hardware experiment (single injected phase). A brief remark on how the ibm_marrakesh design sits inside that framework would help.","section":"Identifiability Matrix; Results"},{"comment":"Table I reports |θ̂ − θ*| but not uncertainties or the number of shots entering each row; adding standard errors (or noting they match the stated shot-noise floor) would make the table self-contained.","section":"Table I"},{"comment":"Typographical inconsistencies: 'twirldestroys' / 'twirl labels—the' spacing in the abstract/intro, 'ibm marrakesh' vs 'ibm_marrakesh', and 'Cram´ er–Rao' accent encoding. Also 'The twirldestroyscoherent-error information' in the Introduction.","section":"Abstract; Introduction"},{"comment":"The claim that the labeled estimator 'eventually surpasses coherent accumulation on generic circuits' (Depth Scaling) is supported numerically in Fig. 1(b) for α = 1; a one-line condition on α (already partly stated) next to that sentence would clarify the analytic scope.","section":"Depth Scaling; Eq. (9)"}],"recommendation":"minor_revision","confidential_remarks":"The central mathematics is standard Fisher-information reasoning applied in a place the community had not looked; I do not see a load-bearing error. The two major comments are scope/wording tightenings around QFI language and the noise class for unbiasedness—appropriate for minor revision, not a barrier to eventual acceptance. Fit for a quant-ph theory-and-methods venue is good. No concerns about circularity or hidden fitting: injected θ* and the marginal-vs-labeled contrast are clean. Future-dated arXiv stamp and AI-assistance disclosure are noted but do not affect the scientific assessment."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple and useful: the usual story that randomized compiling destroys coherent-error information is wrong. The information is moved into the per-shot twirl labels that everyone already generates and then throws away. Keep the labels, form the label–outcome correlation, correct by the visibility read from the same-run marginal, and you recover the coherent angles at the quantum Fisher limit with no extra circuits.\n\nWhat is actually new is the exact conservation law (Theorem 1) plus the complementarity relation that at full twirl F_marg goes to zero while Δ goes to F0. That is standard score-identity and chain-rule work, cleanly applied. Theorem 2 then gives an explicitly unbiased estimator once v = λ_Y is taken from the PTM diagonal of the twirled channel. The paper positions itself correctly against Wallman–Emerson, Pauli learnability, GST, and hidden inverses; it is the coherent-sector complement to the stochastic learnability results.\n\nThey back it properly. Conservation is checked to machine precision (bias ≤ 2e-8) across twelve circuit families in a self-contained numpy simulator. Hardware on ibm_marrakesh recovers injected phases to within ~0.006 rad while the marginal estimator stays near zero at every depth; visibility is flat. Code and data are promised. That is real evidence, not hand-waving.\n\nSoft spots are real but bounded. The theorems assume θ-independent labels (true by construction for experimenter-chosen twirls) and that standard incoherent channels dominate so that v extracted from the marginal is unbiased. Hardware validates recovery of known injected phases, not unknown native coherent errors, so the calibration claim is one step short of fully in-situ. Depth scaling and multi-qubit numerics look fine; the single-qubit hardware demo is narrow but sufficient for the claim as stated. None of this breaks the central argument.\n\nThis is for people who already run RC or care about NISQ characterization. It deserves a serious referee. I would bring it to reading group and I would cite the conservation + label estimator. Send it out.","headline":"Clean Fisher conservation argument that RC relocates coherent-error information into the discarded twirl labels, with a zero-extra-cost estimator that works on hardware for injected phases.","tokens_in":6771,"tokens_out":529,"would_cite":true,"duration_ms":9294,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.67.Lx","03.67.Pp","03.65.Wj"],"model":"grok-4.5","headline":"Randomized compiling does not erase coherent-error information; it moves that information into the discarded twirl labels, which recover the errors at the quantum limit with no extra circuits.","keywords":["randomized compiling","coherent errors","Fisher information","twirl labels","noise tailoring","quantum characterization","Pauli transfer matrix"],"falsifier":"Inject a known coherent phase on a device, run standard randomized compiling while logging the per-shot Pauli labels, and check whether the label–outcome estimator recovers the injected value to within shot noise while the label-discarding marginal estimator remains consistent with zero.","tokens_in":6774,"feed_emoji":"⚛️","tokens_out":771,"duration_ms":14533,"temperature":0.7,"pith_summary":"Randomized compiling is used to turn systematic gate over- and under-rotations into ordinary stochastic noise so that errors no longer grow quadratically with depth. The usual reading is that the random Pauli twirl destroys the coherent-error signal once the shots are averaged. This paper shows the opposite with an exact conservation law for Fisher information: the information removed from the averaged outcome distribution is preserved completely in the classical record of which random gates were chosen on each shot. Keeping those labels and correlating them with the measured outcomes recovers the coherent phases at the quantum Fisher-information bound, unbiased by standard incoherent noise, and at zero extra circuit cost. The conservation identity is proved, checked to machine precision on twelve circuit families, and demonstrated on a 127-qubit processor where the labeled estimator tracks injected phases to a few milliradians while the ordinary marginal estimator sees nothing.","feed_headline":"Twirl labels recover the coherent errors RC was thought to erase","feed_subtitle":"Keeping the random gate choices yields unbiased coherent phases at the quantum limit, free.","key_machinery":"Fisher-information conservation law (Theorem 1): the joint Fisher information of outcomes and labels equals the conditional Fisher information and splits as F_marg + Δ, where Δ ≻ 0 is exactly the information recoverable from the labels; at full twirl F_marg vanishes while Δ saturates the quantum bound.","core_discovery":"The coherent-error information that randomized compiling removes from the averaged output distribution is relocated, not destroyed, into the per-shot twirl labels. Retaining those labels and forming a simple label–outcome correlation recovers the coherent parameters at the quantum Fisher-information limit, unbiased under standard incoherent channels, with no additional circuits.","pith_inferences":["Existing open RC and Pauli-learnability datasets could be re-analyzed offline to produce coherent-error maps that were never intended by the original experiments.","Adaptive choice of the twirl ensemble could be used to isolate crosstalk subspaces that ordinary full twirling mixes together.","If label logging becomes standard, routine RC runs double as continuous, zero-overhead coherent calibration for the whole device."],"forward_implications":["Any group already running randomized compiling can extract a coherent-error map simply by logging the labels it currently discards.","One RC dataset simultaneously yields stochastic Pauli rates from the marginal and coherent angles from the labels.","Labeled Fisher information grows linearly with circuit depth and can surpass the bounded accumulation of an untwirled coherent circuit.","The recovered angles can be fed back as compile-time virtual-Z corrections without new characterization experiments.","Twirl-group design controls which linear combinations of coherent errors remain separately identifiable."],"fun_headline_variants":["Twirl labels keep the coherent errors RC only seems to erase","RC moves coherent-error info into the twirl labels it discards","Retain twirl labels to recover coherent phases at the quantum limit","Label-outcome correlation restores coherent errors from RC averages","Coherent-error information survives intact in per-shot twirl labels"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The random twirl labels must be chosen independently of the unknown coherent errors, and the signal shrinkage factor must be readable without bias from the same run’s ordinary averages.","fun_headline_variants_meta":{"raw":{"variants":["Twirl labels keep the coherent errors RC only seems to erase","RC moves coherent-error info into the twirl labels it discards","Retain twirl labels to recover coherent phases at the quantum limit","Label-outcome correlation restores coherent errors from RC averages","Coherent-error information survives intact in per-shot twirl labels"]},"model":"grok-4.5","effort":"low","cost_usd":0.00446,"raw_usage":{"total_tokens":1251,"prompt_tokens":707,"num_sources_used":0,"completion_tokens":71,"cost_in_usd_ticks":44604000,"prompt_tokens_details":{"text_tokens":707,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":473,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":707,"tokens_out":71,"duration_ms":7655,"temperature":1.0,"reasoning_tokens":473,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T22:00:12.575605+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Inject a known coherent phase on a device, run standard randomized compiling while logging the per-shot Pauli labels, and check whether the label–outcome estimator recovers the injected value to within shot noise while the label-discarding marginal estimator remains consistent with zero.","supporting_citations":[],"review_version":1}