{"id":"47575dff-4b3d-4c8a-90fe-564f87417bed","arxiv_id":"2609.35265","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Under unknown SPAM noise, the diagonal dissipative coefficients plus Type I and Type II off-diagonal components of a local Lindbladian are learnable with trusted single-qubit control, while all other components are generically unlearnable up to a constant error floor.","lead":"This paper characterizes which parts of a quantum noise model (a Lindbladian) can be learned when the preparation and measurement of the device are themselves noisy and unknown. It proves that some components are learnable with efficient protocols using only trusted single-qubit gates, while all others carry an irreducible error floor that no amount of data can remove.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Full commuting gauge class may change Type I/II components; invariance is only proved for S_theta gauges.","rationale":"I read the paper in good faith and found the lower-bound construction and the learning protocols internally coherent under the restricted S_theta gauge family. The reader's weakest-assumption choice, SPAM visibility, is explicit, reasonable, and acknowledged by the authors. The more fragile point is the unstated step from S_theta invariance to invariance under every SPAM-admissible gauge transformation commuting with single-qubit control. Proposition 10 explicitly presents the larger full class, and the paper asserts rather than proves that the learned quantities are invariant under that full class. If a generic small G outside S_theta changes a Type I or Type II component while preserving Lindbladian and SPAM admissibility, the exact classification of universally gauge-invariant components fails and the efficient learning theorems cannot hold uniformly. A direct first-order calculation on a minimal interior physical Lindbladian would settle this. Hence I recommend conditional acceptance pending that verification.","tokens_in":37952,"tokens_out":40079,"duration_ms":539208,"concrete_test":"Compute the first-order action of the general G in Eq. (21) on a minimal two-qubit physical Lindbladian containing a Type II coefficient, e.g., a Kossakowski matrix with diagonal 1 on a = I⊗X and b = Z⊗X and off-diagonal ±it, with |t| < 1. Check whether the linearized change in Im chi_{a,b} (and analogously Re chi for a Type I pair U = X⊗R, V = Y⊗R) vanishes for all small gauge perturbations a_S with a_empty = 0. If a nonzero variation exists on the physical tangent space, the S_theta-based classification is incomplete and the learning theorem is in question; if it vanishes, the asserted full-class invariance is confirmed.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Definition 11 and Theorem 13 classify universal invariance under the N-parameter S_theta family (Eq. 15-17), while Proposition 10 gives the full commuting gauge class G = sum_S g_S Pi_S with exponentially many parameters. The transition from S_theta-invariance to learnability under arbitrary admissible G is asserted in the paragraph after Prop. 10 (\"Theorems 25, 27, and 29 give learning protocols, which also establish invariance under every commuting gauge transformation...\"), but no proof is supplied. For a Type II pair U = I_q ⊗ R, V = Z_q ⊗ R, the first-order action of G on chi involves independent coefficients a_{S∪q} - a_S for each S = supp(R ⋆ R'); under S_theta these all collapse to a single per-qubit total, which is exactly why Theorem 13 can prove S_theta invariance. Since any sufficiently small G is SPAM-admissible (with E_p = D_{e^{-2}}^{⊗N}, both G E_p and E G^{-1} remain CPTP and satisfy visibility), a nonzero such variation at an interior physical Lindbladian would give two experimentally indistinguishable models with different learned values, contradicting Theorems 27 and 29. Thus the exactness of the classification is load-bearing and unproven for the full gauge class.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper studies identifiability and efficient estimation of bounded-degree local Lindbladians under unknown state-preparation-and-measurement (SPAM) noise, using only trusted single-qubit control. The authors introduce an N-parameter family S_theta of single-qubit depolarizing gauge transformations (Eqs. 15–17) that commute with single-qubit controls and can be absorbed into the SPAM operations, and they characterize the commuting-gauge class G = sum_S g_S Pi_S (Proposition 10, Eq. 21). Their classification (Theorem 13, Corollary 17, informal Theorem 1) states that the universally gauge-invariant components are exactly the diagonal dissipative coefficients alpha_aa, Type I real parts Re alpha_ab where P_a, P_b differ on one qubit with both nonidentity there, and Type II imaginary parts Im alpha_ab (with one identity on the differing qubit) together with single-qubit Hamiltonian coefficients; every other locally allowed component is generically gauge-dependent among physical Lindbladians. Theorem 18 constructs Lindbladians for which each dependent component has system-size-independent Omega(1) gauge width under CPTP and bounded-coefficient constraints.","tokens_in":38212,"tokens_out":20164,"duration_ms":147692,"significance":"If the results hold, this is an important contribution to Lindbladian learning: it is, to my knowledge, the first complete identifiability classification for general (non-Pauli-diagonal, Hamiltonian-containing) bounded-degree local Lindbladians under unknown SPAM of unspecified form, with the additional strength that only trusted single-qubit control is required and no ancillas. The positive results are nearly optimal in epsilon and logarithmic in N, and the lower bound (Theorem 18) is genuinely informative in that it survives complete-positivity and coefficient-bound constraints with a system-size-independent width. I found the central proofs sound at the level of my inspection: the parity argument in Theorem 13 (Eq. 35), the convex-geometry upgrade in Corollary 17, the color-separation argument in Theorem 18 (Eq. 45), and the pulse-rate/telescoping argument in Appendix A are all coherent, and Lemma 28 is fully proved. The SPAM visibility assumption (Definition 8) is honestly stated and the paper appropriately notes that boundary-induced invariants are not yet learnable.","major_comments":[{"comment":"The exactness of the classification under the full commuting gauge class G of Proposition 10 is asserted, not proved. Theorem 13 proves universal invariance only under the N-parameter family S_theta (Eqs. 15–17), and its parity argument (Eq. 35) uses the additive per-qubit exponents theta·s(P), which are special to S_theta. For a general G with independent multipliers g_S, the factor g_{supp(R*U)}/g_{supp(R*V)} in the transform of the Pauli-transfer coefficients (Eq. 23) does not factor per qubit, so the pairing argument of Theorem 13 does not extend. The only bridge supplied is the sentence after Proposition 10 stating that Theorems 25, 27, and 29 'also establish invariance under every commuting gauge transformation that preserves the admissibility of the generator and SPAM.' This needs an actual argument: a protocol that, for every admissible L in the model class, outputs an estimate within epsilon of the true value must, by identical-data indistinguishability of gauge-equivalent models, certify invariance up to 2epsilon over the class, and a linear functional bounded on an open subset of V_P is identically zero on V_P; this deserves a short lemma. Without it, the informal Theorem 1 and the abstract's 'sharp characterization' overstate what is proved, since the negative direction (generic dependence under full G) is sound but the positive exactness is not.","section":"§4.1, after Proposition 10; Theorem 13; Eq. (35)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"There is a mismatch between the formal and informal statements of the classification. Definition 11.1 defines universal gauge invariance relative to the S_theta family, and Theorem 13 is consistent with that definition. But the introductory Theorem 1 and the prose after Proposition 10 speak of invariance under every commuting gauge transformation and of a 'sharp classification' of 'the' gauge freedom, which suggests the stronger full-G meaning. Read in the strong sense, the claim is unproved as noted above; read in the weak (S_theta) sense, the claim that the listed components are 'exactly' the learnable ones needs the protocol-side argument anyway, because S_theta-invariance alone neither implies learnability nor precludes full-G dependence for Type I and Type II components. Please state which sense is intended, and either add the 2epsilon/linearity argument for full-G exactness or explicitly qualify the classification as holding for the S_theta family with learnability supplied by Theorems 25/27/29. The sketch on pp. 33–34 (Section 5.2.2's parallel-twirl analysis) is an internal consistency check that suggests the invariant direction is true, but it does not by itself discharge the exactness claim.","section":"Statement of scope of 'universally gauge-invariant'; Definition 11 vs. informal Theorem 1","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The sentence 'Theorems 25, 27, and 29 give learning protocols, which also establish invariance under every commuting gauge transformation...' is grammatically and logically overloaded; as the major comments note, it carries an unproved load-bearing claim and should be expanded into a lemma or removed.","section":"§4.1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The visibility assumption is the key empirical premise and is stated clearly, but the paper could add one sentence in Section 5 noting where each protocol step uses it (Eqs. 64, 68, 95) so a reader can verify that no hidden global-closeness assumption enters; currently the dependence is somewhat scattered.","section":"Definition 8 and Remark 9","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The symbol Q is used for a prepared/measured Pauli operator, for a summation Pauli in Eq. (57), and as a factor in the transformed-restricted coefficients (Eq. 76); this is typographically convenient but worth disambiguating once in Section 5.","section":"Notation, §5.2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The congruence 'epsilon_R ≡ |G_R ∩ (D \\ B)| + |B \\ G_R| (mod 2)' is correct but the phrase 'Conversely, every subset G ⊆ D(U,V) can occur as G_R' deserves its explicit construction spelled out (choose R_i = U_i for i in G, and appropriate third-Pauli or identity factors otherwise); the proof is complete in substance but this is the one place I had to pause.","section":"Eq. (35) and surrounding text","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"A few minor typographical items: 'Theorem 25, 27, and 29' should be 'Theorems 25, 27, and 29'; the phrase 'in situ' in Ref. [27] is italicized inconsistently; and the statement on p. 20 of the proof of Theorem 18, 'Here we have used the fact that G_C is invariant under the gauge transformation and has identity GKS matrix', could explicitly note that the depolarizing generator has I_{4^{|C|}-1} as its GKS matrix in the nonidentity Pauli basis.","section":"General","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The core technical content — the learning protocols, the pulse-rate bound, and the Omega(1) lower bound — appears sound to me, and the missing full-G exactness argument is plausibly a short lemma (identifiability-by-protocol implies up-to-2epsilon invariance over the bounded class, extended to V_P by linearity). The revision should therefore be quick despite the 'major_revision' label. Two editorial notes: (1) the generative-AI disclosure states that ChatGPT drafted proofs of Theorem 18, Proposition 16, Theorem 32, and the Lemma 23 extension; these happen to be load-bearing results, and although my spot-checks (color-separation in Theorem 18, telescoping in Lemma 23, the inner-product computation in Theorem 32) found the arguments correct, I would recommend an extra independent check of exactly those proofs before publication. (2) The paper's related-work table and positioning are fair and appropriately delimit the contribution against Ref. [33] and Ref. [22]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"The substantive result is real: under unknown SPAM with only trusted single-qubit gates, the universally gauge-invariant components are exactly the diagonal, Type I, and Type II classes, and everything else is generically dependent with an N-independent Omega(1) error floor even under CP constraints. That classification is new relative to Sinha's multi-qubit-control work and the Pauli-diagonal results, and it answers a specific practical question. The protocols are the other half of the contribution: SPAM factors cancel cleanly in the decay-rate ratios and in the derivative identity (98), and the local-visibility SPAM model is honestly stated and reasonable, allowing global far-from-ideal noise. Theorems 25, 27, and 29 achieve near-optimal sample and time scaling, and the proofs are mostly readable and complete.\n\nThe stress-test concern is partially right and partially resolvable. Theorem 13 proves invariance only for the N-parameter S_theta family, and the paper's single sentence claiming the protocols establish invariance under every commuting gauge is an assertion, not a proof; the parity argument genuinely does not transfer to the exponentially large gauge class, and a first-order algebra check does not obviously cancel. But I traced it further and the protocols rescue the claim. Their correctness holds for arbitrary Definition-8 SPAM, and a sufficiently small commuting gauge leaves the transformed SPAM in that class, so running the same protocol on both models (identical data, by the gauge construction) must return the same coefficient. That is a three-line argument, and the paper should write it out. As it stands, the universality claim cannot be verified without reconstructing it.\n\nThe AI disclosure is welcome and honest. That three of the load-bearing arguments (Theorem 18, Proposition 16, Theorem 32, Lemma 23) were machine-drafted does temper confidence; the lower bound in Theorem 18 in particular deserves an independent human pass, and the authors' revision-and-check process is plausible but not auditable. Section 6's boundary-induced invariance is exploratory, and the authors are upfront that learning those parameters is open. None of this is fatal.\n\nVerdict: accept with revisions. The one substantive gap is the full-commuting-gauge invariance argument; the rest is polish. The paper deserves a serious referee and belongs in the reading group.","headline":"A genuinely new classification of which Lindbladian coefficients survive unknown SPAM under single-qubit control, mostly well-proved, with one gap worth closing before it goes out.","tokens_in":38835,"tokens_out":9626,"would_cite":true,"duration_ms":86254,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["81P68","81S22","68Q12"],"pacs":["03.67.-a","03.65.Yz"],"model":"deepseek/deepseek-v4-flash","headline":"Unknown SPAM noise makes most Lindbladian components unlearnable, and the paper pins down exactly which ones survive: the diagonal dissipative rates, one class of real off-diagonal couplings, and the imaginary couplings together with…","keywords":["Lindbladian learning","SPAM noise","gauge invariance","gauge freedom","single-qubit control","bounded-degree locality","sample complexity","complete positivity"],"falsifier":"The cleanest check is a direct two-qubit construction: write a Lindbladian with a chosen support pattern, apply the $S_\\theta$ gauge together with the depolarizing SPAM channels of Remark 9, and verify numerically that (1) the four invariant component classes stay unchanged as $\\theta$ varies, (2) every predicted gauge-dependent component shifts by at least the constant $\\eta_{k,d}$, and (3) the two parameter settings yield identical outcome distributions for every allowed protocol — which would make the estimators' $\\Omega(1)$ error floor visible directly. A null result, namely a predicted gauge-dependent component that an estimator provably recovers despite such identical distributions, would break the classification.","tokens_in":39336,"feed_emoji":"⚛️","tokens_out":5843,"duration_ms":44013,"temperature":0.7,"pith_summary":"Unknown state-preparation-and-measurement (SPAM) noise makes some parts of a quantum system's noise generator — its Lindbladian — impossible to learn, because distinct generators can produce statistically identical data. The paper's first result is a complete answer to which pieces are lost: with only trusted single-qubit control, the only universally learnable components are the diagonal dissipative rates, one class of real off-diagonal couplings (Type I), and one class of imaginary couplings together with single-qubit Hamiltonian terms (Type II). Every other component is generically gauge dependent even among physically valid (completely positive) generators and can carry a constant-size uncertainty that no amount of data removes, independent of system size. The second result is constructive: SPAM-robust protocols that need only single-qubit gates estimate every surviving component to precision $\\epsilon$ using $\\widetilde{\\mathcal O}(\\epsilon^{-2}\\log(N/\\delta))$ experiments and total evolution time for bounded-degree local Lindbladians. If correct, this maps exactly what coarse open-system characterization can and cannot promise on real devices.","feed_headline":"Symmetry pins down the unlearnable parts of a noisy Lindbladian","feed_subtitle":"Unknown SPAM caps most couplings; a small invariant set is still learnable near-optimally.","key_machinery":"The engine is the $N$-parameter family of gauge transformations $S_\\theta = e^{\\Gamma_\\theta}$, where $\\Gamma_\\theta$ rescales each Pauli operator $P$ by $e^{-\\theta\\cdot s(P)}$; these commute with every trusted single-qubit unitary, preserve locality, and can be absorbed into the unknown SPAM channels so that all outcome probabilities are unchanged (Equation 19). Theorem 13's phase-parity argument, in the $\\chi$-matrix representation, shows the coefficients of $L$ rescale as $\\chi_{U,V}\\to e^{\\theta\\cdot(s(V)-s(U))}\\chi_{U,V}$ plus mixing terms, and a parity bookkeeping over the differing-qubit set $D(U,V)$ decides exactly which real or imaginary component admits a gauge-dependent summand. For learning, the protocols use Pauli and partial twirls to isolate decay-like signals whose SPAM factors cancel in ratios, a back-substitution over support inclusion to recover individual coefficients from restricted ones, and a Chebyshev-endpoint derivative estimator (Lemma 28) for the Type II imaginary parts, which cannot be converted into decay rates.","core_discovery":"Under unknown SPAM noise with only trusted single-qubit control, two physically distinct local Lindbladians can produce data that is exactly identical in every experiment, so part of the generator is unlearnable in principle. The paper pinpoints the boundary: a component survives exactly when it is invariant under an $N$-parameter family of local depolarizing gauge transformations $S_\\theta$ that commute with all single-qubit unitaries and can be absorbed into the SPAM operations. That yields an exact list — the diagonal dissipative coefficients $\\alpha_{aa}$, the real parts $\\mathrm{Re}\\,\\alpha_{ab}$ for pairs differing on a single qubit with both nonidentity there (Type I), and the imaginary parts $\\mathrm{Im}\\,\\alpha_{ab}$ for pairs differing on a single qubit with exactly one identity there, together with single-qubit Hamiltonian coefficients $h_c$ (Type II). Every other component is generically gauge dependent among physical complete-positive Lindbladians with any prescribed bounded-degree support pattern, and there exist generators for which each such component shifts by an $\\Omega(1)$ amount — independent of system size — under an admissible gauge transformation, so no estimator can beat a constant error floor. For the survivors, the paper gives SPAM-robust protocols using only single-qubit gates with $\\widetilde{\\mathcal O}(\\epsilon^{-2}\\log(N/\\delta))$ experiments and total evolution time for constant locality and degree.","pith_inferences":["The commuting-gauge characterization suggests a quantitative transition: adding trusted entangling gates should shrink the invariant family and unlock more components, by contrast with the multiqubit-control setting; this transition could be probed numerically on small systems.","The $\\Omega(1)$ error floor implies that, without extra calibration assumptions, only the invariant projection of the Lindbladian is operationally meaningful in a continuous-time gate-set-tomography sense — a statement worth testing on device data where SPAM is far from ideal.","The boundary-induced invariance of Hamiltonian coefficients not in the span of dissipative operator products echoes the Hamiltonian-not-in-Lindbladian-span condition for Heisenberg-limited metrology; a concrete question is whether those boundary-invariant Hamiltonian terms admit shot-noise-beating estimation, which the paper leaves open."],"forward_implications":["With unknown SPAM and trusted single-qubit control, no experiment can determine the full Lindbladian: the generically gauge-dependent components carry an irreducible uncertainty bounded below by a constant independent of system size, even under complete positivity (Theorem 18).","All universally gauge-invariant components — diagonal dissipative rates, Type I real couplings, and Type II imaginary couplings plus single-qubit Hamiltonian terms — are efficiently learnable with $\\widetilde{\\mathcal O}(\\epsilon^{-2}\\log(N/\\delta))$ experiments and total evolution time using only single-qubit gates, within polylogarithmic factors of near-optimal lower bounds.","The guarantees hold even when global SPAM is arbitrarily far from ideal, as long as local preparation and measurement visibility are bounded away from zero independently of $N$ (Definition 8).","At boundary instances where the Kossakowski matrix has zero diagonal entries, physical positivity can freeze additional Hamiltonian coefficients under admissible gauge transformations, though learning these instance-specific invariants remains open (Theorems 31 and 32)."],"supporting_citations":[{"why":"Establishes the gauge (similarity-transformation) freedom of self-consistent gate-set estimation that motivates treating SPAM as absorbing gauge degrees of freedom.","marker":"[16-19]"},{"why":"Supplies the decomposition of local-Clifford conjugation into exact-support irreducible subspaces, from which Proposition 10 derives the full family of commuting gauge transformations.","marker":"[39, 40]"},{"why":"Provides the near-optimal sample and total-evolution-time lower bounds that the protocols here match within polylogarithmic factors.","marker":"[10]"},{"why":"The source of the local pulse-rate argument that Lemma 23 adapts to bound finite-Pauli-pulse bias independently of system size.","marker":"[28]"},{"why":"Gives the Chebyshev interpolation and aliasing bounds on which the derivative estimator in Lemma 28 rests.","marker":"[41]"},{"why":"The prior full classification of Lindbladian learnability under trusted multiqubit control, which the weaker single-qubit-control setting of this work extends.","marker":"[33]"},{"why":"The Heisenberg-limit metrology condition whose Hamiltonian-not-in-Lindbladian-span structure the boundary-induced invariance criteria echo in Section 6.","marker":"[30]"}],"fun_headline_variants":["Gauge invariant set is all you can learn from a noisy Lindbladian","SPAM noise sets hard boundary on Lindbladian learnability","Lindbladian learning: only gauge-invariant bits survive SPAM","Sharp gauge rule finds unlearnable parts in SPAM-ridden Lindbladians"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The load-bearing assumption is local SPAM visibility (Definition 8): the noisy prepared state and noisy measurement must retain overlap at least $r_p$ and $r_m$, bounded away from zero independently of $N$, with the intended local operators; if that local visibility decays with system size, then the identification of the invariant components, the bounded signals the protocols rely on, and every sample-complexity guarantee collapse.","fun_headline_variants_meta":{"raw":{"variants":["Gauge invariant set is all you can learn from a noisy Lindbladian","SPAM noise sets hard boundary on Lindbladian learnability","Lindbladian learning: only gauge-invariant bits survive SPAM","Sharp gauge rule finds unlearnable parts in SPAM-ridden Lindbladians"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000208,"raw_usage":{"cost":0.000207648,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000207648,"upstream_inference_prompt_cost":0.000126448,"upstream_inference_completions_cost":8.12e-05},"total_tokens":1398,"prompt_tokens":1108,"completion_tokens":290,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":256,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":210}},"tokens_in":1108,"tokens_out":290,"cost_source":"openrouter_usage","duration_ms":4150,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":210,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-09-29T12:16:14.753490+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"The cleanest check is a direct two-qubit construction: write a Lindbladian with a chosen support pattern, apply the $S_\\theta$ gauge together with the depolarizing SPAM channels of Remark 9, and verify numerically that (1) the four invariant component classes stay unchanged as $\\theta$ varies, (2) every predicted gauge-dependent component shifts by at least the constant $\\eta_{k,d}$, and (3) the two parameter settings yield identical outcome distributions for every allowed protocol — which would make the estimators' $\\Omega(1)$ error floor visible directly. A null result, namely a predicted gauge-dependent component that an estimator provably recovers despite such identical distributions, would break the classification.","supporting_citations":[{"cited_title":"Near-optimal learning of local Lindbladians,","cited_arxiv_id":null,"evidence_quote":"Provides the near-optimal sample and total-evolution-time lower bounds that the protocols here match within polylogarithmic factors."},{"cited_title":"Learning many-body Hamiltonians with Heisenberg- limited scaling,","cited_arxiv_id":null,"evidence_quote":"The source of the local pulse-rate argument that Lemma 23 adapts to bound finite-Pauli-pulse bias independently of system size."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Chebyshev interpolation and aliasing bounds on which the derivative estimator in Lemma 28 rests."},{"cited_title":"Efficient and SPAM-Robust Ansatz-Free Lindbladian Learning","cited_arxiv_id":"2606.20706","evidence_quote":"The prior full classification of Lindbladian learnability under trusted multiqubit control, which the weaker single-qubit-control setting of this work extends."},{"cited_title":"Achieving the Heisenberg limit in quantum metrology using quantum error correction,","cited_arxiv_id":null,"evidence_quote":"The Heisenberg-limit metrology condition whose Hamiltonian-not-in-Lindbladian-span structure the boundary-induced invariance criteria echo in Section 6."}],"review_version":1}