{"id":"3ac8a42e-f1bd-48ec-8f96-87bb4c841969","arxiv_id":"2607.29382","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For Hamiltonian families with known generators and hidden parameters, the exact query cost of implementing the inverse is determined by spectral sumset relations and representation-theoretic reduction, yielding polynomial or constant bounds for several many-body families.","lead":"This paper asks how many calls to a forward Hamiltonian evolution are needed to implement its exact inverse when the interaction form is known but coupling strengths are not. It shows that algebraic structure, such as eigenvalue relations and symmetry sectors, can reduce the cost from exponential to polynomial or constant in the number of qubits.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the clean-protocol convention is explicit, and the Fourier-support lower bound is robust to relaxing ancilla reset.","rationale":"The reader's ACCEPT verdict is well supported. I focused on the fixed-eigenbasis exact-query theorem, since it is the strongest and most formally load-bearing claim. The proof has two components: lower bound via Fourier support and upper bound via spectral routing. The lower-bound argument that a unit-modulus finite Fourier sum is a single exponential is correct; the support intersection then forces the sumset condition. The upper-bound construction explicitly handles multiplicities and restores ancillas. The Wedderburn reduction appears correct: the split/load/unload compilers preserve query number in both directions, and the phase-synchronization circuits satisfy the stated clean convention. I also checked the application certificates: the Tavis-Cummings active dimensions are independent of n under the stated cutoff, the collective-spin sum of block inversion costs is O(n^3), and the passive-link hierarchy matches the dimension counts and character sets. The only plausible soft spot is the clean-protocol assumption. I considered whether allowing dirty ancillas could reduce the exact query number. For a genuine unitary on the system, the output must be product with an input-independent ancilla factor; the frequency support of that ancilla factor is constrained to the same intersection over eigenvalues, so the sumset criterion is unchanged. Therefore the central claim is not undercut by this modeling choice. I recommend no change to the reader's verdict. agreement_with_reader is partial because the reader identified the clean assumption as the weakest point, and I agree it is the only candidate, but I do not regard it as a load-bearing concern.","tokens_in":28445,"tokens_out":48369,"duration_ms":586015,"concrete_test":"Independently re-derive the Theorem S2 lower bound under the relaxed model where the final ancilla is allowed to be any x-dependent state |a(x)> unentangled with the system. Show that for every eigenvalue λ the support of |a(x)> must be contained in λ+Σ_q(Λ), hence the intersection ∩_λ(λ+Σ_q) is nonempty, which is equivalent to ∃c with c−Λ⊆Σ_q. A symbolic check on a small nonsymmetric spectrum such as Λ={0,1,3} should confirm that no one-query protocol exists even without ancilla reset.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I attempted to find a load-bearing flaw in the central claim (Theorem 1 / Theorem S2: exact fixed-eigenbasis query number equals κ(Λ)). The lower bound is sound: any exact clean q-query circuit gives a finite Fourier-polynomial global phase with unit modulus, forcing a single frequency c with c−Λ⊆Σ_q(Λ), and the sufficiency routing is explicit and multiplicity-independent. The only candidate weakness is the clean-protocol restriction highlighted by the reader. It does not land: a protocol that implements a unitary on the system while leaving ancillas unreturned must still be product with an input-independent ancilla state; its frequency support must lie in ∩_λ(λ+Σ_q), whose nonemptiness is exactly the existence of a common c. Thus relaxing ancilla reset does not relax the sumset condition. The Wedderburn reduction (Theorem 2 / Theorem S8) and the three application bounds are consistent with the stated clean model, and the acknowledged physical limitations (dissipation, ideal unitary queries) are explicitly scoped. No internal inconsistency or unsupported central step was found.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies deterministic exact inversion of Hamiltonian evolution U(x)=exp(i∑ x_j H_j) with known generators H_j but unknown parameters x. It introduces a query-complexity measure for 'clean' protocols that return all ancillas. For one-parameter families with a fixed eigenbasis, Theorem 1 characterizes the exact query number as κ(Λ) = min{q: ∃c, c−Λ ⊆ Σ_q(Λ)}, with explicit constructions and a matching lower bound based on finite exponential polynomials. The result extends to commuting multiparameter families. For general (noncommuting) families, Theorem 2 uses Wedderburn decomposition to show that passive multiplicities do not affect the query number, and gives an automatic blockwise compiler with phase synchronization. Applications to Tavis-Cummings, collective-spin, and passive multimode systems yield structure-dependent upper bounds (constant, O(n^3), and O(N n^2) respectively) that contrast with dimension-only exponential benchmarks.","tokens_in":28694,"tokens_out":21060,"duration_ms":231858,"significance":"The paper gives the first exact characterization of query complexity for structured Hamiltonian inversion, moving beyond worst-case dimension-dependent bounds. The main theorems are accompanied by explicit circuit constructions and matching lower bounds, and the proofs in the Supplemental Material are detailed and self-contained. The Wedderburn reduction is a useful general tool that rigorously separates passive multiplicities from active degrees of freedom. The applications demonstrate that physically motivated symmetric families can admit polynomial or even constant reversing cost. I found no load-bearing technical errors. The clean-protocol convention is explicit and standard; the lower-bound Fourier-support argument does not rely on ancilla reset in an essential way, so this is not a hidden restriction.","major_comments":[],"minor_comments":[{"comment":"The definition of h# around Eq. (S45) is ambiguous; the bar over h may have been lost in typesetting. Clarify that h# denotes the coefficientwise conjugate polynomial and that the subsequent conjugation identities are taken in that sense.","section":"Supplemental Material, Lemma S4"},{"comment":"The phrase 'closed (q_β+1)-call sequence' is not defined in the main text; the reader must infer it from Lemma S16 of the Supplemental Material. Add a one-sentence explanation or a pointer to the lemma for readability.","section":"Main text, Theorem 2 proof sketch"},{"comment":"The legend mixes exact optimal values (ring coupling, bright mode) with constructive upper bounds (arbitrary X). State explicitly which curves are exact and which are upper bounds, and note that the bright-mode q=2 and ring-coupling q=n are fixed-eigenbasis optima.","section":"Figure 2"},{"comment":"The paper's model assumes clean deterministic protocols and ideal unitary queries; this is explicit in the definitions, but an explicit 'scope and limitations' sentence in the Discussion would help avoid misinterpretation (e.g., dissipative effects are not reversed, and approximate inversion is not considered).","section":"Discussion"}],"recommendation":"accept","confidential_remarks":"This is a strong contribution to the quantum query-complexity literature. The reader's concern about the clean-protocol assumption is addressed by the lower-bound robustness: the system amplitudes themselves are finite exponential polynomials, so ancilla reset is not needed for the necessity argument. The paper's use of the same group's prior universal inverter (Ref. [14]) is appropriate and not circular. I recommend acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hi — this one is worth your time. The paper asks: given forward calls to U(x)=exp(i∑x_j H_j) with known generators and unknown parameters, how many calls to implement U(x)^† exactly. The main result (Thm 1) is an exact characterization for one-parameter families with a fixed eigenbasis: the query number is the smallest q such that some c satisfies c−Λ⊆Σ_q(Λ), where Λ is the set of distinct eigenvalues and Σ_q is the q-fold sumset. That is clean and new as far as I know. The necessity argument (finite Fourier polynomial with unit modulus must be a single mode) is standard but correctly applied. The sufficiency construction is explicit and handles multiplicities via a label register.\n\nThe second result (Thm 2) is a Wedderburn reduction: repeated symmetry sectors don't affect the exact query complexity. The proof uses a split/load/unload compiler that preserves query count in both directions. That is a genuinely useful structural statement. The applications are well derived: Tavis-Cummings O_{M,N}(1) vs O(n^{2N}), collective spin O(n^3) vs O(4^n), passive links O(n^2) vs O(n^4). The exact small cases (circulant q=n, bright mode q=2) are worked out with matching lower bounds.\n\nWhere are the soft spots? The clean-protocol convention (all ancillas returned) is explicit, and the stress-test note confirms it doesn't loosen the bound — so it's not a hidden assumption. The paper uses its own earlier universal inverter [14] as a black box in applications; that's legitimate, not circular. The \"practical relevance\" is somewhat idealized (ideal unitary calls, no dissipation), but the paper says so. My only real quibble is that the main text is dense and the proofs are all in the supplement; you'll want the supplement open. The supplement is careful — explicit circuits, dimension counts, and the ILP-based synchronization framework.\n\nVerdict: this deserves a serious referee. The results are novel, the proofs are detailed enough to check, and the separations are concrete. I'd bring it to a reading group and would cite it. If I were refereeing, I'd ask for a bit more discussion of the constant factors from the universal inverter (they're large) and a clearer separation of 'structured bound' vs 'practical overhead', but no load-bearing flaw.","headline":"Solid theory paper: exact sumset characterization of query complexity for Hamiltonian inversion plus a Wedderburn reduction; proofs are detailed, and the separations are real.","tokens_in":29155,"tokens_out":3110,"would_cite":true,"duration_ms":33314,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68"],"pacs":["03.67.Lx"],"model":"deepseek-v4-flash","headline":"The exact cost of reversing an unknown Hamiltonian evolution is set by additive relations among its eigenvalues, and for general families by a symmetry-sector reduction — often dropping from exponential to polynomial or constant.","keywords":["exact unitary inversion","reversing cost","query complexity","additive sumset","Hamiltonian families","Wedderburn decomposition","symmetry sectors","Tavis-Cummings model"],"falsifier":"For the explicit five-level spectrum Λ = {−5, −4, 1, 3, 5} analyzed in the Supplemental Material, the formula gives κ(Λ) = 3 while the universal bound is K−1 = 4. Exhaustively searching all clean two-query inversion circuits — or explicitly constructing one — would settle Theorem 1 in this case: a clean two-query protocol would refute the sumset formula, and a certified impossibility would support it. A broader falsification would be any clean exact q-query inversion with q < κ(Λ) for any fixed-eigenbasis family.","tokens_in":28387,"feed_emoji":"⚛","tokens_out":12691,"duration_ms":126602,"temperature":0.7,"pith_summary":"Reversing a quantum evolution U(t)=e^{iHt} normally costs as much as inverting an arbitrary unitary — Θ(d²) forward calls in dimension d. This paper asks whether knowing the structure of the Hamiltonian, while leaving its parameters hidden, lowers that cost for deterministic exact inversion. For one-parameter families with a fixed eigenbasis, it proves that the optimal number of forward calls is exactly the smallest additive-sumset completion of the distinct eigenvalues (Theorem 1), a number that ignores eigenspace degeneracy and never exceeds K−1. For noncommuting generators, a Wedderburn decomposition shows that repeated symmetry sectors are irrelevant to the query count and provides an automatic construction that assembles blockwise inverses with synchronized phases. On the Tavis–Cummings, collective-spin, and passive multimode families, these results replace exponential dimension-only benchmarks with O(1), O(n³), and O(N n²) bounds, so a rapidly growing many-body Hilbert space need not imply a harder inversion problem.","feed_headline":"Eigenvalue sums set the exact cost of reversing hidden dynamics","feed_subtitle":"Additive spectral structure fixes the minimum forward calls for exact inversion — no parameter estimation needed.","key_machinery":"The load-bearing object is the additive sumset Σ_q(Λ) of the distinct eigenvalues: in a fixed eigenbasis every forward call routes the state through one eigenspace and contributes that eigenvalue to the total phase, so a q-query protocol can produce phase c−λ only if c−λ ∈ Σ_q(Λ); κ(Λ) is the smallest q for which every λ can be completed. The universal K−1 construction rests on a cyclic-shift twirling identity: conjugating U by all powers of a cyclic shift over representative eigenstates accumulates every eigenvalue except the input one, so K−1 queries always suffice. For noncommuting generators, the Wedderburn decomposition of the algebra generated by the known H_j splits the space into pas","core_discovery":"For a one-parameter family U(x)=e^{iHx} with distinct eigenvalues Λ, each forward call contributes one eigenvalue phase, so q-query protocols accumulate phases in the q-fold sumset Σ_q(Λ). The minimum number of calls to implement U(x)^† up to a global phase is exactly κ(Λ)=min{ q : ∃c, c−Λ ⊆ Σ_q(Λ) }: every eigenvalue must be completed to one common frequency c. Routing input components through chosen eigenspaces proves sufficiency; a unit-modulus finite exponential polynomial has a single frequency, proving necessity. The same vector-valued criterion covers commuting multiparameter families, and κ(Λ) ≤ K−1 regardless of degeneracy. For noncommuting families, Wedderburn decomposition removes","pith_inferences":["Because exactness is what makes the sumset condition binding, the advertised speedups are expected to be sensitive: generically perturbing eigenvalues should push κ(Λ) back near K−1, so the gains are tied to the exact spectral relations of the idealized model (the paper notes that symmetry-breaking perturbations may remove the exact advantage, without quantifying the generic value).","The clean-ancilla assumption bounds the lower bound: allowing protocols to leave or consume ancilla states could in principle beat κ(Λ). Checking whether a 'catalyst' inversion exists for the five-level spectrum Λ={−5,−4,1,3,5} would test how tight the clean model is.","κ(Λ) is the optimum of an explicit integer program, so the criterion could be used in reverse as a design tool — compute the completion number for a proposed family's spectrum before searching circuit layouts; the paper does not address the complexity of evaluating κ.","A natural testbed is the parity-refined Lipkin–Meshkov–Glick subfamily mentioned in the Supplemental Material: computing its trace-vector synchronization certificate would show whether the O(n³) bound can be improved below cubic for a physically standard model."],"forward_implications":["For any commuting Hamiltonian family, the reversing cost is bounded by K−1, where K counts distinct eigenvalues; eigenspace degeneracy plays no role, so highly symmetric systems with few distinct energies are cheap to reverse.","Loschmidt echoes, OTOC sequences, and echo-verification circuits can run an exact backward branch built purely from forward calls to the same device, without estimating or recalibrating the unknown coupling strengths.","The Tavis–Cummings family restricted to at most N excitations admits a clean exact inverse with O_{M,N}(1) forward calls, independent of the number n of emitters — versus the O(n^{2N}) dimension-only benchmark.","The collective-spin (Ising/LMG-type) family is reversible in O(n³) calls instead of O(4^n), and passive n-mode links in O(N n²) calls instead of exponential-in-truncated-dimension benchmarks.","The Wedderburn result says that repeated copies of the same unknown dynamics — multiplicity — never increase the exact query cost; only inequivalent active blocks matter."],"supporting_citations":[{"why":"Defines the clean quantum-comb query model (auxiliary registers restored) in which the reversing cost is counted.","marker":"[13]"},{"why":"Supplies the universal d-dimensional unitary inverter with O(d²) queries used as the local block primitive in the Wedderburn-based constructions.","marker":"[14]"},{"why":"Gives the dimension-dependent lower bounds that set the Θ(d²) worst-case benchmark for arbitrary-unitary inversion.","marker":"[15]"},{"why":"Fixes the structure of clean universal inverters (q+1 = ds, residual phase det^s) used for determinant phase synchronization across blocks.","marker":"[25]"},{"why":"States the finite-dimensional Wedderburn theorem used to decompose the generated algebra into passive multiplicities and active blocks.","marker":"[24]"},{"why":"Supplies Schur–Weyl duality, which identifies the repeated (passive) symmetry sectors in the Tavis–Cummings and collective-spin applications.","marker":"[32]"},{"why":"Provides the Tavis–Cummings OTOC protocol whose backward branch requires the exact inverse the paper bounds.","marker":"[28]"},{"why":"Provides the collective-spin echo-verification setting whose inverse sequence is assembled from forward calls.","marker":"[29]"},{"why":"Provides the passive multimode-link setting (single-particle mode transformation) whose photon-number sectors are synchronized by the paper's construction.","marker":"[30]"}],"fun_headline_variants":["Spectral sums set exact query cost for reversing dynamics","Additive eigenvalue relations fix inversion call count","Exact inversion needs fewest calls when eigenvalues align","Hidden Hamiltonian reversal cost equals eigenvalue sumset size","Query complexity for inversion solved by eigenvalue geometry"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The exact optimality claims hold only for 'clean' inversion protocols, meaning every auxiliary register must be returned to its initial state; if a protocol were allowed to leave ancillas altered or entangled, the minimum number of forward calls could in principle be smaller.","fun_headline_variants_meta":{"raw":{"variants":["Spectral sums set exact query cost for reversing dynamics","Additive eigenvalue relations fix inversion call count","Exact inversion needs fewest calls when eigenvalues align","Hidden Hamiltonian reversal cost equals eigenvalue sumset size","Query complexity for inversion solved by eigenvalue geometry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000758,"raw_usage":{"total_tokens":3189,"prompt_tokens":714,"completion_tokens":2475,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":458,"completion_tokens_details":{"reasoning_tokens":2404}},"tokens_in":458,"tokens_out":2475,"duration_ms":18311,"temperature":1.0,"reasoning_tokens":2404,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T04:22:50.294087+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the explicit five-level spectrum Λ = {−5, −4, 1, 3, 5} analyzed in the Supplemental Material, the formula gives κ(Λ) = 3 while the universal bound is K−1 = 4. Exhaustively searching all clean two-query inversion circuits — or explicitly constructing one — would settle Theorem 1 in this case: a clean two-query protocol would refute the sumset formula, and a certified impossibility would support it. A broader falsification would be any clean exact q-query inversion with q < κ(Λ) for any fixed-eigenbasis family.","supporting_citations":[{"cited_title":"Higher-order quantum transformations of Hamiltonian dynamics","cited_arxiv_id":"2303.09788","evidence_quote":"Defines the clean quantum-comb query model (auxiliary registers restored) in which the reversing cost is counted."},{"cited_title":"Universal algorithm for transforming Hamiltonian eigenvalues","cited_arxiv_id":"2312.08848","evidence_quote":"Supplies the universal d-dimensional unitary inverter with O(d²) queries used as the local block primitive in the Wedderburn-based constructions."},{"cited_title":"2011 , isbn =","cited_arxiv_id":null,"evidence_quote":"Supplies Schur–Weyl duality, which identifies the repeated (passive) symmetry sectors in the Tavis–Cummings and collective-spin applications."},{"cited_title":"Communications in Mathematical Physics , mendeley-groups =","cited_arxiv_id":null,"evidence_quote":"Provides the Tavis–Cummings OTOC protocol whose backward branch requires the exact inverse the paper bounds."},{"cited_title":"Physical Review Letters , keywords =","cited_arxiv_id":null,"evidence_quote":"Provides the collective-spin echo-verification setting whose inverse sequence is assembled from forward calls."}],"review_version":2}