{"id":"b0e527b5-3400-4adf-acd2-2a1beb139446","arxiv_id":"2512.11619","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"DAQC synthesis of any two-body Hamiltonian can be done in analog time at most √3·T·‖hP⊘hS‖₂, and the paper asserts this bound is tight.","lead":"The paper claims a tight upper bound on the total analog-block time needed to synthesize any two-body Hamiltonian evolution in the digital-analog quantum computing (DAQC) paradigm, improving on previously conjectured bounds. If correct, it would give a clean resource estimate for DAQC circuits, but the proof as written has a major gap for general qubit number.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The all-n induction in the proof of Theorem 1 is asserted, not proved; the inradius bound 1/√3 is verified exactly only for n=3 and numerically for small n.","rationale":"The reader's weakest assumption and my concern coincide: the proof of the inradius lower bound is the load-bearing step, and it is not demonstrated for arbitrary n. The geometric framing is promising and the numerical checks up to n=7 are genuine supporting evidence, but they do not constitute the claimed theorem. I am not asserting the bound is false; I am asserting that the text does not prove it. The algebraic slip in Section II.A is secondary and repairable; the induction gap is not repaired by the manuscript as written. Since a rejected preprint should not be conditionally accepted on the strength of an asserted induction, the reader's REJECT verdict stands unchanged.","tokens_in":10185,"tokens_out":17740,"duration_ms":154854,"concrete_test":"Compute exactly the Euclidean inradius of P_8 for the ZZ case. Using P_n = conv{p_i p_j}_{i<j} with p ∈ {±1}^n modulo global sign, n=8 has 128 vertices in R^28; enumerate the facets of P_8 (or the vertices of its polar) in exact rational arithmetic and compute min |c|/∥a∥ over all facet inequalities. If min < 1/√3, Theorem 1 is false. If min = 1/√3, repeat for n=9, and extend to arbitrary two-body Hamiltonians for n=5, to test the recursive step before accepting the all-n claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central bound ∥t_opt∥_1 ≤ √3∥b∥_2 is equivalent to the assertion that the DAQC sign-pattern polytope P_n has Euclidean distance at least 1/√3 from the origin to every facet. For n=3 this is computed exactly (regular tetrahedron), but for general n the proof proceeds by a recursive/embedding argument. Section II.A contains the explicit admission: 'we assume that the worst problem corresponds to an n=3 all-to-all problem embedded in a larger system, and assume that this holds up to n qubits.' The parameterization step is not a formal induction: it does not bound the new facets introduced when adding qubit n+1, and the treatment of n=4–6 and n≥7 is a sketch. Exact facet information is reported only up to n=7 for ZZ and n=4 for arbitrary two-body Hamiltonians. Hence the theorem's tightness for arbitrary n is not established by the text. The claimed counterexample to the BHK conjecture is also not exhibited, which is consistent with this same gap: the n=3 embedded problem saturates both bounds and does not by itself break the BHK bound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies digital-analog quantum computation (DAQC) and proposes a tight upper bound on the total analog evolution time required to simulate an arbitrary two-body Hamiltonian. After vectorizing Hamiltonians, the compilation problem is reduced to solving M t = b with t ≥ 0, where M is a sign-pattern matrix. The authors' main result, Theorem 1, states that for any compatible target and source Hamiltonian, ∥t_opt∥_1 ≤ T√3 ∥h^P ⊘ h^S∥_2, and that the bound is tight, realized by problems with three non-zero couplings. The proof is based on a geometric interpretation: the columns of M generate a polytope whose inradius would need to be at least 1/√3. The paper verifies the n=3 case exactly, gives numerical evidence for moderate n, and extends the claim to arbitrary two-body Hamiltonians. It also states that the previous Baßler-Heinrich-Kliesch bound is not optimal and that a counterexample is provided.","tokens_in":10465,"tokens_out":8051,"duration_ms":74356,"significance":"If Theorem 1 were rigorously established, the result would be a clean, parameter-free bound on DAQC time resources, scaling linearly with the number of qubits for all-to-all Hamiltonians and only polylogarithmically in the number of couplings in typical cases. The geometric reformulation in terms of the inradius of the DAQC sign-pattern polytope is appealing and could be a useful tool for future work. The paper also contains careful exact or numerical information for small system sizes. However, the central theorem for arbitrary n is not proven: the key inductive step is explicitly assumed, not derived, and no counterexample to the BHK conjecture is actually exhibited. The claimed tightness and the claimed improvement over the previous bound therefore remain unsupported. The numerical experiments are suggestive but cannot replace a proof of the general statement.","major_comments":[{"comment":"The induction proving the bound for all n is asserted, not demonstrated. The text says, 'we assume that the worst problem corresponds to an n=3 all-to-all problem embedded in a larger system, and assume that this holds up to n qubits,' and later 'we can extend the upper bound to arbitrary n.' This does not bound the new facets introduced when adding a qubit, nor does it justify that the n=3 configuration remains the closest facet to the origin. Exact facet information is reported only for n≤7 for ZZ Hamiltonians and n≤4 for arbitrary two-body Hamiltonians. Hence Theorem 1 for general n is not proven by the manuscript.","section":"Section II.A, proof of Theorem 1"},{"comment":"The proof that a boundary point b̃ must have minimal 1-norm equal to 1 contains a scaling error. If M t = b̃ with ∥t∥_1 = ξ < 1, the proposed t' = (2−ξ)t satisfies ∥t'∥_1 = (2−ξ)ξ, not 1. The contradiction therefore does not follow as written. A correct argument could rescale by 1/ξ and use convexity, but that argument is not given. This is a load-bearing step in the geometric reduction.","section":"Section II.A, paragraph beginning 'For any b̃'"},{"comment":"The paper states that the BHK conjecture is 'actually not optimal' and that a counterexample is provided, but no explicit counterexample appears. The n=3 worst-case vector b = (−α,−α,−α) gives ∥t_opt∥_1 = T√3∥b∥_2 = 3Tα, which equals the BHK bound for odd n=3 and is smaller than the BHK bound for larger n. Thus this example does not violate Conjecture 1. No explicit problem is exhibited for which ∥t_opt∥_1 exceeds the BHK bound.","section":"Introduction and Conclusion"},{"comment":"The extension to arbitrary two-body Hamiltonians lacks a verified base case for n=2. The text states that n=2 is non-trivial and identifies candidate worst directions, but it does not compute the facets or the inradius for the 9-dimensional polytope. Since the subsequent argument again relies on the same assumed induction for ZZ Hamiltonians, the general theorem is not established beyond the explicitly checked small cases.","section":"Section II.A, arbitrary two-body Hamiltonians"}],"minor_comments":[{"comment":"The phrase 'linear dependence with the number of couplings' is inaccurate: for equal-magnitude couplings the 2-norm grows as sqrt(N_couplings), not linearly in N_couplings. The scaling is linear in the number of qubits for all-to-all Hamiltonians.","section":"Abstract"},{"comment":"Typo: 'Firsly' should be 'Firstly'.","section":"Section II"},{"comment":"The notation M ∈ M_{d,d'}(±1) is nonstandard; it would be clearer to state that M is a d×d' matrix with ±1 entries and full row rank.","section":"Eq. (7), notation"},{"comment":"The description of the numerical generation of 'green' problems is imprecise ('close to the axes' with 'a maximum of 6 nonzero elements'). Please specify the exact distribution to make the numerical results reproducible.","section":"Section II.B and figures"}],"recommendation":"reject","confidential_remarks":"The paper has a useful geometric reformulation and correct-looking small-n computations, but the main theorem for arbitrary n is not proven; the proof explicitly assumes the key inductive step. The claimed counterexample to the BHK bound is also not exhibited. I would be willing to reconsider if the authors supply a complete proof of the inradius bound for all n and an explicit counterexample, but as submitted the central claims are not established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the bound is new and the geometric angle is right, but the proof of Theorem 1 for arbitrary n is not there. The paper explicitly says so at two key points: \"we assume that the worst problem corresponds to an n=3 all-to-all problem embedded in a larger system\" and \"by brute-forcing the resolution of this problem, we see...\" That is an induction hypothesis plus a numerical check, not a proof. The reader's verdict is correct.\n\nWhat is actually new: the bound ∥t∥1 ≤ √3 T ∥hP⊘hS∥2 is not in the earlier literature; the previous best was linear in n with an ∞-norm. The claim that the Baßler-Heinrich-Kliesch conjecture is not tight is also new, though the counter-example is never exhibited. The geometric picture (projecting onto the DAQC polytope and bounding the inradius) is a natural and likely correct way to attack the problem, and the n=3 case is computed exactly: the regular tetrahedron gives the √3 constant.\n\nWhere it falls apart: the step from n=3 to general n. For n=4–6 the paper parameterizes the new couplings and asserts that setting them to zero is worst. For n≥7 it says the topology reduces to n=6 and \"we can extend the upper bound to arbitrary n.\" That is not a formal induction; it does not bound the facets introduced by the new qubit. The numerical exact facet calculation stops at n=7 for ZZ and n=4 for general two-body, so the tightness for larger n is unverified. There is also a minor slip in the decomposition for n=4: the α_i are not constrained to sum to 3α, so the decomposition does not exactly reproduce b. That is fixable, but it adds to the impression that the induction was never fully written down.\n\nThe citation pattern is fine. The main self-citation [22] is to a prior result that the sign-pattern matrix spans the space and the polytope contains the origin—that is a legitimate base to build on, not a circular argument.\n\nBottom line: this is a plausible conjecture with a suggestive partial proof and good small-n numerics. It deserves a serious referee, but not acceptance in its current form. The authors need to either prove the inradius bound for all n—probably via explicit facet bounds for the recursive polytope structure—or state the result as a conjecture and show the counter-example to BHK directly. As is, I would not cite it as a theorem. If you are working on DAQC resource estimates, it is worth reading for the geometric framing and the n=3 constant.","headline":"The claimed √3 bound is fresh and plausible, but the all-n proof is asserted rather than shown; the paper is a promising draft, not a theorem.","tokens_in":10953,"tokens_out":2292,"would_cite":false,"duration_ms":19507,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68","52B12"],"pacs":[],"model":"deepseek-v4-flash","headline":"Digital-analog quantum computation has a tight worst-case time bound: any compatible two-body Hamiltonian can be implemented in total analog time at most T√3 times the 2-norm of the coupling ratios, and some three-coupling problems require","keywords":["digital-analog quantum computation","DAQC","time-optimal compilation","Hamiltonian simulation","convex polytope","two-body Hamiltonians","tight bound","Trotter decomposition"],"falsifier":"Compute exactly, for the ZZ-Ising polytope with n=8 qubits, the minimum Euclidean distance from the origin to any facet (the columns of M are the vertices, so the closest facet gives the worst-case rescaling). If that distance, after normalizing the columns appropriately, is smaller than the corresponding distance for the embedded n=3 tetrahedron—i.e., if a problem b can be found with ∥t_opt∥_1 > T√3∥hP⊘hS∥2—then the claimed bound fails; the authors' own facet calculations stop at n=7.","tokens_in":10058,"feed_emoji":"⏱️","tokens_out":5796,"duration_ms":54560,"temperature":0.7,"pith_summary":"The paper aims to settle the worst-case time cost of digital-analog quantum computation (DAQC), where a computation is compiled into single-qubit gates interleaved with evolutions under a fixed source Hamiltonian. It claims a tight upper bound: any compatible problem Hamiltonian can be implemented in total analog time at most T√3∥hP⊘hS∥2, and there exist problems—three non-zero couplings among three qubits—that reach this bound. This improves on earlier bounds that grew quadratically with the number of qubits, yielding a linear-in-qubits estimate for the worst case. If correct, resource estimates for DAQC simulations and comparisons with purely digital approaches become rigorous.","feed_headline":"T√3 bound fixes worst-case DAQC total time","feed_subtitle":"Linear-in-qubits resource bound for digital–analog circuits; three-coupling problems hit it exactly.","key_machinery":"The key object is the convex polytope M whose vertices are the columns of the sign matrix M, which encodes the sign flips produced by single-qubit Pauli gates sandwiching each analog block. Solving a DAQC problem b = T h^P ⊘ h^S as M t = b, t ≥ 0, is equivalent to writing b as a positive combination of those vertices; the optimal total time ∥t∥_1 is the scale factor needed to bring b onto the polytope's surface. The argument reduces the worst-case time to the minimal Euclidean distance from the origin to a facet of M, which is verified for small n to occur on the n=3 embedded tetrahedron. The recursive structure M(n+1) = [[L,L'],[M(n),M(n)]] carries the extension, with the new couplings para","core_discovery":"The central claim is Theorem 1: for any problem Hamiltonian H_P compatible with a source Hamiltonian H_S, an optimal DAQC schedule exists with ∥t_opt∥_1 ≤ T√3 ∥h^P ⊘ h^S∥_2, where h^P and h^S are the vectors of the Hamiltonian couplings. The proof treats the columns of the sign matrix M as vertices of a convex polytope containing the origin; any problem vector b is a positive rescaling of a surface point of that polytope, and the optimal total time equals the rescaling factor. The paper identifies the worst facets as those belonging to the smallest non-trivial system: for ZZ Hamiltonians this is n=3, where the polytope is a regular tetrahedron and the closest facets correspond to four sign p","pith_inferences":["If the recursive step holds for all n, the same geometric inradius argument should extend to other gate sets and non-Pauli sign-flipping operations, giving similar √3-type constants wherever the worst-case polytope is the smallest non-trivial one.","The polytope-inradius perspective suggests a practical way to generate hard DAQC instances: problems whose direction is perpendicular to the closest facets of M, which could be found for moderate system sizes by solving the facet-closest-point problem without full facet enumeration.","A direct device-level test is possible on small processors: compiling the identified worst-case problems (three equal-magnitude couplings of opposite sign on three qubits) should reproduce the √3 saturation; deviations would reveal gaps between the compilation model and hardware constraints such as finite gate speed or crosstalk."],"forward_implications":["Any DAQC protocol with a compatible source Hamiltonian can be compiled with total analog time bounded by T√3 times the 2-norm of the coupling ratios, giving a direct resource estimate independent of the number of qubits except through that norm.","The bound is tight: some all-to-all problems with only three non-zero couplings require the full amount, so the constant √3 cannot be improved without restricting the problem class.","The result yields a two-sided characterization T∥hP⊘hS∥∞ ≤ ∥t_opt∥_1 ≤ T√3∥hP⊘hS∥2, with both endpoints achieved by explicit problems.","For all-to-all Hamiltonians, the 2-norm of the coupling ratios grows as the square root of the number of couplings, so the worst-case DAQC time scales linearly with the number of qubits, confirming the earlier conjecture's linear scaling though with a different constant and norm.","It also disproves a previously conjectured sharp bound based on the ∞-norm and n or n−1, providing a concrete counter-example.","The authors argue the result carries over to arbitrary two-body Hamiltonians and to DAQC protocols with fixed or arbitrary single-qubit rotations, since Pauli-based protocols give the worst case."],"fun_headline_variants":["DAQC total time pinned down to tight linear bound","T√3 theorem sets linear limit for DAQC schedules","Digital-analog quantum circuits get exact time bound","New bound cuts DAQC simulation time estimates to linear","Tight DAQC time bound scales with couplings, proof shows"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof depends on an induction step that is asserted rather than fully demonstrated: the authors assume that for any number of qubits, the worst-case problem is the n=3 all-to-all problem embedded in the larger system, with all newly added couplings set to zero; the manuscript states this explicitly and gives only a sketch for n≥8.","fun_headline_variants_meta":{"raw":{"variants":["DAQC total time pinned down to tight linear bound","T√3 theorem sets linear limit for DAQC schedules","Digital-analog quantum circuits get exact time bound","New bound cuts DAQC simulation time estimates to linear","Tight DAQC time bound scales with couplings, proof shows"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000189,"raw_usage":{"total_tokens":1134,"prompt_tokens":666,"completion_tokens":468,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":410,"completion_tokens_details":{"reasoning_tokens":390}},"tokens_in":410,"tokens_out":468,"duration_ms":4404,"temperature":1.0,"reasoning_tokens":390,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T16:49:18.885038+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute exactly, for the ZZ-Ising polytope with n=8 qubits, the minimum Euclidean distance from the origin to any facet (the columns of M are the vertices, so the closest facet gives the worst-case rescaling). If that distance, after normalizing the columns appropriately, is smaller than the corresponding distance for the embedded n=3 tetrahedron—i.e., if a problem b can be found with ∥t_opt∥_1 > T√3∥hP⊘hS∥2—then the claimed bound fails; the authors' own facet calculations stop at n=7.","supporting_citations":[],"review_version":1}