{"id":"9c831903-dbb9-492a-8090-87411176aa48","arxiv_id":"2601.07037","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A new variational quantum eigensolver ansatz combines subspace-selected superpositions with diagonal Walsh operators to prepare CI wavefunctions, demonstrated on molecules.","lead":"This paper proposes a quantum-circuit ansatz that uses Walsh functions to encode configuration-interaction coefficients of molecular wavefunctions. It demonstrates ground-state energies for small molecules on simulators and IBM's Torino processor, with code and data shared on Zenodo.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Post-selection normalization is mishandled: exact-encoding Eq. 5 omits 1/√D and success probability is 1/D, not 50%.","rationale":"The Reader's weakest_assumption correctly identifies that normalization of the subspace superposition is mishandled, but the specific claimed bound (√D |c_k| ≤ 1) is not the real defect: unitarity of Σ± in Eq. 4 only requires |c_k| ≤ 1, which holds for normalized CI coefficients. The actual issue is that Eq. 5 omits the 1/√D factor and the success probability is 1/D, contradicting the paper's 'always 50%' claim. This is a genuine soft spot in the central derivation: as written, the exact-encoding equation and the probability claim are false. However, the underlying VQE idea remains plausible—the normalized post-measurement state is correct, and the numerical evidence is promising. The Walsh-coefficient formula (Eq. 7) also needs correction, using phases arccos(c_k) rather than c_k. These are presentation/correctness errors that can be fixed, so CONDITIONAL is the right verdict. The Reader's conclusion is therefore appropriate, though the supporting argument for why should be revised.","tokens_in":12765,"tokens_out":24055,"duration_ms":235305,"concrete_test":"Build the circuit described by Eqs. 3–5 for H6 (D=400) using a normalized uniform superposition and Σ± as in Eq. 4. Compute the ancilla-0 branch probability and the normalized post-measurement state. If the probability is ≈1/400 and the state matches FCI only after renormalization, Eq. 5's equality and the 50% success claim are refuted. Equivalently, analytically evaluate ||(Σ+ + Σ−)/2 |S⟩||² for arbitrary D; it equals 1/D, not 1/2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central exact-encoding equation is incorrect as written. With a normalized uniform superposition |S⟩ = D^{-1/2} Σ_k |ψ_k⟩, the unnormalized ancilla-0 branch is (Σ+ + Σ−)/2 |S⟩ = diag(c_k)|S⟩ = D^{-1/2} Σ c_k|ψ_k⟩ = D^{-1/2}|Ψ⟩. Eq. 5 claims this equals |Ψ⟩, dropping the D^{-1/2} factor. The post-measurement state is then |Ψ⟩ only after renormalization, and the success probability is ||diag(c_k)|S⟩||² = Σ|c_k|²/D = 1/D, not the claimed 1/2. For H6 (D=400) this is 0.25%, not 50%. The Reader's worry about a bound √D |c_k| ≤ 1 is a red herring: unitarity of Σ± only requires |c_k| ≤ 1, which normalized CI coefficients satisfy. Separately, Eq. 7 sets Walsh coefficients to the transform of c_k, but the diagonal entries of Σ± are e^{±i arccos(c_k)}; a correct expression would use arccos(c_k). Thus the printed theory does not support the claimed circuit or its constant 50% success probability.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a variational quantum eigensolver (VQE) ansatz for molecular ground states. A uniform superposition over selected Slater determinants is acted on by a diagonal unitary built from Walsh operators, dilated with one ancilla; the resulting amplitudes are supposed to encode the CI coefficients. The authors claim exact FCI/SCI state preparation with a number of CNOT gates linear in the number of determinants D, a constant 50% success probability, and absence of barren-plateau overparameterization. They present simulations for H2, H6, H2O, and a hardware demonstration for H2 on IBMQ Torino.","tokens_in":13067,"tokens_out":13062,"duration_ms":137590,"significance":"The numerical demonstrations are encouraging, and the idea of a systematically improvable, diagonal, subspace-selected ansatz is potentially interesting. The paper also ships code and data on Zenodo, which is a concrete strength. However, the central exact-encoding derivation is not self-consistent: the normalization of the subspace superposition is mishandled in Eq. (5), the success probability is not 50% for general D, and Eq. (7) identifies the Walsh coefficients with the transform of the CI coefficients rather than with the transform of the phase angles required by Eq. (6). These are load-bearing issues for the main claim, and the scaling statements in the abstract and conclusions are also inconsistent. The method may be salvageable after a substantial revision, but the paper as written does not establish its central theoretical claims.","major_comments":[{"comment":"The exact-encoding equation omits the normalization of |S>. With |S> = D^{-1/2} Σ_k |ψ_k>, the ancilla-|0> branch before postselection is (1/2)(Σ+ + Σ-)|S> = Σ|S> = D^{-1/2} Σ_k c_k |ψ_k> = D^{-1/2}|Ψ>, not |Ψ>. The postselection probability is ||Σ|S>||^2 = (1/D) Σ_k |c_k|^2 = 1/D, not 50%, except for D=2. The sentence claiming each conditioned wavefunction has unit norm and that each outcome occurs half the time is therefore incorrect. For H6 (D=400) the success probability would be 0.25%, which materially affects the practical claim.","section":"Theory and Methods, Eqs. (3)-(5)"},{"comment":"The Walsh coefficients a_j in Eq. (7) are defined as the Walsh-Hadamard transform of the CI coefficients c_k, but the diagonal entries of U in Eq. (6) are exp(i Σ_j a_j (-1)^{<k,j>}). To represent Σ+ = diag(c_k + i sqrt(1-c_k^2)), the phases must be arccos(c_k), so the correct relation is a_j = 2^{-r} Σ_k arccos(c_k) (-1)^{<k,j>}. As written, Eq. (7) does not provide a unitary whose diagonal is Σ±, and the exact-encoding claim is not derivable. The full-rank/oversampling analysis is performed on the linear map c -> a, not on the actual state-preparation map, so it does not establish expressibility of the final wavefunction.","section":"Theory and Methods, Eqs. (6)-(7)"},{"comment":"The scaling claims are inconsistent: the abstract and Results state O(D) CNOTs for the Walsh ansatz, while Conclusions states O(D log D) CNOTs. Moreover, no circuit decomposition is supplied for Eq. (6) that justifies either count. A product over O(D log D) Walsh operators implemented with standard Gray-code methods would cost O(r D log D) two-qubit gates in general. Table I reports numerical counts, but no formula or derivation is given. Since the claimed linear scaling is one of the central selling points, this needs to be stated precisely and proved or corrected.","section":"Results and Conclusions, scaling claims"},{"comment":"The paper reports that the quantum FCI result matches classical FCI to machine precision, but it does not clarify whether Eq. (7) was used as an initialization or as an exact encoding, nor how many VQE iterations were required. Given the inconsistency between Eqs. (6) and (7), the numerical exactness cannot be attributed to the Walsh-series construction from the written theory. The role of Eq. (7) in the optimizer should be stated explicitly, and the numerical results should be re-analyzed under the corrected Walsh-phase relation.","section":"Results, Table II and VQE optimization"}],"minor_comments":[{"comment":"The expression inside the square root is typeset as '1 - c_k/||c_k||', which is dimensionally inconsistent. It should presumably be 1 - (c_k/||c||)^2, with ||c|| the norm of the full CI vector. Please correct the notation.","section":"Eq. (4)"},{"comment":"The text says the Torino noisy simulator is shown with blue circles and the Torino device with green triangles, but the legend lists 'ibmq_torino' (blue) and 'fake_torino' (green). The colors are swapped relative to the text.","section":"Figure 2"},{"comment":"The text in the Results section says the example is H2O in a 6-31G basis, but Figure 3 is captioned as the dissociation of H2 in 6-31G. Please correct the mismatch in the text, figure caption, or both.","section":"Results, Figure 3"},{"comment":"The manuscript states both that the O(D log D) Walsh circuit is 'overparameterized' and that the ansatz 'introduces no overparameterization of the chosen subspace.' These statements should be explicitly separated into the QR mode and the random-oversampling mode.","section":"Theory/Conclusions"},{"comment":"There are several typos, e.g., 'resuling' in the abstract, 'probabalistical' in the introduction, and 'dissocation' in the Figure 2 caption. A careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper has a reproducible code artifact and promising small-molecule demonstrations, but the central theoretical derivation needs substantial correction. I recommend major revision rather than rejection because the Walsh-series approach may be salvageable: the authors should fix the subspace normalization and success probability, replace Eq. (7) with the correct arccos-based Walsh-phase relation, and provide a precise gate-count derivation. If these corrections cannot be made, the central claims would not survive."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Promising method and good numerics, but the theory section has a normalization error that breaks the exact-encoding claim. The paper combines a subspace-selected uniform superposition with a partial Walsh series to prepare CI wavefunctions. The oversampling trick for a full-rank WFT is useful, and the H2, H6, H2O, and IBM Torino results are encouraging. Code and data are on Zenodo.\n\nThe main issue: Eq. (5) says postselecting the ancilla gives |Ψ⟩, but with the normalized uniform superposition |S⟩ = D^{-1/2} Σ|ψ_k⟩, the unnormalized branch is diag(c_k)|S⟩ = D^{-1/2}|Ψ⟩. So the success probability is 1/D, not 50% as claimed in the abstract and conclusions. For H6 (D=400) that's 0.25%, which changes the resource picture. Eq. (7) also maps Walsh coefficients to CI coefficients directly, while the diagonal entries of Σ± are exp(±i arccos(c_k)); the correct transform should use phases. These are load-bearing errors in the derivation.\n\nThat said, the numerics suggest the underlying method may work—the energy curves match FCI, and the hardware demo is credible. The errors are likely fixable. The reader's bound worry about √D|c_k| ≤ 1 is a red herring; unitarity only needs |c_k| ≤ 1. The gate-scaling claim is also sloppy: O(D) in the abstract versus O(D log D) for the oversampled approach.\n\nThis deserves a serious referee, because the empirical evidence is solid and the ansatz family is a useful addition if the theory is corrected. As it stands, I wouldn't cite it yet. I'd bring it to reading group to discuss how a plausible circuit can have a flawed derivation.","headline":"Promising ansatz and strong numerics undercut by a normalization error in the exact-encoding claim and a wrong 50% success probability.","tokens_in":13578,"tokens_out":3978,"would_cite":false,"duration_ms":39708,"reading_group":"yes","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A variational quantum eigensolver ansatz encodes a configuration-interaction wavefunction as a uniform superposition over selected determinants followed by a diagonal Walsh-operator product, yielding exact full-CI energies with a gate count","keywords":["variational quantum eigensolver","configuration interaction","Walsh operators","Walsh-Fourier transform","subspace state preparation","selected CI","quantum chemistry","diagonal unitary"],"falsifier":"Take the paper's H2 FCI example: two determinants, coefficients c ≈ 0.99 and c ≈ 0.14. Compute Σ±k from Eq. (4): the first entry has magnitude √2·0.99 ≈ 1.40 > 1, so the matrix is not a valid unitary. A direct numerical check of the circuit's dilation therefore fails, which would disprove the exact-encoding claim as written unless an alternative normalization is supplied.","tokens_in":12650,"feed_emoji":"⚛️","tokens_out":8102,"duration_ms":70978,"temperature":0.7,"pith_summary":"This paper proposes a variational quantum eigensolver ansatz that maps a configuration-interaction (CI) wavefunction onto a quantum circuit in two steps: prepare a uniform superposition over a chosen set of Slater determinants, then apply a diagonal unitary built from a partial Walsh series to imprint the CI coefficients. The result is a systematically improvable, non-overparameterized ansatz in which the number of variational parameters equals the number of determinants, and the CNOT count grows linearly with that number. If correct, this would give a way to reach full-CI or selected-CI ground-state energies without classical matrix diagonalization, and to do so on near-term hardware because the ansatz is a product of commuting diagonal operators that does not form a 2-design and therefore avoids the usual barren-plateau scaling. The paper demonstrates chemical accuracy for small molecules on simulators and on quantum hardware, and reports that a full-rank Walsh transform can be obtained by randomly oversampling D log D Walsh functions.","feed_headline":"One parameter per determinant reaches exact full-CI energies","feed_subtitle":"The diagonal Walsh-series ansatz bypasses classical matrix diagonalization and keeps the parameter count equal to the number of determinants","key_machinery":"The central object is the dilated diagonal unitary U = [[Σ+ , 0], [0, Σ−]] acting on the system plus one ancilla, where Σ+ and Σ− are conjugate diagonal matrices whose average reproduces the CI coefficient matrix. U is implemented as a product of Walsh operators e^{i a_j w_j}, with Walsh coefficients a_j obtained from the CI coefficients by a restricted Walsh-Fourier transform over the D selected determinants. This identity is what converts a nonunitary coefficient-encoding problem into a concrete circuit: a superposition over the subspace, a Walsh-operator product, and a post-selected ancilla measurement.","core_discovery":"The central claim is that any CI wavefunction over a chosen subspace of Slater determinants can be prepared exactly on a quantum computer: uniformly superpose the D bitstrings of the subspace, then apply a diagonal operator whose eigenvalues are the CI coefficients, embedded as a unitary on an extra ancilla qubit via a dilation. Because the diagonal operator is written as a product of Walsh operators (diagonal Pauli strings), it can be implemented with RZ rotation angles given by the Walsh transform of the coefficient vector, using O(D) CNOT gates and exactly D rotation parameters. The paper argues this gives an exact representation of the FCI state when all determinants are included, near-e","pith_inferences":["A direct extension the paper only hints at: replacing the uniform superposition with a symmetry-adapted one (translational, parity) would let the same Walsh encoding prepare ground states of lattice models, not just molecules.","The random oversampling result suggests an empirical rule: roughly half of D log D Walsh functions suffice for full rank. Testing this on larger D could turn a probabilistic statement into a practical heuristic with a proven success probability.","Because the ansatz is diagonal, it could be composed with non-diagonal cluster operators to form a hybrid ansatz for multireference coupled-cluster, a direction the paper mentions as future work."],"forward_implications":["With a full CI expansion, the ansatz yields the exact ground state with a circuit whose CNOT count is linear in the number of Slater determinants, matching or beating conventional hardware-efficient VQE circuits.","Within a selected-CI subspace, energies improve systematically as the subspace grows, with fidelity rising from 0.991 to 0.994 to 1.000 in the paper's H2O example as the threshold tightens.","For large problems, choosing D log D random Walsh functions instead of D exactly yields a full-rank transform with probability approaching 1 as D grows, trading a small amount of overparameterization for an O(D^2 log D) classical preprocessing cost.","Because the ansatz is diagonal and commutative, it does not form a 2-design, which the paper argues mitigates barren plateaus and spurious local minima during the VQE optimization.","The subspace-selection step is independent of the ansatz, so the method can impose particle-number, spin, or other symmetries by choosing the superposition, and extends to any Hamiltonian written in a qubit basis."],"fun_headline_variants":["Exact full-CI energies from Walsh-series ansatz","One parameter per determinant reaches exact ground states","Quantum CI ansatz skips diagonalization, uses D parameters","Walsh-transform VQE: exact CI without classical matrices","Subspace-selected VQE: exact energies via diagonal Walsh ops"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The claim that the circuit exactly encodes the CI wavefunction rests on an unstated normalization condition: after the subspace superposition, every target coefficient must satisfy |c_k| ≤ 1/√D (in magnitude, up to the ancilla dilation), and the paper's own H2 example exceeds this bound.","fun_headline_variants_meta":{"raw":{"variants":["Exact full-CI energies from Walsh-series ansatz","One parameter per determinant reaches exact ground states","Quantum CI ansatz skips diagonalization, uses D parameters","Walsh-transform VQE: exact CI without classical matrices","Subspace-selected VQE: exact energies via diagonal Walsh ops"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000169,"raw_usage":{"total_tokens":1062,"prompt_tokens":664,"completion_tokens":398,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":408,"completion_tokens_details":{"reasoning_tokens":316}},"tokens_in":408,"tokens_out":398,"duration_ms":4874,"temperature":1.0,"reasoning_tokens":316,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T11:12:45.650622+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the paper's H2 FCI example: two determinants, coefficients c ≈ 0.99 and c ≈ 0.14. Compute Σ±k from Eq. (4): the first entry has magnitude √2·0.99 ≈ 1.40 > 1, so the matrix is not a valid unitary. A direct numerical check of the circuit's dilation therefore fails, which would disprove the exact-encoding claim as written unless an alternative normalization is supplied.","supporting_citations":[],"review_version":1}