{"id":"c3a4dfd9-41c2-4572-83e2-3a2876aaa633","arxiv_id":"2607.15425","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"ON-VQE estimates molecular energies from quantum-measured occupation numbers alone, reducing VQE measurement settings to a single qubit-wise commuting group.","lead":"This paper proposes ON-VQE, a hybrid quantum-classical algorithm that measures only occupation numbers from a quantum device and feeds them into classical natural-orbital functionals to estimate molecular energies. It claims to cut the number of distinct quantum measurement settings from O(M^4) to a single commuting group, with a hardware demo on an 8-hydrogen cluster.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unverified premise: the self-consistently optimized orbitals are assumed to diagonalize the measured 1RDM, but off-diagonal elements are never measured, so the 'ONs' may not be natural occupations.","rationale":"The paper's mathematical core—that occupation numbers are obtained from single-qubit Z measurements and that all such measurements form one QWC group—is correct. The measurement-cost reduction from O(M^4) to O(M/2) is a genuine contribution. However, the central claim 'accurate electronic energies can be obtained from quantum measurements of ONs alone' is only valid if the measured diagonal elements are true ONs, i.e., if the optimized orbital basis C diagonalizes the 1RDM of the prepared state. The paper does not verify this. I examined Algorithm 1 and the surrounding text: the objective function used for both the inner (θ) and outer (C) optimization is E_NOF[n,C] with n the measured diagonal of ⟨Z_p⟩. No term in this objective depends on off-diagonal 1RDM elements. Therefore a stationary point of the loop does not imply γ_pq = 0. For a general UpCCD state, off-diagonal elements are generically non-zero at first order in the pair amplitudes. Thus the assumption is not a harmless approximation; it is a correctness condition that could fail. The noiseless H8 validation (Fig. S1) is a valuable empirical check and suggests the method works for that case, but it does not justify the general claim. My proposed test directly measures the off-diagonal elements and compares the diagonal-only NOF energy against the correct NOF energy from the full 1RDM. This settles whether the concern lands. I agree with the reader's weakest_assumption; no other issue appears as load-bearing. The verdict should remain CONDITIONAL: the paper is promising but must verify or justify the natural-orbital diagonalization before the central claim is accepted.","tokens_in":10239,"tokens_out":8843,"duration_ms":91976,"concrete_test":"Run ON-VQE in the noiseless simulator on a multi-orbital system where the UpCCD ansatz is non-trivial, e.g., N2 or H2O in STO-3G, and at convergence compute the full 1RDM γ_pq = ⟨a†_p a_q⟩ in the optimized basis C (using the same quantum circuit). (1) If max_{p≠q} |γ_pq| > 0.01, then C is not the natural-orbital basis. (2) Diagonalize γ to obtain true ONs and NOs; evaluate the NOF energy (e.g., PNOF7) using these true ONs/NOs and compare with the ON-VQE energy. A difference beyond ~1 mHa would show that the diagonal-only evaluation is uncontrolled. This test directly settles whether the measured quantities are ONs.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the orbital basis C optimized in Algorithm 1 is the natural-orbital basis of the prepared state, i.e., that the one-particle RDM γ_pq = ⟨a†_p a_q⟩ is diagonal in C. The paper measures only ⟨Z_p⟩ (Eq. 4) and never checks γ_pq for p≠q. The self-consistent loop minimizes E_NOF[n(C,θ), C], a function of only the diagonal n_p and the integrals in C; no term involves off-diagonal 1RDM elements. There is therefore no variational or algorithmic force driving γ_pq → 0. For the UpCCD ansatz (Eq. 8), even first-order perturbation theory gives γ_ia ∝ t_i^a ≠ 0 when the basis is not the natural basis. If C is not the eigenbasis of γ, the measured values are diagonal elements in an arbitrary basis, not ONs, and the NOF energy E_NOF[n,C] is not the energy of the state under the NOF functional. The H8 validation (Fig. S1) shows the method reproduces PyNOF for that system, but it does not establish that the self-consistent C diagonalizes γ in general. This is the load-bearing weakness: the advertised 'ONs alone' claim depends on an unverified premise.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes ON-VQE, a hybrid quantum-classical algorithm that measures only single-qubit Z expectation values, which in the Jordan–Wigner representation give occupation numbers in the currently chosen orbital basis. These occupations are fed into a natural orbital functional (NOF), which is evaluated classically together with a self-consistently optimized orbital matrix C. The quantum circuit is a unitary pair-double-excitation ansatz (UpCCD). The authors claim that this reduces the measurement overhead from O(M^4) to O(M/2) measurement settings, validate the method on small molecules in STO-3G, and report a hardware demonstration on the cubic H8 cluster with post-selection.","tokens_in":10634,"tokens_out":10128,"duration_ms":108483,"significance":"If the central premise is correct, the reduction of the measurement problem to a single qubit-wise-commuting group of Z operators is a practically important step for NOF-based quantum algorithms. The derivation of n_p = (1 - <Z_p>)/2 is correct and the workflow is clearly presented. The idea of performing error mitigation directly in occupation-number space is original and appealing. However, the paper does not establish that the self-consistently optimized orbital basis is actually the natural-orbital basis of the prepared state, and the numerical demonstrations are mostly in a regime where all tested NOFs coincide. The hardware validation is also statistically under-reported. These issues prevent the paper from fully supporting its headline claim.","major_comments":[{"comment":"The central measurement target is not established. Eq. (4) is correct, but it gives the population of spin-orbital p in the current basis C, not necessarily the natural occupation. Algorithm 1 measures only <Z_p> and evaluates E_NOF[n,C]; the objective contains no off-diagonal 1-RDM element, so the ADAM update of C has no mechanism to drive <a†_p a_q> (p≠q) to zero. For the unitary state generated by Eq. (8), off-diagonal 1-RDM elements are generically nonzero unless C is exactly the natural-orbital basis. The manuscript asserts that C is 'determined self-consistently' but never verifies diagonality. A concrete test is to compute <a†_p a_q + h.c.> in the optimized basis for a nontrivial system, or to add a penalty/constraint that enforces diagonality; without this, the measured quantities are not necessarily occupation numbers and the NOF energy is uncontrolled.","section":"Eq. (4) and Algorithm 1"},{"comment":"The numerical validation does not support the accuracy claim. Table I uses only STO-3G with Ng≤2, where each electron-pair subspace has at most one weakly occupied orbital and n_p = 1 - n_g, so all tested NOFs reduce to very similar expressions. The article itself concedes this. Fig. S1 validates the noiseless ON-VQE implementation against the same PNOF7 functional used in the energy evaluation, which checks code consistency, not physical accuracy. A meaningful validation should include at least one system with Ng≥3 or a larger basis, and should compare against FCI/CCSD or conventional VQE energies, not only against the same NOF.","section":"Table I and Fig. S1"},{"comment":"The O(M/2) claim counts QWC groups or the number of independently measured observables, but the total shot cost is not analyzed. All occupations share a single QWC group, but the number of shots required to reach a given precision depends on the variance of <Z_p>, which is not discussed. The comparison of 1198 groups vs 1 group is striking, but the manuscript should define the cost metric explicitly and provide a sampling-error analysis; otherwise the headline 'measurement cost' reduction is ambiguous and could be misleading.","section":"Measurement-cost claim"},{"comment":"Figure 3 reports no error bars, and each point is the average of only ten post-selected hardware executions. The mitigation protocol in Table SI involves numerous tuned parameters (k_sigma, pair-sum tolerance, maximum ON/pair corrections, polarization thresholds, lambda_pol). Without cross-validation, sensitivity analysis, and reporting of the fraction of discarded samples, the agreement with the noiseless curve may be partly a result of selection bias. Please provide confidence intervals, acceptance rates, and a sensitivity study for the mitigation hyperparameters.","section":"Hardware results and Fig. 3"}],"minor_comments":[{"comment":"The text refers to noiseless ON-VQE, but the figure legend appears to label the curve as NOF-VQE (PNOF7). Please reconcile the terminology.","section":"Fig. S1"},{"comment":"The relation between the geminal parameters n_g in Eq. (6) and the variational angles theta_p^g in Eq. (8) is not specified. It is unclear whether thetas are constrained to reproduce the desired occupations or are free variational parameters.","section":"Eqs. (6)-(8)"},{"comment":"The caption, 'Representative values obtained for the largest molecular systems considered,' is incomplete. Please list the molecules, basis, and the exact number of QWC groups for each method.","section":"Fig. 2"},{"comment":"Reference 36 points to a single Python file in a GitHub repository. A versioned release or a DOI would be preferable for reproducibility.","section":"Reference 36"},{"comment":"The step 'Infer complementary occupations from the electron-pair constraints (when applicable)' is not defined in the main text. Please specify exactly when and how this inference is applied, as it can mask measurement errors.","section":"Algorithm 1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is likely salvageable. The decisive technical issue is the diagonality assumption: a quick numerical check of off-diagonal 1-RDM elements in the optimized basis would resolve whether the method is measuring true occupation numbers. The limited benchmarks and the heavily post-processed hardware demonstration are secondary but should be addressed in a revision. This is a feasible major revision rather than a reject."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere is my read. The paper's core trick is real and clean: measuring ⟨Z_p⟩ gives occupation numbers directly, and the UpCCD ansatz keeps those Z's in one qubit-wise commuting group, so the whole NOF energy comes from a single measurement setting. That is a legitimate reduction from NOF-VQE's O(M²) 1RDM reconstruction, and the benchmark numbers in Table I support the claim that the same set of ONs can feed PNOF5, PNOF7, and GNOF without extra quantum work. The authors also deserve credit for shipping code and for being upfront that the STO-3G benchmarks with Ng≤2 make the NOFs nearly equivalent, so those numbers do not oversell the physics.\n\nThe soft spots are real but I would not call any of them fatal. The measurement math, Eq. (4), is textbook—the Jordan-Wigner string cancels, so n_i = (1−⟨Z_i⟩)/2 is exact regardless of the basis. The actual concern is the unverified premise that the classically optimized orbital coefficient matrix C diagonalizes the measured 1RDM. The self-consistent loop only minimizes E_NOF[n(C,θ), C], which depends only on the diagonal n_p; nothing in the objective involves off-diagonal γ_pq. So there is no variational force pushing γ_pq to zero, and the measured values are only true ONs if C happens to be the eigenbasis. The H8 noiseless validation against PyNOF checks implementation, not this premise, because PyNOF itself uses the same NOF functional. This is a load-bearing weakness, and the paper's own text acknowledges the orbitals are 'determined self-consistently' without ever verifying the off-diagonal elements. Minor: the hardware curve has no error bars, and the error-mitigation protocol has several tuned parameters (kσ, tolerances, λpol), so the hardware demonstration is suggestive, not definitive.\n\nWho is this for? People working on NISQ measurement reduction and NOF-based quantum algorithms. It is a legitimate incremental extension of the authors' NOF-VQE line, and the single-QWC-group observation is worth knowing. I would not treat the 'ONs alone' claim as established until the off-diagonal 1RDM elements are either measured or a formal argument shows the optimal C diagonalizes γ. That said, the paper deserves a serious referee: the core idea is sound, the presentation is honest, and the missing verification is testable.\n\nRecommendation: send to peer review. Ask for a check of the off-diagonal γ_pq in a non-minimal basis, and error bars on the hardware points.","headline":"ON-VQE is a genuine but incremental step: a clean single-QWC measurement trick for NOF energies, with a load-bearing unverified assumption about the orbital basis diagonalizing the 1RDM.","tokens_in":11086,"tokens_out":680,"would_cite":true,"duration_ms":9255,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A hybrid quantum-classical algorithm called ON-VQE claims that accurate molecular energies can be obtained from measured occupation numbers alone, reducing VQE's measurement cost from O(M^4) to O(M/2).","keywords":["occupation numbers","natural orbital functional","variational quantum eigensolver","reduced density matrix","qubit-wise commuting measurement","measurement cost reduction","quantum hardware","strong correlation"],"falsifier":"Measure the off-diagonal elements of the one-particle density matrix in the optimized natural orbital basis (via full 1RDM tomography) and compare them with the measured occupations; if they are non-negligible for a correlated system such as stretched H8 or N2, then ignoring them means the 'occupation numbers' are not the NOF variables and the energy is uncontrolled.","tokens_in":10147,"feed_emoji":"⚛️","tokens_out":6636,"duration_ms":62246,"temperature":0.7,"pith_summary":"ON-VQE is a hybrid quantum-classical scheme whose thesis is that the quantum computer only needs to deliver occupation numbers, not the Hamiltonian expectation value or the full one-particle density matrix. In the natural orbital basis, each occupation number equals 1/2(1 - <Z_i>), a local single-qubit observable, and all such observables commute, so a single measurement setting gives all the data the energy needs. The paper argues this reduces the measurement cost of VQE from O(M^4) to O(M/2) while keeping energies close to conventional VQE, and supports the argument with noiseless simulations of seven small molecules and with quantum-hardware results for the dissociation curve of cubic H8. If the scheme holds, near-term quantum devices can spend their limited shots on precision rather than on measuring many noncommuting Hamiltonian terms, and classical improvements in natural orbital functionals can be adopted without changing the quantum workflow.","feed_headline":"Occupation numbers alone cut VQE measurements from O(M^4) to O(M/2)","feed_subtitle":"The quantum computer only measures single-qubit Z values; classical natural orbital functionals turn them into energies.","key_machinery":"The load-bearing object is the occupation-number identity n_i = 1/2(1 - <Z_i>), which turns each occupation into a local observable and makes the full set of occupations simultaneously measurable in one qubit-wise commuting group. Around it, the method places a natural orbital functional energy expression E_NOF[n,C] that depends only on the diagonal of the one-particle density matrix in the natural orbital basis, a pair-double-excitation ansatz that keeps the state-preparation circuit shallow while generating the needed occupations, and a self-consistent classical optimization of the orbital coefficient matrix C. The NOF supplies the correlation energy classically; the quantum circuit is res","core_discovery":"The central claim is that accurate electronic energies can be extracted from occupation numbers alone, without ever reconstructing a reduced density matrix or measuring the Hamiltonian. The paper identifies the occupation number n_i with the expectation value of a single local Z operator, n_i = 1/2(1 - <Z_i>), a simplification that follows from the Jordan-Wigner transform once the one-particle RDM is diagonal in the natural orbital basis. Because these operators are diagonal in the computational basis, all occupations lie in a single qubit-wise commuting measurement group, and with a pair-double-excitation ansatz all occupation numbers are read from one collection of bitstrings. The authors","pith_inferences":["If only occupation numbers are needed, the objective for the quantum circuit could be reframed as preparing a state whose natural-occupation spectrum is correct, rather than one whose energy is directly minimized; this could simplify variational optimization and connect ON-VQE to reduced-density-matrix functional theory more broadly.","The O(M/2) scaling and single-QWC-group property rely on the electron-pair ansatz and NOF pair structure; for states that are not seniority-zero, the off-diagonal blocks of the 1RDM may not be negligible, and ONs alone would likely be insufficient.","A natural stress test is to push ON-VQE into basis sets and geometries where NOF approximations are known to lose accuracy; if discrepancies appear, they will reveal whether the error comes from the functional or from the diagonalization assumption on the measured density matrix.","Since the measured quantities are simple Z expectation values, the same occupation-data pipeline could in principle be applied to any state-preparation method, not only pair-correlated ansatze, provided the natural orbital basis is known."],"forward_implications":["The number of measurement settings for energy evaluation drops to a single QWC group, so the entire energy can be assembled from one set of computational-basis bitstrings.","The same measured occupation numbers can be substituted into several NOFs without any additional quantum measurements, separating quantum data acquisition from classical energy evaluation.","Because occupation numbers obey known N-representability constraints, hardware errors can be corrected directly in occupation space by projection, polarization recovery, and post-selection, rather than through RDM reconstruction.","For spin-singlet electron-pair states, spin symmetry halves the number of distinct occupations that must be measured, giving the O(M/2) scaling.","The reduction in measurement overhead frees shots to be spent on statistical precision and noise resilience on near-term hardware."],"fun_headline_variants":["Quantum VQE measure only occupation numbers: O(M^4) to O(M/2)","Single-qubit Z readouts replace full VQE Hamiltonian measurement","Occupation-number VQE: one measurement group for all energies","H8 demo: O(M/2) measurements instead of O(M^4) in VQE","No RDM reconstruction: natural orbitals from qubit Z reads"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise, stated where the paper says the NOs are 'determined self-consistently' (Algorithm 1), is that the classically optimized orbital coefficient matrix exactly diagonalizes the measured one-particle density matrix, so the unmeasured off-diagonal elements can be ignored and the diagonal Z expectations are true occupation numbers; if that diagonalization is imperfect, the NOF energy is evaluated with quantities that are not occupation numbers and the result","fun_headline_variants_meta":{"raw":{"variants":["Quantum VQE measure only occupation numbers: O(M^4) to O(M/2)","Single-qubit Z readouts replace full VQE Hamiltonian measurement","Occupation-number VQE: one measurement group for all energies","H8 demo: O(M/2) measurements instead of O(M^4) in VQE","No RDM reconstruction: natural orbitals from qubit Z reads"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001301,"raw_usage":{"total_tokens":5152,"prompt_tokens":760,"completion_tokens":4392,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":504,"completion_tokens_details":{"reasoning_tokens":4293}},"tokens_in":504,"tokens_out":4392,"duration_ms":27851,"temperature":1.0,"reasoning_tokens":4293,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T23:23:46.334347+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the off-diagonal elements of the one-particle density matrix in the optimized natural orbital basis (via full 1RDM tomography) and compare them with the measured occupations; if they are non-negligible for a correlated system such as stretched H8 or N2, then ignoring them means the 'occupation numbers' are not the NOF variables and the energy is uncontrolled.","supporting_citations":[],"review_version":1}