{"id":"3a6466c8-396a-424d-9c72-ebadbb252432","arxiv_id":"1908.07927","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"A quantum gradient-descent (power iteration) algorithm with LCU implementation is proposed for molecular ground-state energies, with numerical demos for H2, LiH, H2O, and NH3.","lead":"This paper proposes a quantum algorithm called the full quantum eigensolver that tries to find molecular ground-state energies by repeatedly applying a simple update to a quantum state, with all optimization steps on a quantum computer. It reports simulations for four small molecules and claims faster convergence than the standard variational quantum eigensolver, but the complexity claims and the comparison rest on several questionable assumptions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The convergence proof's assumption that molecular Hamiltonians have only negative eigenvalues is false and is load-bearing for the O(log(N/epsilon)) iteration-depth claim.","rationale":"The reader identified the same load-bearing weakness: the convergence proof assumes all molecular Hamiltonian eigenvalues are negative, which is false and unstated in the main text. My independent reading of the Supplemental Material confirms that the ordering by absolute value does not imply |1 - gamma*lambda_i| < |1 - gamma*lambda_1|; a separate spectral condition is required. This flaw directly undermines the headline claim of logarithmic iteration depth and the claim that FQE generally converges to the ground state. The paper does provide reproducible small-molecule numerics with STO-3G and STO-6G bases, and the LCU implementation is plausible, but the proof of the central complexity claim is not. Because this is the same core concern that led the reader to REJECT, no verdict adjustment is needed.","tokens_in":13670,"tokens_out":5894,"duration_ms":62079,"concrete_test":"Exact-diagonalize the four STO-3G qubit Hamiltonians used in Figs. 2 and 5, and list all eigenvalues in the full Fock space and in the N-electron subspace used by the HF initial state. For each molecule, with gamma = 1, compute r = max_{i>=2} |1 - gamma*lambda_i| / |1 - gamma*lambda_1| and check whether r < 1. Also check whether any eigenvalue is nonnegative. If any N-electron eigenstate with nonzero overlap from the HF state has r > 1, simulate the LCU power iteration from that HF state and verify whether it fails to converge to the ground state. If all tested molecules satisfy r < 1, the proof still needs an explicit spectral condition for general N, since the stated 'all eigenvalues less than 0' assumption is false even for these small Hamiltonians.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central convergence claim rests on the Supplemental Material's proof that repeated application of Hg = I - gamma H drives |X(t)> to the ground state. The proof orders eigenvalues by decreasing absolute value and then states: 'In the case of molecule Hamiltonian H, all of the eigenvalues are less than 0.' That assertion is not generally true. In the Jordan-Wigner representation the vacuum state has eigenvalue 0, and finite-basis molecular Hamiltonians in a fixed N-electron sector can have positive eigenvalues, especially when high-energy virtual orbitals are included. More importantly, the actual convergence condition is not eigenvalue ordering by magnitude; it is |1 - gamma*lambda_i| < |1 - gamma*lambda_1| for every i >= 2. With gamma = 1, as used in the numerics, any eigenvalue with lambda_i > 2 - lambda_1 (with lambda_1 < 0) makes the ratio exceed 1, so the iteration can amplify an excited component or fail to converge. The main text's Eq. (13) and the error bound in Section II.B state k = O(log(N/epsilon)) without this spectral condition. Since the proof as written requires all eigenvalues to be negative, and this is not established or generally true, the claimed logarithmic depth and the claim that FQE generally converges to the molecular ground state are not supported. The small-molecule numerics may still be correct, but they do not repair the missing condition in the proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a full quantum eigensolver (FQE) for molecular ground-state chemistry. The method replaces the classical optimizer of VQE with a quantum gradient descent: it repeatedly applies the non-unitary operator Hg = I - γH, implemented via linear combination of unitaries (LCU), to a Hartree-Fock initial state, and claims that the number of iterations needed to reach precision ε is O(log(N/ε)). The authors present numerical simulations for H2, LiH, H2O, and NH3 in STO-3G and STO-6G bases, a noise robustness study, an extension based on perturbation theory, and a comparison with VQE.","tokens_in":13867,"tokens_out":14946,"duration_ms":141230,"significance":"If the complexity claim were rigorously established, FQE would offer a fully quantum alternative to VQE that avoids classical optimization loops and potentially converges in depth logarithmic in system size. The paper's numerical results are consistent with the intended power iteration and are compared against exact diagonalization, which is a strength. The explicit LCU circuit and the perturbation-theory shortcut are useful contributions. However, the central convergence proof contains a false spectral assumption, so the logarithmic-depth claim is not currently supported.","major_comments":[{"comment":"The convergence proof in the Supplemental Material assumes that 'all of the eigenvalues are less than 0' for the molecular Hamiltonian. This assumption is not generally true: in the Jordan-Wigner representation the vacuum state has eigenvalue 0, and excited states in a finite basis can have positive eigenvalues. The convergence condition for the iteration Hg = I - γH is max_{i≥2} |1 - γλ_i| < |1 - γλ_1|, not |λ_1| > |λ_2| as stated. If some λ_i > 2/γ - λ_1 (with λ_1 < 0), the amplification factor for that excited component exceeds that of the ground state, and the iteration may converge to an excited state or diverge. The error bound and the iteration depth k = O(log(N/ε)) in Eq. (13) therefore do not follow from the proof as written. The authors should either prove the result under the correct spectral condition or shift the Hamiltonian by a constant cI so that all eigenvalues of the shifted Hamiltonian are negative, and then correct the reported energies accordingly.","section":"Supplemental Material A and Section II.B"},{"comment":"The paper claims that FQE 'does not need to make measurements of the expectation values of Hamiltonian during each iteration procedure,' yet the stopping criterion introduced in the same section is ε = |⟨x_t|H|x_t⟩ - ⟨x_{t+1}|H|x_{t+1}⟩|/⟨x_t|H|x_t⟩. Checking this criterion requires energy measurements at successive iterations, and measuring the energy destroys the iteration state, so it is unclear how the criterion can be applied without restarting the computation. In addition, setting the number of iterations k in advance requires knowledge of the spectral gap (through the ratio (1 - γλ_2)/(1 - γλ_1)), which is not generally known. The paper should clarify how k is chosen in practice and reconcile the claim about energy measurements with the proposed stopping rule.","section":"Section II.B"}],"minor_comments":[{"comment":"The error bound expression contains typographical issues: the ratio (1 - γλ_i)/(1 - γλ_1) should appear squared (because the expansion involves (1 - γλ_i)^{2k}), and the coefficients should be |a_i|^2 rather than a_i; the current expression is dimensionally inconsistent, although the logarithmic scaling in k is unchanged.","section":"Supplemental Material A"},{"comment":"The statement 'we let H′ be equal to Hg' conflicts with the earlier definition of H′ as containing only σ_x and σ_y Pauli terms; since Hg = I - γH includes the identity and σ_z terms, the perturbation expansions in Eqs. (15)-(18) need to be clarified.","section":"Section II.C"},{"comment":"In Table I, the H2O second-order value is listed as '75.0032' without a minus sign, inconsistent with the other negative energy entries.","section":"Table I"},{"comment":"The comparison with VQE in Fig. 3 is not persuasive: only two molecules are shown, and the VQE learning rate is fixed at γ = 10^{-3} without evidence that this is a fair or optimal choice; the conclusion that 'FQE generally converges faster than VQE' is broader than the data support.","section":"Section III.A"},{"comment":"The noise model is described too vaguely: the state noise term |δx⟩ added to |x_k⟩ and the subsequent renormalization is not tied to a realistic error model, and the amplitudes for H2O/NH3 are stated without explaining how they were chosen.","section":"Section III.A"}],"recommendation":"major_revision","confidential_remarks":"The main technical concern is the spectral assumption in the convergence proof; this is fixable by shifting the Hamiltonian, but the authors must also address the stopping-time issue. The novelty is somewhat incremental given the existing quantum power iteration literature, but the manuscript does make a concrete proposal for quantum chemistry. I would support major revision rather than rejection because the central idea is sound and the numerical evidence is consistent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a power iteration for ground states, implemented via LCU, dressed up as a “full quantum eigensolver.” The numerics for H2, LiH, H2O, and NH3 are clean and match exact diagonalization, and the noise robustness tests are a nice practical touch. The LCU circuit itself is standard but clearly explained. If you need a worked example of LCU-based ground-state search for a reading group, this paper is serviceable. But the headline claims do not survive close reading. The convergence proof in the Supplemental Material explicitly assumes all eigenvalues of the molecular Hamiltonian are negative. That is not true: the vacuum state has eigenvalue 0, and a finite basis in a fixed particle sector can produce positive eigenvalues above the Fermi level. The actual condition for power iteration to converge to the ground state is |1−γλ_i| < |1−γλ_1| for every i ≥ 2. With γ=1, as used in the simulations, any eigenvalue greater than 2−λ_1 (with λ_1 the most negative) will cause an excited component to grow. So the O(log(N/ε)) iteration depth in Eq. (13) is unsupported. This is not a minor detail; it is the load-bearing assertion of the paper. The complexity analysis also contradicts the summary: per iteration costs O(N^4 log N) basic gates, so with k = O(log(N/ε)) the total is polynomial in N, not the claimed polylogarithmic. The perturbation theory section has a separate conceptual slip: it says “we let H′ be equal to Hg,” but H′ is supposed to be the non-diagonal part of H, not I−γH. That identification is unjustified and the perturbative formulas that follow are therefore not actually computing what the text claims. The VQE comparison likewise uses a weak finite-difference gradient descent as the classical optimizer, so the “faster than VQE” conclusion is not meaningful. None of this is fatal to the basic idea—power iteration with a shifted Hamiltonian is a valid route to ground states if the spectral conditions are stated and respected. But as written, the central proof is wrong, the complexity claim is inflated, and the perturbative shortcut is not well founded. A serious referee should engage with it, because the errors are instructive and the numerics are reproducible, but the paper needs major revision before it can be trusted. I would not cite it in its current form.","headline":"The algorithm is a repackaged power method that converges in the small-molecule examples, but the paper's main complexity proof rests on a false spectral assumption and the perturbation shortcut is conceptually muddled.","tokens_in":14454,"tokens_out":1847,"would_cite":false,"duration_ms":19482,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68","81V55","65F15"],"pacs":["03.67.Ac","31.15.A-"],"model":"deepseek-v4-flash","headline":"The paper proposes FQE, a fully quantum gradient-descent eigensolver that finds molecular ground states without a classical optimizer.","keywords":["full quantum eigensolver","quantum gradient descent","linear combination of unitaries","ground-state energy","molecular Hamiltonian","variational quantum eigensolver","perturbation theory","quantum chemistry simulation"],"falsifier":"Run the FQE iteration on a small Hamiltonian with a known zero or positive excited eigenvalue, starting from a mean-field state, and check whether the final energy equals the exact ground-state energy after $O(\\log(N/\\epsilon))$ steps; a concrete instance is the vacuum state, whose energy is zero, appearing in the spectrum. If the iteration selects that level or oscillates instead of reaching the ground state, the convergence proof's all-negative-eigenvalue assumption is load-bearing.","tokens_in":13397,"feed_emoji":"⚛️","tokens_out":10502,"duration_ms":100260,"temperature":0.7,"pith_summary":"FQE aims to replace the hybrid loop of the variational quantum eigensolver with a fully quantum one: instead of sending Hamiltonian expectation values to a classical optimizer, the algorithm applies the operator $H_g = I - \\gamma H$ repeatedly to a mean-field initial state using a linear combination of unitaries. The paper argues that after $k = O(\\log(N/\\epsilon))$ iterations the state converges to the molecular ground state, with no energy measurement needed during the iteration, and that the iteration converges faster than VQE in numerical tests on $H_2$, LiH, $H_2O$, and $NH_3$. If the claim holds, FQE gives a quantum chemistry solver whose entire optimization runs on the quantum computer, at a depth that grows only logarithmically with system size and inverse precision. The paper also shows that a perturbation-theory version needs only one circuit iteration to reach chemical precision, which matters for near-term hardware.","feed_headline":"Quantum gradient descent finds molecular ground states","feed_subtitle":"FQE runs the entire optimization on the quantum computer, reaching chemical accuracy for H2, LiH, H2O, and NH3.","key_machinery":"The load-bearing object is the shifted Hamiltonian $H_g = I - \\gamma H$, a sum of Pauli unitaries whose repeated application performs gradient descent on the energy landscape. It is implemented by the linear-combination-of-unitaries (LCU) trick: an ancilla register is prepared in a superposition weighted by the coefficients $\\beta_i$, controlled unitaries $H_i^g$ entangle the ancilla with the work register, Hadamard gates recombine the branches, and postselection on the ancilla state $|0\\rangle$ projects the work system onto $H_g|x(t)\\rangle$. The convergence proof rests on the ratio $(1-\\gamma\\lambda_2)/(1-\\gamma\\lambda_1)$, kept below one by choosing the learning rate $\\gamma$; the iteration depth then scales as $O(\\log(N/\\epsilon))$.","core_discovery":"On the paper's own terms, the discovery is that quantum gradient descent can serve as a complete eigensolver for molecular Hamiltonians. Starting from a product state $|x(0)\\rangle$ with good overlap with the ground state, each iteration applies $|x(t+1)\\rangle = H_g|x(t)\\rangle$ with $H_g = I - \\gamma H$, implemented as a linear combination of the Pauli terms of $H$. Because the iteration is a power method on $H_g$, the error after $k$ steps is bounded in terms of $((1-\\gamma\\lambda_2)/(1-\\gamma\\lambda_1))^k$, giving $k = O(\\log(N/\\epsilon))$; the paper's simulations show convergence to the diagonalized ground-state energy for $H_2$, LiH, $H_2O$, and $NH_3$ within chemical precision, with FQE converging before VQE for comparable learning rates. The algorithm is called 'full' because, unlike VQE, it does not outsource the optimization step to a classical computer.","pith_inferences":["Because FQE is a power iteration on $I - \\gamma H$, a natural continuation is to use it as a subroutine inside a larger algorithm, for instance to refine a state prepared by another method or to update a variational ansatz in the final iterations where gradient signals are small.","The perturbation version suggests a testable near-term protocol: run one LCU iteration on hardware, read off the off-diagonal matrix elements of $H'$, and compare the second-order energy with full diagonalization; the four molecules here provide the numerical baseline.","The same circuit can be applied to any Hamiltonian expressed as a Pauli sum, not only fermionic chemical systems, so the scheme could be tested directly on spin-model ground-state problems of comparable dimension.","If one wants the same convergence guarantee for spectra with zero or positive eigenvalues, a constant shift of the Hamiltonian before defining $H_g$ would restore the all-negative-spectrum condition without changing eigenstates; this adjustment is not analyzed in the paper."],"forward_implications":["No classical optimizer is needed: parameter updates are generated by repeated quantum operations, so the hybrid VQE loop can be replaced by a single quantum procedure.","The iteration depth needed for a target precision grows logarithmically with the number of basis states and inversely with the error threshold, so the main per-iteration cost is the circuit for the linear combination of unitaries rather than the number of iterations.","Energy expectation values do not have to be measured during the iteration, which avoids the state-destroying measurements that slow VQE; the depth can be preset from the error bound.","The perturbation-theory route computes first- and second-order corrections from one pass through the same circuit and reaches chemical precision for the four molecules tested, so a single iteration can suffice on noisy hardware.","Under small random or Gaussian noise the simulated energies still converge to chemical precision, while under ten-fold larger noise convergence degrades or oscillates, indicating where the method's noise tolerance ends."],"supporting_citations":[{"why":"Defines the variational quantum eigensolver that FQE removes the classical optimizer from.","marker":"[18]"},{"why":"Supplies the fermion-to-qubit mapping used to turn the molecular Hamiltonian into a sum of Pauli terms.","marker":"[25]"},{"why":"Proposes the general quantum interference and linear-combination-of-unitaries principle used to implement the non-unitary update.","marker":"[45]"},{"why":"Provides oblivious amplitude amplification used to boost the success probability of the ancilla readout step.","marker":"[51]"},{"why":"Shows how a non-unitary evolution can be embedded unitarily in a larger space, justifying the circuit construction.","marker":"[53]"},{"why":"Gives the efficient initialization scheme used to prepare the weighted ancilla superposition in wave division.","marker":"[54]"},{"why":"Supplies the quantum gradient-descent framework and the measurement-complexity analysis that FQE builds on.","marker":"[55]"},{"why":"Provides the iterative-eigenvalue error analysis from which the convergence bound and iteration depth are derived.","marker":"[58]"}],"fun_headline_variants":["All-quantum eigensolver beats VQE on molecules","Quantum gradient descent solves molecular ground states","FQE: full quantum eigensolver, no classical step","Quantum power method hits chemical accuracy","Faster convergence: quantum eigensolver for chemistry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that every eigenvalue of the molecular Hamiltonian is negative, so the ratio $(1-\\gamma\\lambda_2)/(1-\\gamma\\lambda_1)$ stays below one; in a real finite basis the vacuum state sits at energy zero and excited levels can be positive, and if that happens the iteration can converge to an excited state or fail to converge.","fun_headline_variants_meta":{"raw":{"variants":["All-quantum eigensolver beats VQE on molecules","Quantum gradient descent solves molecular ground states","FQE: full quantum eigensolver, no classical step","Quantum power method hits chemical accuracy","Faster convergence: quantum eigensolver for chemistry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000186,"raw_usage":{"total_tokens":1320,"prompt_tokens":934,"completion_tokens":386,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":550,"completion_tokens_details":{"reasoning_tokens":313}},"tokens_in":550,"tokens_out":386,"duration_ms":8871,"temperature":1.0,"reasoning_tokens":313,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:54:13.896054+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the FQE iteration on a small Hamiltonian with a known zero or positive excited eigenvalue, starting from a mean-field state, and check whether the final energy equals the exact ground-state energy after $O(\\log(N/\\epsilon))$ steps; a concrete instance is the vacuum state, whose energy is zero, appearing in the spectrum. If the iteration selects that level or oscillates instead of reaching the ground state, the convergence proof's all-negative-eigenvalue assumption is load-bearing.","supporting_citations":[{"cited_title":"A variational eigenvalue solver on a photonic quantum processor","cited_arxiv_id":null,"evidence_quote":"Defines the variational quantum eigensolver that FQE removes the classical optimizer from."},{"cited_title":"About the pauli exclusion principle","cited_arxiv_id":null,"evidence_quote":"Supplies the fermion-to-qubit mapping used to turn the molecular Hamiltonian into a sum of Pauli terms."},{"cited_title":"General quantum interference principle and duality computer","cited_arxiv_id":null,"evidence_quote":"Proposes the general quantum interference and linear-combination-of-unitaries principle used to implement the non-unitary update."},{"cited_title":"Simulating hamiltonian dy- namics with a truncated taylor series","cited_arxiv_id":null,"evidence_quote":"Provides oblivious amplitude amplification used to boost the success probability of the ancilla readout step."},{"cited_title":"Restricted allowable generalized quantum gates","cited_arxiv_id":null,"evidence_quote":"Shows how a non-unitary evolution can be embedded unitarily in a larger space, justifying the circuit construction."},{"cited_title":"Efficient scheme for initializing a quantum register with an arbitrary superposed state","cited_arxiv_id":null,"evidence_quote":"Gives the efficient initialization scheme used to prepare the weighted ancilla superposition in wave division."},{"cited_title":"Quantum gradient descent and newtons method for constrained polynomial optimization","cited_arxiv_id":null,"evidence_quote":"Supplies the quantum gradient-descent framework and the measurement-complexity analysis that FQE builds on."},{"cited_title":"Iterative Methods for Computing Eigenvalues and Eigenvectors","cited_arxiv_id":"1105.1185","evidence_quote":"Provides the iterative-eigenvalue error analysis from which the convergence bound and iteration depth are derived."}],"review_version":1}