{"id":"fb8367e2-34cd-4a1a-944e-ed7797558a59","arxiv_id":"2607.09476","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"An orthogonal quantum Krylov algorithm (OQKD) implements classical Lanczos recursion via GQSP polynomial transformations, avoiding overlap-matrix regularization but inheriting an exponential GQSP normalization overhead.","lead":"This paper proposes OQKD, a quantum algorithm that builds an orthogonal Krylov basis from polynomial transformations of the Hamiltonian, mimicking the classical Lanczos method without an ill-conditioned overlap matrix. A smart generalist might read it because it promises more stable quantum eigensolvers and a state-preparation route for quantum phase estimation.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Measurement noise in the moments feeding Eq. (14) is never analyzed; finite-shot errors in β̃_i will break orthogonality and may reintroduce the ill-conditioning OQKD claims to eliminate.","rationale":"The paper's algebraic core is sound: Eq. (14) follows from the Lanczos three-term recurrence, and the GQSP construction in Appendix C correctly block-encodes the polynomial. The numerical experiments demonstrate that in exact arithmetic the method reproduces Lanczos. However, the selling point is numerical stability without overlap regularization in a quantum setting. That requires the recursion coefficients to be known accurately. Finite measurement statistics couple directly into the orthogonality of the generated basis. This is the same mechanism that makes classical Lanczos lose orthogonality under floating-point round-off; here the perturbation is statistical and can be several orders of magnitude larger than machine epsilon. No analysis or simulation addresses this. The reader's verdict already flagged this; my read agrees. A second gap—the promised Appendix D analysis of restart success probability—is also real but secondary, since ROQKD inherits the same coefficient noise. I therefore see no reason to alter the verdict: CONDITIONAL remains appropriate until the noise sensitivity is characterized.","tokens_in":18881,"tokens_out":7133,"duration_ms":77284,"concrete_test":"Use a classical state-vector simulation of Algorithm 1 for the 4×4 J1–J2 model of Fig. 2. Replace the exact moments at each iteration with noisy estimates: m̃_i = m_i + ε_i, ᾱ̃_i = ᾱ_i + δ_i, where ε_i, δ_i are independent Gaussians with variance chosen to model M shots (e.g., σ_m = ‖H̃²‖/√M, σ_α = ‖H̃‖/√M). Run M = 10^4, 10^6, 10^8. For each M, compute (i) the condition number κ(S) of the overlap matrix between the actually prepared states, and (ii) the ground-state energy error vs D up to D=30. Also record any occurrence of negative RHS in Eq. (14). If κ(S) grows substantially above 1 or the energy error deviates from the classical Lanczos curve at D<20 even for M=10^6, the claim that OQKD eliminates overlap-matrix regularization under realistic measurement noise is falsified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"OQKD's status as an 'orthogonal quantum analog of Lanczos' rests on Eq. (14), which converts measured diagonal moments into the Lanczos coefficient β̃_{i+1}. In any real run, the moments ⟨ψ_i|H̃|ψ_i⟩ and ⟨ψ_i|H̃²|ψ_i⟩ are estimated from finitely many shots, so the α̃_i and β̃_i used in the recursion (and in the polynomial-coefficient recurrence, Appendix B) carry statistical errors. These errors are not a small perturbation: because state preparation at step i uses polynomials built from all previous noisy coefficients, the actual state |ψ_i⟩ differs from the exact Lanczos vector, and the overlap matrix S_ij = ⟨ψ_i|ψ_j⟩ is no longer δ_ij. The projected Hamiltonian is no longer tridiagonal, so one cannot avoid measuring off-diagonal elements, and S may become ill-conditioned—the very problem OQKD claims to eliminate. The recursion is also fragile near small β_{i+1}: a negative fluctuation in the RHS of Eq. (14) gives an imaginary β and breaks the algorithm. The paper contains no error propagation or shot-noise analysis; Fig. 2 and Fig. 4 are exact-arithmetic. Thus the central practical claim is untested.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces Orthogonal Quantum Krylov Diagonalization (OQKD), a quantum subspace method that constructs an orthonormal Lanczos basis by representing Lanczos vectors as polynomial transformations of a rescaled Hamiltonian. The key recursion, Eq. (14), determines the Lanczos normalization coefficient β̃_{i+1} from diagonal moments ⟨ψ_i|H̃²|ψ_i⟩, α̃_i and β̃_i, avoiding off-diagonal Hamiltonian measurements and overlap-matrix regularization. The authors derive this recursion in Appendix A, provide a GQSP/block-encoding implementation in Section III.B and Appendix C, and benchmark the method on a 4×4 J1–J2 Heisenberg model, reporting convergence identical to classical Lanczos and κ(S)≈1 in exact-arithmetic simulations. They further introduce a restarted protocol (ROQKD) intended to keep the GQSP success probability nearly constant while iteratively improving a trial state, and they argue that OQKD removes the initial-state-overlap dependence of the measurement complexity. The central theoretical construction is arithmetically sound, but the practical claim that OQKD eliminates the numerical instability of QKD/CQKD is established only in exact arithmetic; finite-shot measurement noise is never analyzed or simulated.","tokens_in":19164,"tokens_out":4749,"duration_ms":58113,"significance":"If the central claim holds, OQKD would be a conceptually important bridge between classical Lanczos diagonalization and quantum Krylov methods, providing an orthonormal Krylov basis without overlap-matrix inversion or Hadamard tests. The derivations in Appendices A and C are clean: the β-recursion follows directly from the Lanczos three-term identity, and the GQSP realization of the polynomial P_n(H̃) is correctly described with an explicit normalization condition. The exact-arithmetic numerical demonstration on the J1–J2 model is a useful proof of principle, and the idea of restarting with a GQSP-implemented approximate ground state is a promising direction for state preparation. However, the load-bearing advantage of OQKD over existing methods is numerical stability in a realistic quantum setting, and the manuscript does not provide any finite-shot-noise analysis, error propagation for Eq. (14), or simulation with measurement noise. The exponential decrease of the GQSP success probability is acknowledged but is not reconciled with the claimed query-complexity equivalence to Chebyshev-based methods. The contribution is therefore significant in exact arithmetic but presently condit","major_comments":[{"comment":"The central practical claim — that OQKD preserves orthogonality and eliminates overlap-matrix regularization — is not established under finite-shot measurement noise. In a real run, α̃_i and ⟨ψ_i|H̃²|ψ_i⟩ are estimated from a finite number of shots, so β̃_{i+1} computed from Eq. (14) carries statistical error. Because P_{i+1} is built from all previous noisy coefficients, the actual prepared state |ψ_i⟩ differs from the exact Lanczos vector, orthogonality is lost, and the projected Hamiltonian is no longer tridiagonal. The RHS of Eq. (14) can also become negative, yielding an imaginary β and breaking the algorithm. Neither an error-propagation bound nor a noisy simulation is provided; Figs. 2 and 4 use exact arithmetic. This is load-bearing because the advantage over CQKD is precisely numerical stability.","section":"§III.A, Eq. (14); Algorithm 1 steps 3(b)–3(c)"},{"comment":"The abstract claims 'the same asymptotic query complexity as Chebyshev-based QKD methods,' but this is only true for the circuit depth of a single polynomial application, not for the expected cost of successfully preparing a Lanczos vector. The GQSP success probability is p_succ = 1/Λ_n², which Fig. 3 shows decreases exponentially with the Lanczos dimension. With amplitude amplification, the expected number of walk-operator calls becomes O(Λ_n n), not O(n), whereas CQKD has p_succ ≈ constant. The effective cost in Eq. (34) contains 1/p_succ(D), so the asymptotic comparison in Eq. (33) vs. Eq. (34) is misleading unless the success-probability overhead is included in the stated query complexity. The manuscript should either qualify the claim or analyze the total query complexity including postselection/amplification.","section":"§III.B and §III.D, Eqs. (22)–(25), (34)"},{"comment":"The restarted protocol's main assertion is that the restart polynomial Q^(k) remains well-conditioned and that the accumulated success probability Λ^(R) is nearly constant. This is supported only by the 4×4 J1–J2 example in Fig. 4; Appendix D, which is described as a 'detailed analysis,' contains no bound on Λ^(R), no system-size scaling of the success probability, and no argument beyond the statement that Q^(k) is a refined ground state. Since the entire advantage of ROQKD over direct OQKD rests on this point, a concrete analysis or a set of larger-system simulations is needed. The promised detailed analysis in Appendix D is absent.","section":"§IV and Appendix D"},{"comment":"The measurement-complexity comparison assumes that, in OQKD, only diagonal moments need to be measured and that the projected Hamiltonian is exactly tridiagonal. This is true in exact arithmetic, but it is precisely the property that finite-shot noise destroys. The error bound in Eq. (28) for CQKD explicitly includes statistical sampling error and thresholding effects; no analogous bound is derived for OQKD that includes the statistical error of the moments entering Eq. (14) and its propagation into the Ritz values. Without this, the claimed Θ(1/ε²) per-matrix-element cost and the removal of the |γ0| dependence are not substantiated as a practical complexity statement.","section":"§III.D, Eqs. (28)–(34)"}],"minor_comments":[{"comment":"There are several typos and inconsistent notations: 'OKQD' appears in §IV.A; 'straightworward' in §III.B; 'rationalization' should be 'orthogonalization' in §II; 'it is seems suited' in §IV. The tildes on α and β are sometimes dropped in the text surrounding Eq. (13) and Appendix A; please make the scaled/unscaled notation consistent.","section":"General"},{"comment":"The exponential decrease of the success probability is shown, but the horizontal axis is the Lanczos subspace dimension on a 4×4 model. Reporting the same quantity for a larger system or, better, the scaling of Λ_n with n for a simple Hamiltonian would make the limitation quantitative rather than illustrative.","section":"§III.C, Fig. 3"},{"comment":"The complementary-polynomial construction and recursive phase-factor extraction follow Ref. [41], but the text should note that the numerical optimization of the complementary polynomial has no provable runtime guarantee in general. This is relevant because the claimed O(n) classical overhead assumes successful convergence of this step.","section":"Appendix C.1"},{"comment":"Panel (b) shows that the success probability is 'nearly constant,' but the curves appear to be for one small model and a small range of restart counts. Please specify the system parameters, the restart count range, and the numerical precision used; otherwise the claim of near-constancy is not robustly supported.","section":"§IV.B, Fig. 4"},{"comment":"Ref. [26] is a preprint without a DOI; consider citing the published version if available. Also, the text mentions 'Q-SENSE' but the reference list does not include the original Q-SENSE paper; please add a proper citation.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The exact-arithmetic contribution is solid and the paper addresses a real problem in quantum Krylov methods. The gap between the exact-arithmetic identity and a practical noisy algorithm is the main risk: the title and abstract claim that OQKD reproduces classical Lanczos orthogonality and eliminates overlap regularization, but this is not demonstrated under measurement noise. The missing finite-shot analysis is not a cosmetic issue; it is central to the method's advertised advantage. I would be willing to reconsider after the authors add a noise analysis or clear error propagation bounds, and after they substantially qualify the query-complexity claim to include the success probability. A minor point for the editor: the paper's framing of the exponential p_succ as 'technical rather than fundamental' underplays the fact that the same exponential appears in the total query complexity of the only proposed implementation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the short version: OQKD is a real idea, the math in the appendices checks out, and the paper deserves a serious referee. The key contribution is closing the Lanczos three-term recursion with Eq. (14), so you can build an orthonormal Krylov basis from diagonal moments only and implement the Lanczos polynomials with GQSP. That is a genuine integration of known pieces, and it does eliminate the overlap-matrix headache in exact arithmetic. The derivation in Appendix A is clean, the Chebyshev coefficient recursion in Appendix B is correct, and the GQSP realization in Appendix C is standard but properly spelled out. I also give them credit for stating the exponential growth of the normalization factor Lambda_n up front; that is the dominant practical cost and they don't hide it.\n\nThe soft spot is exactly where the stress-tester points: none of the numerical demonstrations include measurement noise. Equation (14) is the load-bearing wall, and it turns measured moments into all future betas and polynomials. With finite shots, alpha and beta carry statistical errors, the prepared state drifts off the true Lanczos vector, the overlap matrix stops being identity, and the thing that made OQKD special - no overlap regularization - can quietly come back. The paper's Fig. 2 and Fig. 4 are exact-arithmetic simulations on a 4x4 chain, so the headline numerical-stability claim is essentially untested outside noiseless arithmetic. This is not a fatal flaw; the method may still be stable under realistic shot counts. But the authors need to provide error propagation or at least a noisy simulation before the claim stands. The restart protocol's nearly constant success probability is supported by Fig. 4(b) for one small model and a few n_Lanczos values; the promised analytic treatment in Appendix D is really just the Kaniel-Paige bound plus a scaling argument for D = O(N^2), not a bound on Q^(k) conditioning. So that part is currently heuristic.\n\nOne more thing: the measurement-complexity comparison is a bit generous. They compare their 1/p_succ(D) overhead against CQKD's overlap-dependent overhead, but if D itself grows like N^2, the exponential in Lambda_n couples to N as well. The qualitative point - no |gamma_0|^{-4} factor - still holds, but the full accounting is more pessimistic than the abstract suggests.\n\nWho is this for? People working on quantum subspace methods and state preparation for QPE will find the construction useful and worth a careful read. My recommendation: send it to review, and make the noise analysis a required revision, not an optional one.","headline":"The recursion is right and the idea is genuinely new, but the numerical-stability story skips the one thing that can break it: finite-shot noise in the measured moments.","tokens_in":19689,"tokens_out":3580,"would_cite":true,"duration_ms":35199,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68","65F15"],"pacs":["03.67.Ac"],"model":"deepseek-v4-flash","headline":"The paper claims that the classical Lanczos recursion can be reconstructed on a quantum processor from diagonal expectation values alone, yielding an orthonormal Krylov basis without overlap-matrix regularization.","keywords":["Quantum Krylov subspace","Lanczos algorithm","block encoding","quantum signal processing","overlap matrix","eigenvalue estimation","state preparation","Heisenberg model"],"falsifier":"Run the OQKD recursion on a small, exactly solvable Hamiltonian using moments corrupted by controlled noise (e.g., rounding to k bits or adding Gaussian shot noise), and track the condition number of the implicitly generated overlap matrix or the ground-state energy error. If small moment errors cause β̃_i to drift and orthogonality to degrade at modest Krylov dimension, the practical claim collapses; the paper reports no such noise experiment.","tokens_in":18754,"feed_emoji":"⚛️","tokens_out":4897,"duration_ms":44551,"temperature":0.7,"pith_summary":"The paper proposes a quantum algorithm that reproduces the classical Lanczos procedure without ever forming the non-orthogonal Krylov basis that plagues earlier quantum Krylov methods. The key move is to rewrite each Lanczos vector as a polynomial of the rescaled Hamiltonian acting on a fixed initial state, and to close the Lanczos recursion from two diagonal expectation values per step. The authors argue this yields an orthonormal basis by construction, a tridiagonal projected Hamiltonian, and the same convergence as classical Lanczos, so no overlap-matrix regularization is needed. They support this with exact-arithmetic simulations on a 4x4 J1-J2 Heisenberg model, an asymptotic measurement-complexity analysis, and a restarted protocol for state preparation.","feed_headline":"Quantum eigensolver restores Lanczos orthogonality","feed_subtitle":"Rebuilds the Lanczos recursion from diagonal moments, matching classical convergence without overlap regularization.","key_machinery":"The recursive coefficient identity β̃²_{i+1} = ⟨ψ_i|H̃²|ψ_i⟩ − ᾱ_i² − β̃_i² (Eq. 14), derived from the tridiagonal structure of the projected Hamiltonian, is the mechanism that lets the algorithm compute the next Lanczos coefficient from diagonal measurements alone. It feeds the polynomial recurrence P_{n+1}(x) = ((x − ᾱ_n)P_n(x) − β̃_n P_{n−1}(x))/β̃_{n+1}, whose Chebyshev coefficients are then implemented as a GQSP circuit acting on a block-encoded walk operator.","core_discovery":"The central claim is that the classical Lanczos three-term recurrence can be lifted from vectors to operators: with H̃ = H/α, the nth Lanczos vector is |ψ⊥_n⟩ = P_n(H̃)|Φ0⟩, and the normalization coefficient in the recurrence is fixed by β̃²_{n+1} = ⟨ψ⊥_n|H̃²|ψ⊥_n⟩ − ᾱ²_n − β̃²_n (Eq. 14). This identity, a consequence of the tridiagonal form of H in the Lanczos basis, means the entire basis can be generated from diagonal moments ⟨ψ|H̃|ψ⟩ and ⟨ψ|H̃²|ψ⟩, with no overlap matrix, no Hadamard tests, and no quantum memory. The authors show the Lanczos polynomials can be realized with block encoding and Generalized Quantum Signal Processing, with the same query complexity as Chebyshev-based Krylov","pith_inferences":["If noise in the measured moments propagates through the recursion, the orthogonality that OQKD gains 'by construction' could be lost in practice; a finite-shot error-propagation analysis is the natural next test, and the paper does not provide one.","Because the Λ_n growth is tied to the Lanczos polynomial coefficients, the exponential overhead may be inherent to orthogonalization in any polynomial encoding, not just GQSP; exploring alternative encodings or approximate orthogonal polynomials could reveal whether the trade-off can be shifted.","The restarted protocol can be viewed as iterative polynomial filtering; combining it with error-mitigation techniques might extend it to excited states or noisy devices."],"forward_implications":["No overlap-matrix regularization: the overlap matrix is the identity by construction, so the generalized eigenvalue problem reduces to a standard tridiagonal eigenproblem.","Convergence matches classical Lanczos: in exact-arithmetic simulations on J1-J2 Heisenberg, ground-state energy error follows the classical Lanczos curve to machine precision, while QKD/CQKD stall due to conditioning.","Measurement complexity decouples from initial-state overlap: per matrix element it is order 1/ε², independent of |γ0|, whereas prior Chebyshev QKD inherits exponential dependence on system size through |γ0|.","Same asymptotic query complexity as Chebyshev-based QKD.","Restarted protocol gives a state-preparation route for Quantum Phase Estimation with nearly constant success probability per restart."],"fun_headline_variants":["Quantum Krylov goes orthogonal, dumps overlap matrix","OQKD: orthogonal quantum Krylov without regularization","No overlap matrix, same convergence: OQKD","Restart protocol aids state prep in orthogonal quantum Krylov"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The recursion stays numerically stable when the expectation values ⟨ψ_i|H̃|ψ_i⟩ and ⟨ψ_i|H̃²|ψ_i⟩ are estimated from finitely many noisy measurements; the paper's simulations use exact arithmetic and never test how shot noise degrades the recursively built orthonormal basis.","fun_headline_variants_meta":{"raw":{"variants":["Quantum Krylov goes orthogonal, dumps overlap matrix","OQKD: orthogonal quantum Krylov without regularization","No overlap matrix, same convergence: OQKD","Restart protocol aids state prep in orthogonal quantum Krylov"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000272,"raw_usage":{"total_tokens":1525,"prompt_tokens":857,"completion_tokens":668,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":601,"completion_tokens_details":{"reasoning_tokens":604}},"tokens_in":601,"tokens_out":668,"duration_ms":6909,"temperature":1.0,"reasoning_tokens":604,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T07:33:25.742369+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the OQKD recursion on a small, exactly solvable Hamiltonian using moments corrupted by controlled noise (e.g., rounding to k bits or adding Gaussian shot noise), and track the condition number of the implicitly generated overlap matrix or the ground-state energy error. If small moment errors cause β̃_i to drift and orthogonality to degrade at modest Krylov dimension, the practical claim collapses; the paper reports no such noise experiment.","supporting_citations":[],"review_version":2}