{"id":"0ab2bc89-7e56-4f26-a0f0-b6571b27598e","arxiv_id":"2507.18970","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Two Schrödingerization-based Hamiltonian simulation algorithms for multiscale linear transport are proposed, but the claimed O(N_v N_x^2 log N_x) query complexity undercounts the auxiliary grid dimension.","lead":"This paper designs two quantum Hamiltonian simulation algorithms for multiscale linear transport equations, combining Schrödingerization with exponentially integrated asymptotic-preserving schemes and incoming boundary conditions. It claims a query complexity of O(N_v N_x^2 log N_x), better than earlier quantum and classical solvers, though the proof's accounting of the auxiliary grid dimension appears to omit factors that scale with the simulated time.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The complexity proofs conflate the p-grid count with the momentum spectral symbol; under the domain-length requirement of Section 3.1.1 the claimed O(N_v N_x^2 log N_x) is not derived, and Theorem 4.1 gains an extra log.","rationale":"The reader correctly identifies a load-bearing gap in the treatment of the Schrodingerization p-grid, and the advertised complexity is indeed not established by the proofs as written. However, the reader's specific consequence, O(N_v N_x^4 poly-log N_x), follows only if one keeps the paper's erroneous identification ||D_eta||_max = N_eta while also enforcing the domain-length growth. With the correct Fourier symbol ||D_mu||_max = pi/Delta p, the iterative method's query bound O(N_v N_x^2 log N_x) is actually recoverable; the steady-state method, by contrast, still picks up an extra log from T_stea = O(N_t log(1/delta)), so Theorem 4.1's claimed complexity is not supported. This is enough to keep the reader's REJECT verdict: the central claim, that both algorithms achieve O(N_v N_x^2 log N_x) and outperform prior quantum algorithms, remains unproven. The paper has plausible design ideas and the numerical checks are self-consistent at small scale, so a revised complexity analysis might restore a weaker result, but the current manuscript does not justify the headline.","tokens_in":28814,"tokens_out":28617,"duration_ms":292452,"concrete_test":"Recompute chi in Theorems 3.1 and 4.1 using the relations from Section 3.1.1: N_p = (R+L)/Delta p, L,R = Theta(lambda_max(A1) * T_ev), mu_k = 2*pi*(k - N_p/2)/(R+L), and the paper's own estimates N_t = O(N_x^2), T_stea = O(N_t log(1/delta)), S = O(N_v). For both algorithms, evaluate chi = S * ||H||_max * T_ev with ||H||_max = O(||D_mu||_max) and with the alternative ||D_mu||_max = N_eta. If the corrected computation gives O(N_v N_x^2 log N_x) for both theorems, the reader's concern is overridden; if Theorem 4.1 gives O(N_v N_x^2 log^2 N_x), the abstract's claim for the steady-state method is unsupported. Additionally, estimate T_stea directly from the Jordan form of H (eigenvalues all 1) to check whether T_stea can be improved to O(N_t), which would remove the extra log.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorems 3.1 and 4.1 both assert Q_query = O~(N_v N_x^2 log N_x), but the derivations are internally inconsistent about the Schrodingerization p-grid. Section 3.1.1 fixes the p-domain by L,R > lambda_max(A1) * T_ev, so N_p = (L+R)/Delta p = Theta(T_ev / Delta p), while the proofs set N_eta = N_p = O(log(1/delta)) independently of T_ev and identify ||D_eta||_max with N_eta. With the Fourier eigenvalues mu_k = 2*pi*(k - N_p/2)/(R+L) as defined in Section 3.1.1, ||D_mu||_max = pi/Delta p = O(log(1/delta)) for smooth initial data, so the number of grid points enters only through the Hilbert-space dimension m(H) = O(log N_x). If, instead, one adopts the paper's identification ||D_eta||_max = N_eta while enforcing L,R = Theta(T_ev), one gets chi = O(N_v N_eta^2) = O(N_v N_x^4 log^2 N_x), the reader's pessimistic bound; but the physically correct computation gives a different result. For the iterative method, T_ev = N_t and chi = O(N_v * log(1/delta) * N_t) = O(N_v N_x^2 log N_x), so the advertised query count can be recovered. For the steady-state method, the paper's own estimate is T_stea = O(N_t log(1/delta)) (Section 4.2.3); then chi = S * ||H||_max * T_stea = O(N_v * log(1/delta) * N_t log(1/delta)) = O(N_v N_x^2 log^2 N_x), one log factor above the claim. Thus the headline complexity is not established by the proofs as written, and the steady-state method's bound is off by a log even after correcting the Fourier-symbol error.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two quantum Hamiltonian simulation algorithms for the one-dimensional multiscale linear transport equation with incoming boundary conditions, combining the Schrödingerization method with exponential-integrator asymptotic-preserving (AP) discretizations. The first algorithm treats the discrete time-marching scheme as a linear iterative system and solves it via Schrödingerization; the second reformulates the all-time linear system as a steady-state problem for an ODE and solves it by Hamiltonian simulation. Both algorithms are claimed to achieve query and gate complexity O~(N_v N_x^2 log N_x), improving on the HHL-based solver of He, Jin and Yu [11] and on classical methods. The paper includes a detailed error analysis based on Laplace transforms, several lemmas on matrix norm estimates, and numerical experiments for three test problems.","tokens_in":29253,"tokens_out":19627,"duration_ms":166208,"significance":"If the complexity claims were correct, the iterative method would provide a polynomial improvement in N_v over the existing HHL-based quantum solver, and both methods would avoid HHL-style deep circuits, which is relevant for near-term hardware. The combination of Schrödingerization with AP schemes for multiscale transport is a natural and potentially useful extension of the authors' prior framework, and the Laplace-transform-based bound for the matrix exponential is a technically interesting contribution. The numerical experiments are consistent with the algorithms' correctness as proof-of-concept. However, the complexity analysis as written contains several load-bearing inconsistencies, so the headline complexity claims are not established.","major_comments":[{"comment":"The p-grid size N_eta is not O(log(1/delta)) as used in the proofs. Section 3.1.1 requires the p-domain to satisfy L,R > lambda_max(A1) T, and Remark 3.2 together with the CFL condition gives T = s_iter = N_t = O(N_x^2). Hence the number of grid points is N_p = (L+R)/Delta p = O(N_x^2 log(1/delta)). The proofs instead set N_eta = O(log(1/delta)) and identify ||D_eta||_max = N_eta. This is inconsistent with the definition D_mu = diag(2π(k - N_p/2)/(R+L)) in Section 3.1.1, from which ||D_mu||_max = pi/Delta p = O(log(1/delta)) independent of N_p. The derivations as written are therefore invalid; they should be redone with the correct Fourier multiplier norm and with N_p entering only through the Hilbert-space dimension m(H).","section":"Section 3.1.1; proofs of Theorems 3.1 and 4.1"},{"comment":"The query complexity claimed for the steady-state method is not established even after the Fourier-symbol correction. Section 4.2.3 gives T_stea = O(N_t log(1/delta)) = O(N_x^2 log N_x) for delta = O(1/N_x). With the corrected ||H||_max = O(log(1/delta)), one obtains chi = O(N_v log(1/delta) * T_stea) = O(N_v N_x^2 log^2 N_x). The proof's displayed equality chi = O(N_v N_t N_eta log(1/delta)) = O(N_v N_x^2 log N_x) is arithmetically false when N_eta = O(log(1/delta)). Thus Theorem 4.1's O(N_v N_x^2 log N_x) is unsupported; the best bound derivable from the paper's own estimates has an extra logarithmic factor.","section":"Section 4.2.3; Theorem 4.1 proof"},{"comment":"Both proofs substitute chi into Lemma 3.1 but ignore the factor log(chi/delta) in the query complexity. Lemma 3.1 states Q_query = O(chi log(chi/delta)/log log(chi/delta)). Even under the paper's own chi = O(N_v N_x^2 log N_x) and delta = O(1/N_x), this yields Q_query = O~(N_v N_x^2 log^2 N_x), not O~(N_v N_x^2 log N_x). The claimed Q_query and Q_gate in Theorems 3.1 and 4.1 are therefore off by at least a logarithmic factor under the paper's own assumptions.","section":"Lemma 3.1 application in both theorem proofs"},{"comment":"The abstract and the conclusions assert that both algorithms achieve O(N_v N_x^2 log N_x) and outperform existing quantum and classical algorithms for this problem. In light of the previous comments, this statement is not supported by the proofs. The paper should report the corrected complexity bounds and adjust the comparison with He et al. [11] accordingly.","section":"Abstract and Section 6"}],"minor_comments":[{"comment":"After Eq. (2.20), the paper says that subsequent variables rn and jn represent the post-processed variables, but the hat notation is dropped inconsistently; later sections should clearly state that the matrices A1, A2, B1, B2 in Sections 3 and 4 refer to the preprocessed matrices.","section":"Section 2.5"},{"comment":"The sentence 'in which δ is the desired accuracy' appears after the definition of L and R but is not connected to those definitions; this is likely a typo and should be rephrased.","section":"Section 3.1.1"},{"comment":"The numerical experiments use L = R = N_x, whereas the theoretical requirement in Section 3.1.1 is L,R ~ N_t for large T; the experiments therefore operate in a regime where the p-grid size does not reflect the asymptotic complexity analysis. A note acknowledging this would be helpful.","section":"Section 5"},{"comment":"Lemma 4.1 is cited from Hu et al. [13] without proof, yet it plays a central role in the error analysis of the steady-state method. A proof or a more precise statement should be included.","section":"Lemma 4.1"},{"comment":"The paper defines \\tilde{O} as ignoring log log terms, but the complexity expressions also drop polylogarithmic factors such as log(chi/delta) and log^{2.5}(chi/delta). The notation should be clarified or the standard convention for \\tilde{O} (hiding all polylog factors) should be adopted.","section":"Notation for \\tilde{O}"}],"recommendation":"major_revision","confidential_remarks":"The paper presents a promising algorithmic framework, but the complexity analysis currently contains multiple serious gaps: the p-grid dimension is not tracked correctly, the steady-state method's evolution time introduces an additional logarithm, and the application of Lemma 3.1 ignores the log(chi/delta) factor. These issues affect the central claims in the abstract and theorems. The algorithms themselves may still be valid and the corrected complexity (e.g., O~(N_v N_x^2 log^2 N_x) for the iterative method and a further log for the steady-state method) would still improve on HHL in N_v, so the paper is potentially salvageable. I would ask the authors to redo the complexity analysis carefully and restate the theorems and comparisons accordingly. I also note that several key lemmas are cited from the authors' own prior work without proof in the manuscript; the editor may wish to verify that those results are indeed established in the cited literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things worth knowing about arXiv:2507.18970. First, the iterative Schrödingerization method in Section 3 has a plausible query-complexity claim that probably survives a corrected p-grid accounting. Second, the steady-state method in Section 4, as analyzed, is one log factor above its advertised O~(N_v N_x^2 log N_x).\n\nThe paper is the first to combine Schrödingerization with the exponential-integrator AP diffusive relaxation scheme for multiscale transport, and it incorporates incoming boundary conditions, which earlier work skipped. The preprocessing step in Section 2.5—rescaling j by N_x and symmetrizing the velocity quadrature—is a genuine trick that removes a factor of N_v. The Laplace-transform bound on ||e^{-HT}|| in Section 4.2.2 is also a solid piece of analysis. The numerical tests at small N_x show the methods work.\n\nThe soft spot is the complexity analysis, and it is load-bearing. Section 3.1.1 requires L,R > λ_max(A1) T, so for the iterative method T = N_t = O(N_x^2) forces the p-domain length to grow like N_x^2. The proofs instead take N_η = O(log(1/δ)) and write ||D_η||_max = N_η, conflating the grid count with the momentum symbol norm. Once you separate those, the iterative method's chi becomes O(N_v · ||D_μ||_max · N_t) = O(N_v log(1/δ) · N_x^2), which gives the advertised O(N_v N_x^2 log N_x). So the result likely survives, but the proof as written is wrong. The steady-state method is worse: their own Section 4.2.3 gives T_stea = O(N_t log(1/δ)), and plugging that in gives chi = O(N_v N_t log^2(1/δ)) = O(N_v N_x^2 log^2 N_x). Theorem 4.1 is off by a log, and the abstract's claim that both algorithms achieve O(N_v N_x^2 log N_x) is not supported.\n\nMinor point: the p-grid growth also adds a log N_x to the gate-count dimension m(H), which the paper drops without comment.\n\nWho should read this? People working on quantum algorithms for kinetic and transport PDEs. The AP-preserving Schrödingerization design is worth engaging with. I'd send it to peer review—with the steady-state theorem restated at the higher complexity or tightened—rather than desk reject. The reader's rejection, based on O(N_v N_x^4), overstates the damage; the real problem is an extra log, not a polynomial loss.","headline":"Iterative method's complexity likely survives a corrected p-grid accounting; steady-state method's headline claim is off by a log—worth a serious referee, not a desk reject.","tokens_in":29804,"tokens_out":6944,"would_cite":false,"duration_ms":62963,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68","65M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proposes two Hamiltonian simulation algorithms for the multiscale linear transport equation with incoming boundary conditions; both achieve query complexity $\\tilde{O}(N_v N_x^2 \\log N_x)$, independent of the mean free path…","keywords":["quantum simulation","linear transport equation","Schrödingerization","exponential integrators","asymptotic-preserving scheme","Hamiltonian simulation","multiscale problems","incoming boundary conditions"],"falsifier":"Take a representative one-dimensional test problem, for instance the constant-coefficient case with $N_x = 64$ and $N_t = N_x^2$, compute the spectrum of $A_1$, and ask whether the $p$-domain $[-L,R]$ with $L=R=N_x$ used in the numerical section satisfies $L > \\lambda^-_{\\max}(A_1) N_t$; if the answer is no, or if the simulation error does not decay like $O(1/N_x)$ when $N_p$ is held at $O(\\log N_x)$, then the claimed query complexity has not been established.","tokens_in":28525,"feed_emoji":"⚛️","tokens_out":8473,"duration_ms":85764,"temperature":0.7,"pith_summary":"The paper claims that the multiscale linear transport equation, the phase-space model of particles moving through a scattering medium, can be simulated on a quantum computer in $\\tilde{O}(N_v N_x^2 \\log N_x)$ queries, with the cost independent of the mean free path $\\varepsilon$. It offers two Hamiltonian simulation algorithms, one iterative and one steady-state, both built from the Schr\\\"odingerization method, a variable change that turns a non-Hermitian linear evolution into a Hermitian Schr\\\"odinger-type equation, combined with an exponential-integrator, asymptotic-preserving discretization. Incoming boundary conditions are included, which earlier quantum solvers for this equation omitted. If the bounds are right, the second algorithm also produces all intermediate time states, making it useful for time-dependent boundary and interface problems.","feed_headline":"Two quantum solvers hit O~(Nv Nx² log Nx) for transport","feed_subtitle":"The methods keep cost independent of the tiny mean free path ε, beating earlier quantum and classical solvers.","key_machinery":"Schr\\\"odingerization: for a linear ODE $du/dt = Au + b$, the change of variables $\\hat{u} = e^{-p}u$ extends the system in an auxiliary $p$-direction and rewrites the evolution as $\\partial_t \\hat{u} = -A_1 \\partial_p \\hat{u} + iA_2 \\hat{u}$ with Hermitian $A_1, A_2$; after a Fourier spectral discretization in $p$ this becomes unitary dynamics $-iH\\hat{u}$ with $H = D_\\mu \\otimes A_1 - I \\otimes A_2$, simulable by a Hamiltonian-simulation oracle. The paper pairs this with an exponential-integrator diffusive-relaxation discretization of the transport equation that is asymptotically preserving, a preprocessing step that rescales the odd-parity flux $j$ by $1/N_x$ to bring all matrix elements to $O(1)$, and a Laplace-transform bound on $\\|e^{-HT}\\|_2$ that fixes the evolution time for the steady-state solver. These pieces jointly deliver the $\\tilde{O}(N_v N_x^2 \\log N_x)$ query count.","core_discovery":"On the paper's own terms, the discovery is a way of combining three existing ingredients, Schr\\\"odingerization, exponential-integrator diffusive relaxation schemes, and a preprocessing rescaling of the odd-parity variable, so that the multiscale transport equation can be solved by Hamiltonian simulation at query complexity $\\tilde{O}(N_v N_x^2 \\log N_x)$. This improves on the previous quantum linear-system solver's $O(N_v^2 N_x^2 \\log N_x)$ by a factor of $N_v$ and is exponentially better in $N_x$ than classical solvers. The key technical step in the steady-state variant is a Laplace-transform-based estimate of the matrix exponential norm $\\|e^{-HT}\\|_2$ that gives a tight evolution time $T = O(N_t \\log(1/\\delta))$ instead of a crude norm bound. Both algorithms are claimed to be the first to combine Schr\\\"odingerization with an effective asymptotic-preserving scheme for this equation, and their complexity is claimed to be independent of $\\varepsilon$.","pith_inferences":["A direct test of the quoted complexity would check whether the $p$-domain length required by the paper's own condition $L > \\lambda^-_{\\max}(A_1) T$ stays bounded when $T = O(N_x^2)$; if it does not, the $\\tilde{O}(N_v N_x^2 \\log N_x)$ rate would need an extra factor of $N_x^2$.","The Laplace-transform estimate of the matrix exponential is not specific to transport; it could sharpen evolution-time bounds in other steady-state quantum linear solvers where naive norm bounds overestimate $T$.","Because the discretization is asymptotic-preserving, the same algorithms should remain well-conditioned as $\\varepsilon \\to 0$; one testable prediction is that measured query counts stay flat as $\\varepsilon$ drops from $10^{-1}$ to $10^{-8}$.","A natural next test is to replace the Fourier $p$-grid by an adaptive or wavelet representation: if the $p$-direction can be compressed, the $\\varepsilon$-independent bound would hold with a smaller constant and less dependence on smoothness assumptions."],"forward_implications":["Both proposed algorithms solve the multiscale transport equation with incoming boundary conditions at $\\tilde{O}(N_v N_x^2 \\log N_x)$ queries, removing one factor of $N_v$ relative to the earlier quantum linear-system approach and an exponential factor in $N_x$ relative to classical methods.","The complexity is independent of the mean free path $\\varepsilon$, so the same circuit budget covers kinetic regimes and the diffusion limit.","The steady-state variant returns all intermediate time states, making it usable for time-dependent boundary and interface problems where an iterative time-marcher would need repeated solves.","Being Hamiltonian simulation rather than a linear-system solver, the algorithms are more compatible with near-term hardware.","This is the first combination of Schr\\\"odingerization with an asymptotic-preserving scheme for multiscale transport, opening a template for other stiff multiscale equations."],"supporting_citations":[{"why":"Supplies the sparse Hamiltonian simulation lemma used to convert Hamiltonian data into the query and gate counts in Theorems 3.1 and 4.1.","marker":"[3]"},{"why":"Establishes the baseline quantum algorithm for the same equation, whose $O(N_v^2 N_x^2 \\log N_x)$ complexity the paper improves by a factor of $N_v$.","marker":"[11]"},{"why":"Provides the Schr\\\"odingerization framework for discrete linear dynamical systems on which the iterative algorithm is built.","marker":"[20]"},{"why":"Gives the smooth-initialization result used to set the auxiliary $p$-grid size $N_\\eta = O(\\log(1/\\delta))$, fixing the logarithmic factors in the complexity.","marker":"[25]"},{"why":"Introduces the Schr\\\"odingerization method that maps non-Hermitian linear ODEs to Hermitian Schr\\\"odinger dynamics.","marker":"[30]"},{"why":"Supplies the diffusive relaxation asymptotic-preserving discretization and the incoming-boundary treatment that the exponential integrator uses.","marker":"[32]"},{"why":"Provides the quantum IMEX framework that the steady-state method adapts to multiscale transport.","marker":"[14]"}],"fun_headline_variants":["Quantum transport solver hits O(Nv Nx² log Nx) cost","First AP Schrödingerization scheme for transport equations","Multiscale transport via quantum exponential integrators","Quantum scheme for transport independent of mean free path","Schrödingerization plus exponential integrators for transport"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument's load-bearing premise is that the auxiliary grid used in the unitary transformation can stay logarithmically small even when the simulation time is $O(N_x^2)$; the paper's own domain-size condition suggests the grid may need to grow proportionally to that time.","fun_headline_variants_meta":{"raw":{"variants":["Quantum transport solver hits O(Nv Nx² log Nx) cost","First AP Schrödingerization scheme for transport equations","Multiscale transport via quantum exponential integrators","Quantum scheme for transport independent of mean free path","Schrödingerization plus exponential integrators for transport"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000854,"raw_usage":{"total_tokens":3707,"prompt_tokens":941,"completion_tokens":2766,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":2688}},"tokens_in":557,"tokens_out":2766,"duration_ms":21472,"temperature":1.0,"reasoning_tokens":2688,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:04:31.526022+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a representative one-dimensional test problem, for instance the constant-coefficient case with $N_x = 64$ and $N_t = N_x^2$, compute the spectrum of $A_1$, and ask whether the $p$-domain $[-L,R]$ with $L=R=N_x$ used in the numerical section satisfies $L > \\lambda^-_{\\max}(A_1) N_t$; if the answer is no, or if the simulation error does not decay like $O(1/N_x)$ when $N_p$ is held at $O(\\log N_x)$, then the claimed query complexity has not been established.","supporting_citations":[{"cited_title":"Berry, Andrew M","cited_arxiv_id":null,"evidence_quote":"Supplies the sparse Hamiltonian simulation lemma used to convert Hamiltonian data into the query and gate counts in Theorems 3.1 and 4.1."},{"cited_title":"Time complexity analysis of quantum difference methods for the multiscale transport equations","cited_arxiv_id":null,"evidence_quote":"Establishes the baseline quantum algorithm for the same equation, whose $O(N_v^2 N_x^2 \\log N_x)$ complexity the paper improves by a factor of $N_v$."},{"cited_title":"Quantum simulation of discrete linear dynamical systems and simple iterative methods in linear algebra","cited_arxiv_id":null,"evidence_quote":"Provides the Schr\\\"odingerization framework for discrete linear dynamical systems on which the iterative algorithm is built."},{"cited_title":"Quantum simulation of partial differential equations via schr¨ odingerization.Physical Review Letters, 133:230602, 2024","cited_arxiv_id":null,"evidence_quote":"Introduces the Schr\\\"odingerization method that maps non-Hermitian linear ODEs to Hermitian Schr\\\"odinger dynamics."},{"cited_title":"Uniformly accurate diffusive relaxation schemes for multiscale transport equations","cited_arxiv_id":null,"evidence_quote":"Supplies the diffusive relaxation asymptotic-preserving discretization and the incoming-boundary treatment that the exponential integrator uses."},{"cited_title":"Schr¨ odingerization-based quantum implicit-explicit schemes: From time-dependent pdes to multiscale equations, 2025","cited_arxiv_id":null,"evidence_quote":"Provides the quantum IMEX framework that the steady-state method adapts to multiscale transport."}],"review_version":1}