{"id":"6326faeb-d0dd-4878-a816-ff907c6811b4","arxiv_id":"2507.20599","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Measuring a quantum state's Fourier coefficients, rather than its grid-point amplitudes, recovers smooth amplitude-encoded functions with shot count independent of grid size, preserving quantum speedups for CAE readout.","lead":"This paper proposes a quantum-classical readout method that extracts smooth functions from quantum states by measuring their Fourier coefficients instead of grid-point amplitudes. The scheme's cost depends on the function's smoothness and target accuracy, not on grid size, which could preserve quantum speedups for computer-aided engineering simulations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The formal complexity bound in Table 1 is not established because the sign-determination step is excluded from the proof: Appendix C.2 derives Eqs. (19)-(20) after assuming signs are correctly determined, while the actual FSR protocol can misclassify coefficients near the sampling threshold.","rationale":"The reader's weakest assumption was the smoothness/regularity requirement for Fourier-coefficient decay, which is a necessary condition for the N-independence of M. That concern is valid and is explicitly acknowledged in the paper's framing. However, I see a more load-bearing internal gap: the formal proof of the claimed shot complexity assumes away the sign-determination error that the paper's own protocol introduces. Appendix C.2 derives Eqs. (19)-(20) only after stating that signs are correctly determined for simplicity, and the concluding section admits that sign determination is still a problem. This means the central complexity claim in Table 1 is not fully proven even in the favorable smooth regime. The method remains plausible: the numerical experiments suggest N-independence for smooth examples, and the omitted sign-error contribution may indeed be small, but that needs a theorem or a careful simulation. My assessment therefore does not change the reader's CONDITIONAL verdict; it sharpens the reason the verdict should be conditional.","tokens_in":32500,"tokens_out":12354,"duration_ms":148862,"concrete_test":"Add the missing sign-error term to the Appendix C.2 derivation: for each k, let E_k be the event that sgn(g_k + delta) differs from sgn(c_k), and include 4 * sum_k |c_k|^2 * Pr(E_k | Nshot, delta) inside the bound of Eq. (19). Bound Pr(E_k) via Chebyshev on g_k using the binomial variances of d_k and e_k. Then check numerically for the quadratic example with delta = 2/sqrt(Nshot), N = 65536, M = 64, and Nshot = 160000 whether the resulting RMSE still reaches epsilon with M = O(epsilon^{-2/3}) and Nshot = O(epsilon^{-8/3}); if the sign-error term is not O(M/Nshot), Table 1's scaling is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing gap is in the shot-complexity proof, not only in the smoothness hypothesis. In Appendix C.2, the bounds on the RMSE and the normalized error, Eqs. (19) and (20), follow from an estimate that explicitly assumes 'the signs are correctly determined ... for simplicity.' The actual protocol in Sect. 2.3 and Fig. 5 determines signs by comparing g_k = 2e_k - d_k - 1/sqrt(M) with a threshold delta = O(1/sqrt(Nshot)), chosen in Remark 4 as delta = 2/sqrt(Nsum). Coefficients whose magnitude is comparable to or below the sampling error will sometimes receive the wrong sign. The paper's own conclusion concedes that 'the efficient determinations of the signs of the Fourier coefficients still remain a problem' and that sign inversions of small coefficients cause a relatively large variance. The error bounds in Eqs. (19) and (20) contain only a truncation term and a sampling term of size O(beta^2 M/Nshot); they contain no term accounting for mis-signed coefficients. Therefore Table 1's quantum complexity O(polylog N (1/epsilon)^{2+s}) is not a proven consequence of the analysis as written. This concern is independent of the smoothness assumption: even for the quadratic function f1, which satisfies the regularity and boundary hypotheses, the proof omits an actual error source of the protocol. The claim may survive a repair, since the mis-signed coefficients are small and their squared contribution may be absorbable, but presently the central complexity statement is conditional on an unproven sign-error estimate.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes the Fourier space readout (FSR) method for recovering a real-valued function encoded in the amplitudes of a quantum state. The protocol applies an inverse quantum Fourier transform to the extended (even) state, measures in the computational basis, post-selects on the low-frequency registers, and uses a second circuit to determine signs via a shifted-state overlap. The authors claim that the number of quantum measurements and the classical reconstruction cost depend on the function smoothness and error bound but not on the grid number N: quantum complexity O(polylog N (1/ε)^{2+s}), classical O((1/ε)^{2+s}) + O(J(1/ε)^s). The paper provides explicit circuits for 1D and multi-dimensional cases, numerical experiments on Qiskit for up to 65536 grid points (1D) and 512×512 (2D), and a theoretical analysis in Appendices B and C based on Fourier truncation error and binomial shot-noise bounds.","tokens_in":32767,"tokens_out":6079,"duration_ms":63792,"significance":"If the complexity claim holds, the FSR method would be a practically relevant counterexample to the common assertion that full state readout necessarily destroys the quantum speedup in PDE/CAE solvers. The paper's strengths are its explicit, low-depth circuits (fan-out, controlled incrementer, QFT), the clean truncation analysis in Lemma 1 that connects Sobolev regularity to M = O(ε^{-s}), and the numerical demonstration of N-independence for smooth test functions. The adaptive scheme in Sect. 5.2 is a useful practical addition since it removes the need for a priori knowledge of M. The central derivation is largely self-contained and the numerical scalings agree with the theoretical orders for the examples chosen.","major_comments":[{"comment":"The error bound for the FSR method is derived under the explicit assumption that \"the signs are correctly determined ... for simplicity.\" The actual protocol in Sect. 2.3 and Fig. 5 determines signs by comparing g_k = 2e_k - d_k - 1/sqrt(M) with δ = O(1/sqrt(Nshot)) (Remark 4). For coefficients with |c_k| comparable to or below the sampling error, the sign can be misclassified, and the squared-error contribution from such mis-signed coefficients is not included in Eqs. (19)-(20). The paper's own conclusion (Sect. 6) concedes that \"the efficient determinations of the signs of the Fourier coefficients still remain a problem\" and that sign inversions of small coefficients cause \"a relatively large variance.\" Consequently, the Table 1 complexity O(polylog N (1/ε)^{2+s}) is not a proven statement for the complete FSR protocol as written. The authors should either prove that the sign-error contribution is absorbed into the existing error budget (e.g., bounded by O(β^2 M/Nshot) plus higher-order terms) or replace the threshold-based sign determination with a procedure that has rigorous success guarantees (for example, using quantum amplitude estimation or additional ancillas to estimate the sign with controlled failure probability).","section":"Appendix C.2, Eqs. (19)-(20)"},{"comment":"The shot-count analysis does not explicitly account for the post-selection probability in Figs. 4 and 5. The number of successful post-selected samples is Nshot * p_sel, where p_sel is the probability that the most significant n-m+1 (or n-m+2) qubits are measured in |0⟩; this probability equals the total weight of the retained Fourier coefficients and is generally less than 1. The binomial model in Appendix C.2 treats Nshot as the number of trials directly, which is valid only if p_sel is either close to 1 or the constant overhead is absorbed. Since p_sel is independent of N but depends on the function and M, the statement \"Nshot = O((1/ε)^{2+s})\" should be conditioned on a uniform lower bound for p_sel, or the O(·) constant should be understood to include the factor 1/p_sel. This does not affect the N-independence, but it is a gap in the precise complexity accounting that should be clarified.","section":"Sect. 3.3, Table 1 and Appendix C.2"},{"comment":"The adaptive algorithm determines M from the observed support of d_k, and Remark 7 claims that this scheme has the same order as the fixed-M analysis. The justification is heuristic: for finite Nshot, the noise floor is of order 1/sqrt(Nshot), so isolated single-shot events can produce nonzero d_k beyond the true support. The margin parameter Nmar is introduced to suppress such events, but no theoretical guarantee is given that the adaptive M stays within a constant factor of the optimal truncation level. Since the adaptive scheme is presented as the practical version of the FSR method and is used in the numerical reconstructions of Sect. 4, this gap should either be closed with a probabilistic bound or explicitly labeled as an empirical heuristic with the main complexity claim reserved for the fixed-M version.","section":"Sect. 5.2 and Remark 7"}],"minor_comments":[{"comment":"The construction of the even extension U_ext,ψ requires f(x_N) = f(x_0), i.e., f(0) = f(L). This condition is stated only in Remark 1; it should be stated in the main text before Eq. (2), because the circuit in Fig. 3 does not implement Eq. (2) without it.","section":"Sect. 2.3, Eq. (2)"},{"comment":"The phrase \"we use simply a QFT instead a controlled QFT\" should read \"instead of a controlled QFT,\" and the equivalence between the two circuits deserves a brief explanation.","section":"Fig. 5 caption"},{"comment":"Typo: \"we count the the number\" should read \"we count the number.\"","section":"Sect. 5.2, Step 2"},{"comment":"The passage \"If f has the continuous Fourier expansion exactly\" is ambiguous: the equality c_{k,d} = c_{k,c} for k = -N/2,...,N/2-1 requires that the Fourier series of f converges to f at each grid point, which is not guaranteed for arbitrary piecewise W^{1,1} functions at jump discontinuities. Since the final complexity claims concern smooth or piecewise smooth functions, this is a technical point, but the conditions under which the equality holds should be stated precisely.","section":"Appendix C.1, Lemma 1"},{"comment":"The notation Nshot1 and Nshot2 is used in Sect. 2.3, but the complexity discussion mostly uses Nshot; a sentence explicitly stating that the analysis takes Nshot1 = Nshot2 and that the two constants are absorbed into O(·) would improve readability.","section":"Sect. 3.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is a good fit for a quantum-information journal and the core idea is worth publishing if the sign-determination gap is closed. The conclusion currently overstates the result: the phrase \"the readout ... does not ruin the quantum speedup regarding the grid number\" is stronger than what is proven, because the formal error bounds omit a real error source (sign misclassification) that the authors themselves acknowledge in Sect. 6. I would encourage the editor to request a rigorous treatment of the sign step or a clearly separated version of the complexity theorem that assumes correct signs, with the sign issue as a formally analyzed extension rather than a concession."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth your time. The central idea is simple and effective: instead of measuring the amplitude-encoded function at grid points, apply the inverse QFT, read out the dominant Fourier coefficients, and reconstruct classically. For smooth functions this gives a shot count that depends on the error bound and not on the grid number, which is exactly what quantum PDE workflows need. The authors also provide explicit circuits for the even/odd extension and for sign extraction via LCU, plus an adaptive scheme for choosing the truncation level. That is a real step beyond the overlap-based readouts of [64] and [66], and the numerical results, including the 1D and 2D reconstructions, support the claimed N-independence.\n\nThe main soft spot is exactly what the stress-test note says: the formal complexity in Table 1 is not established. Appendix C.2 derives the bound on the assumption that signs are correctly determined, and the actual protocol can mis-sign coefficients near the sampling threshold. The paper's own conclusion admits this. The error bound contains no term for mis-signed coefficients, so as written the O(polylog N (1/eps)^{2+s}) claim is conditional. I think the gap is repairable — the mis-signed coefficients are small, and their squared contribution probably can be absorbed into the existing sampling term — but the repair needs to be written down. This is the one thing that should block acceptance in current form.\n\nOther issues are minor. The abstract overgeneralizes: the method's efficiency relies on the underlying function being smooth enough that M = O(eps^{-s}) with fixed s. The known-normalization assumption on A_N is real but stated clearly. Missing error bars on the numerical plots are annoying but not damning. There is also a likely typo in Section 2.3, where the extended-grid spacing is written as jL/(2N) but should be jL/N.\n\nI agree with the reader's overall assessment: the method is novel and significant within the quantum readout subfield, the derivation is mostly clean, and the central claim is defensible under the stated assumptions. The sign-determination gap is the one thing I would insist on fixing before treating the complexity statement as proven.\n\nWho is this for? Anyone working on readout for quantum simulation of PDEs or CAE. It deserves a serious referee — send it to peer review, but require the sign-error estimate to be either proved or explicitly folded into the stated assumptions, and ask for a slightly less sweeping abstract.","headline":"The core idea is clean and the numerics are persuasive, but the paper's headline claim of N-independent readout cost is not actually proven because the sign-determination step sits outside the formal error analysis.","tokens_in":33358,"tokens_out":1921,"would_cite":false,"duration_ms":21170,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68","65T50","65Y20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Reading out a quantum-encoded function can be done with shots set by smoothness, not grid size.","keywords":["quantum state readout","Fourier space readout","amplitude encoding","quantum Fourier transform","function recovery","quantum linear system algorithms","complexity analysis","computer-aided engineering"],"falsifier":"Take a discontinuous or endpoint-mismatched function and run the FSR circuits at a fixed shot count; for instance, compare the RMSE for $f(x)=x^2$ with $f(x)=\\operatorname{sign}(\\sin(2\\pi x))$ at $N=1024$ and $N=65536$. If the RMSE grows with $N$, or if the inferred $M$ must grow with $N$ rather than with $(1/\\varepsilon)^s$, the central $N$-independence claim is wrong.","tokens_in":32260,"feed_emoji":"⚛️","tokens_out":7190,"duration_ms":75390,"temperature":0.7,"pith_summary":"This paper tries to establish that the readout bottleneck of quantum computing in engineering simulation can be removed for smooth functions. It proposes a quantum-classical hybrid Fourier space readout (FSR): instead of measuring all $N$ grid-point amplitudes of an amplitude-encoded state, apply the inverse quantum Fourier transform, read out the $M$ dominant Fourier coefficients, and reconstruct the function classically. The claimed complexity is $O(\\operatorname{polylog} N\\,(1/\\varepsilon)^{2+s})$ quantum shots and $O((1/\\varepsilon)^{2+s}) + O(J(1/\\varepsilon)^s)$ classical cost, where $s$ is set by the function's smoothness and $J$ is the number of target points. If true, readout no longer ruins the quantum speedup in grid number for CAE-type solutions, and the cost depends on the underlying function and desired accuracy rather than on the discretization.","feed_headline":"Readout cost stops scaling with grid size for smooth quantum states","feed_subtitle":"Shot count and classical cost track the encoded function's smoothness, not the number of grid points.","key_machinery":"The central object is the even-extension operator followed by the inverse quantum Fourier transform. The extension operator doubles the computational domain and makes the amplitude-encoded function center-symmetric, so the resulting quantum Fourier coefficients are real-valued; post-selecting the high-order register leaves only the $M$ dominant coefficients. Their magnitudes come from one Z-basis circuit, and their signs come from a second circuit built with a linear combination of unitaries that shifts the state by a known constant. This carries the argument because the Fourier coefficients of a smooth function decay as $O(k^{-p})$, so $M=O((1/\\varepsilon)^{2/(2p-1)})$ suffices and the total shot count becomes $O((1/\\varepsilon)^{2+2/(2p-1)})$, independent of $N$.","core_discovery":"The paper claims that for a real-valued function $f$ amplitude-encoded as $\\psi_j = f(jL/N)/A_N$, the inverse quantum Fourier transform followed by post-selection yields all $M$ dominant Fourier coefficients at once: their absolute values from one measurement circuit and their signs from an LCU-shift circuit. Because the Fourier coefficients of a regular function decay, $M=O((1/\\varepsilon)^s)$ terms suffice for error $\\varepsilon$, and the measurement repetition count is $O((1/\\varepsilon)^{2+s})$ — both independent of the grid number $N$. The classical reconstruction $f(x)\\approx (A_N/\\sqrt{N})(\\hat c_0+2\\sum_{k=1}^{M-1}\\hat c_k \\cos(k\\pi x/L))$ costs $O(JM)$ for $J$ target points. The paper concludes that readout of function-based quantum states does not ruin the quantum speedup regarding grid number.","pith_inferences":["Beyond the paper, the argument is a compression argument: any basis with a cheap unitary quantum transform and sparse or decaying coefficients for the solution class, such as wavelets, should inherit the same readout scaling; the paper itself notes Chebyshev transforms but flags the difficulty of constructing a unitary inverse.","Beyond the paper, the adaptive truncation rule could serve as a built-in spectral-error indicator: if non-negligible coefficients still appear near the cutoff as the shot count grows, the simulation is under-resolved at that scale.","A testable extension is to push the method to solutions with shocks or interfaces; one would expect a crossover in $N$ beyond which the FSR error starts growing, and a piecewise or dictionary basis might restore $N$-independence."],"forward_implications":["For smooth, boundary-matched solutions of PDEs, the shot count needed to read out the full solution becomes independent of the grid number, so the quantum solver's speedup in $N$ is no longer cancelled by readout.","When only $J$ target points matter, classical reconstruction costs $O(J(1/\\varepsilon)^s)$ plus the histogram cost, so targeted regions of the solution can be extracted much more cheaply than all $N$ grid values.","The $N_{\\mathrm{shot}}$-adaptive scheme lets the algorithm choose the truncation $M$ from the measurement record alone, removing the need to know the function in advance.","The $d$-dimensional version uses $d+1$ ancillary qubits and tensorized circuits, giving the same independence of $N$ in each dimension.","Even and odd extension operators handle periodic and zero-boundary cases, and reconstruction is defined at arbitrary target points, not only at the grid points."],"supporting_citations":[{"why":"Supplies the Fourier-coefficient decay estimate used to bound the truncation parameter $M$ by the error bound and regularity.","marker":"[8]"},{"why":"Defines the quantum linear-systems setting whose readout bottleneck this method targets.","marker":"[16]"},{"why":"Prior orthogonal-function-expansion readout that FSR extends by avoiding iterative classical optimization per basis.","marker":"[66]"},{"why":"Prior overlap-based feature-extraction readout whose per-basis optimization the direct Fourier-coefficient extraction replaces.","marker":"[64]"},{"why":"Provides the linear combination of unitaries construction used in the sign-determination circuit.","marker":"[70]"},{"why":"Supplies the incrementer and modular-adder gates used to build the even-extension and shift operations.","marker":"[69]"}],"fun_headline_variants":["Readout no longer kills quantum speedup for smooth functions","Fourier readout recovers functions without grid-size penalty","Quantum readout that scales with smoothness, not grid points","Readout cost now tracks function smoothness, not grid size"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central grid-number independence rests on the input function having Fourier coefficients that decay quickly enough that a fixed number $M$, set only by the desired error, captures it; for rough, oscillatory, or discontinuous functions the necessary $M$ grows with the grid and the shot count loses its $N$-independence.","fun_headline_variants_meta":{"raw":{"variants":["Readout no longer kills quantum speedup for smooth functions","Fourier readout recovers functions without grid-size penalty","Quantum readout that scales with smoothness, not grid points","Readout cost now tracks function smoothness, not grid size"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000622,"raw_usage":{"total_tokens":2859,"prompt_tokens":901,"completion_tokens":1958,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":517,"completion_tokens_details":{"reasoning_tokens":1888}},"tokens_in":517,"tokens_out":1958,"duration_ms":14541,"temperature":1.0,"reasoning_tokens":1888,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:41:45.331666+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a discontinuous or endpoint-mismatched function and run the FSR circuits at a fixed shot count; for instance, compare the RMSE for $f(x)=x^2$ with $f(x)=\\operatorname{sign}(\\sin(2\\pi x))$ at $N=1024$ and $N=65536$. If the RMSE grows with $N$, or if the inferred $M$ must grow with $N$ rather than with $(1/\\varepsilon)^s$, the central $N$-independence claim is wrong.","supporting_citations":[{"cited_title":"Childs, J","cited_arxiv_id":null,"evidence_quote":"Supplies the Fourier-coefficient decay estimate used to bound the truncation parameter $M$ by the error bound and regularity."},{"cited_title":"Miyamoto, and H","cited_arxiv_id":null,"evidence_quote":"Prior orthogonal-function-expansion readout that FSR extends by avoiding iterative classical optimization per basis."}],"review_version":2}