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REVIEW 3 major objections 5 minor 76 references

Fourier space readout method for efficiently recovering functions encoded in quantum states

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Reading out a quantum-encoded function can be done with shots set by smoothness, not grid size.

desk verdict 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. read the letter →

arxiv 2507.20599 v1 pith:REWGAYD2 submitted 2025-07-28 quant-ph

Xinchi Huang , Hirofumi Nishi , Yoshifumi Kawada , Tomofumi Zushi , Yu-ichiro Matsushita This is my paper · ORCID
classification quant-ph MSC 81P6865T5065Y20
keywords quantumstatereadoutFourierspaceamplitudeencodingtransformfunctionrecoverylinearsystemalgorithmscomplexityanalysiscomputer-aidedengineering
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $N$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [Appendix C.2, Eqs. (19)-(20)] 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).
  2. [Sect. 3.3, Table 1 and Appendix C.2] 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.
  3. [Sect. 5.2 and Remark 7] 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.
minor comments (5)
  1. [Sect. 2.3, Eq. (2)] 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.
  2. [Fig. 5 caption] 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.
  3. [Sect. 5.2, Step 2] Typo: "we count the the number" should read "we count the number."
  4. [Appendix C.1, Lemma 1] 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.
  5. [Sect. 3.2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the FSR complexity derivation is self-contained, with truncation and shot counts set by Fourier decay estimates and error bounds rather than fitted to the claimed result.

full rationale

The paper's central claim — that reading out a smooth function's Fourier coefficients costs O(polylog N (1/ε)^{2+s}) quantum operations and O((1/ε)^{2+s}) + O(J(1/ε)^s) classical work — is derived from standard ingredients: (i) the quantum Fourier transform is unitary and efficient, (ii) Fourier coefficients of piecewise W^{1,1} functions decay as O(k^{-p}) (Lemma 1, Appendix C.1), and (iii) Chebyshev-type sampling bounds convert a target error ε into M = O(ε^{-2/(2p-1)}) and Nshot = O(ε^{-(2+2/(2p-1))}). These parameters are chosen from the error bound and the function's regularity; they are not fitted so as to reproduce the claimed N-independence. The numerical experiments compare measured RMSE scalings against predicted reference lines, but the predicted exponents come from the same Fourier-decay analysis and are not used to define the method's constants. Self-citations in the paper (e.g., Refs. [34], [52], [64], [65]) are contextual and are not load-bearing for the FSR derivation. One substantive caveat, noted in the paper itself, is that Appendix C.2's bound assumes 'the signs are correctly determined ... for simplicity' and therefore omits the error contribution from mis-signed small Fourier coefficients; the conclusion also concedes that 'the efficient determinations of the signs of the Fourier coefficients still remain a problem.' This is a correctness/completeness gap in the proof rather than a circular reduction: the eventual error bound does not define the sign-determination rule, and the sign rule does not assume the bound it is meant to prove. Since no prediction is equivalent by construction to an input, no fitted quantity is renamed as a prediction, and no load-bearing argument reduces to a self-citation, the appropriate circularity score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper's N-independence is purchased by smoothness and Fourier decay, known normalization, efficient state preparation, and boundary compatibility. These are stated assumptions rather than derived results. The method introduces no new physical entities.

free parameters (3)
  • M (truncation level) = adaptive, e.g., 16 to 64 in numerical examples
    Number of retained Fourier coefficients; chosen from error bound and smoothness in theory, adaptively from measured nonzero coefficients in Section 5.2, and fixed to 64 in some scans.
  • delta (sign threshold) = 2/sqrt(Nsum) in numerics
    Threshold deciding whether a Fourier coefficient is positive or negative in Eq. (3); chosen by hand in Remark 4 and affects sign-error rates.
  • Nmar margin parameter = 4 in 2D examples
    Adaptive algorithm ignores isolated nonzero counts within a window of Nmar; chosen ad hoc in Section 5.2 to avoid overestimating M from single-shot noise.
assumptions (5)
  • domain assumption The input quantum state is an amplitude encoding of a real-valued continuous function f with known normalization A_N.
    Defined in Section 2.2; without this, Fourier-coefficient localization and the reconstruction formula Eq. (4) do not apply.
  • domain assumption The function satisfies boundary condition f(0)=f(L) or an equivalent condition so the even extension circuit Eq. (2) is exact.
    Remark 1 restricts to periodic or homogeneous Dirichlet boundary conditions; general boundary conditions need additional gates.
  • standard math Fourier coefficients decay as |c_k|=O(k^{-p}) with p>=1 under the regularity and boundary assumptions of Lemma 1.
    Proved in Appendix C.1 using integration by parts and Sobolev regularity; it bounds M and drives the N-independence claim.
  • domain assumption The state preparation oracle U_psi has gate complexity O(polylog N).
    Assumed in Section 3 and Table 1; otherwise the quantum readout circuit is not logarithmic in N.
  • standard math Measurement outcomes are independent binomial trials, so Chebyshev-type bounds on estimation error apply.
    Appendix C.2 uses standard binomial and Chebyshev arguments for shot-count scaling.

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Pith. "Pith review of Fourier space readout method for efficiently recovering functions encoded in quantum states." pith.science (2026). https://pith.science/paper/REWGAYD2

@misc{pith2026250720599,
  author       = {Pith},
  title        = {Pith review of: Fourier space readout method for efficiently recovering functions encoded in quantum states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/REWGAYD2}},
  note         = {Machine review of arXiv:2507.20599}
}
read the original abstract

Applying quantum computing in the computer-aided engineering (CAE) problems are highly expected since quantum computers yield potential exponential speedups for the operations between extremely large matrices and vectors. Although efficient quantum algorithms for the above problems have been intensively investigated, it remains a crucial task to extract all the grid-point values encoded in the prepared quantum states, which was believed to eliminate the achieved quantum advantage. In this paper, we propose a quantum-classical hybrid Fourier space readout (FSR) method to efficiently recover the underlying function from its corresponding quantum state. We provide explicit quantum circuits, followed by theoretical and numerical discussions on its complexity. In particular, the complexity on quantum computers has only a logarithmic dependence on the grid number, while the complexity on classical computers has a linear dependence on the number of target points instead of the grid number. Our result implies that the achieved quantum speedups are not necessarily ruined when we read out the solutions to the CAE problems.

Figures

Figures reproduced from arXiv: 2507.20599 by the authors.

Figure 1
Figure 1. Quantum circuit for the real space readout. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Quantum circuit for the Fourier space readout with an integer parameter [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. A quantum circuit for the (even) extension operator [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: Quantum circuit for determining the absolute values of the Fourier coefficients in the FSR [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Auxiliary quantum circuit for determining the signs of the dominant Fourier coefficients [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: RMSE plots regarding approximation parameter [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: RMSE plots regarding number of repetitions [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: RMSE plots regarding grid number N for the RSR method and the FSR method. Two subplots for a quadratic function and a linear combination of two Gaussian functions are illustrated. The gray reference lines indicate O [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: Reconstructions by the RSR method and the FSR method for two functions with a small [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: Reconstructions by the RSR method and the FSR method for two functions with a large [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: Reconstructions by the RSR method and the FSR method for a 2D trigonometric function [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: Reconstructions by the RSR method and the FSR method for two functions with a large [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 13
Figure 13. Figure 13: A quantum circuit for the 2D Fourier space readout method. Multi-dimensional cases [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: Explicit quantum circuits using Z-basis measurements for the FSR method in the 2D case. Multi-dimensional cases can be discussed in a similar way. and the input quantum state is given by |ψ⟩ = 1 A N X1−1 j1=0 · · · N Xd−1 jd=0 f  x (1) j1 , . . . , x (d) jd  |jd⟩ ⊗ …
Figure 15
Figure 15. Figure 15: A quantum circuit for an extension operator [PITH_FULL_IMAGE:figures/full_fig_p022_15.png]
Figure 16
Figure 16. Figure 16: A quantum circuit for the odd extension operator [PITH_FULL_IMAGE:figures/full_fig_p025_16.png]
Figure 17
Figure 17. Figure 17: Reconstructions by the RSR method and the FSR method for a quadratic function. [PITH_FULL_IMAGE:figures/full_fig_p032_17.png]
Figure 18
Figure 18. Figure 18: Reconstructions by the RSR method and the FSR method for a linear combination of [PITH_FULL_IMAGE:figures/full_fig_p033_18.png]
Figure 19
Figure 19. Figure 19: Reconstructions of the normalized functions/states by the RSR method and the FSR [PITH_FULL_IMAGE:figures/full_fig_p033_19.png]
Figure 20
Figure 20. Figure 20: Reconstructions of the normalized functions/states by the RSR method and the FSR [PITH_FULL_IMAGE:figures/full_fig_p034_20.png]
Figure 21
Figure 21. Figure 21: Quantum circuits for calculating the value of the [PITH_FULL_IMAGE:figures/full_fig_p036_21.png]

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