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REVIEW 4 major objections 4 minor 35 references

Quantum medical image encoding and compression using Fourier-based methods

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims that loading a medical image through truncated 2D Fourier coefficients into a Fourier-series state prepares an approximate QPIE state with gate counts at least four times smaller than the pixel count.

desk verdict A plausible gate-reduction idea for QPIE, but the reported fidelities contradict the stated first-quadrant DFT truncation, so the central validation is unsupported as written. read the letter →

arxiv 2505.06471 v1 pith:FLXKUJBY submitted 2025-05-09 quant-ph cs.NAmath.NA

Taehee Ko , Inho Lee , Hyeong Won Yu This is my paper · ORCID
classification quant-phcs.NAmath.NA MSC 81P6868Q12 PACS 03.67.-a
keywords quantumimageprocessingFAQPIEdiscreteFouriertransformamplitudeencodingcircuitcompressionmedicalimagingseriesloadergatecomplexity
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

The paper tries to establish that medical images can be loaded into quantum circuits far more cheaply than pixel-by-pixel schemes by taking their two-dimensional discrete Fourier transform and keeping only the low-frequency coefficients. It proposes FAQPIE, which uses a Fourier-series-loading circuit to put those truncated coefficients directly into the amplitudes of a quantum state, in place of the pixel amplitudes of a quantum-probability image encoding (QPIE). If correct, a $2^n \times 2^n$ image needs $O(4^{m+2}+n^2)$ gates, with $m$ chosen from how compressible the image is, so a 1024x1024 surgical frame uses roughly 96% fewer gates than an improved FRQI encoding. On the tested thyroid-surgery images, two further compression strategies, thresholding small uniformly-controlled-rotation parameters and partitioning the image into blocks, lower maximal gate counts by about 80% relative to uncompressed FAQPIE while retaining surgical details.

What carries the argument

The load-bearing object is the FSL (Fourier series loader) circuit, a low-depth circuit that prepares a quantum state whose amplitudes are Fourier coefficients of a classical function. FAQPIE combines FSL with QPIE amplitude encoding: pixel values are replaced by truncated 2D DFT coefficients $C_{xy}$, so the image's compressibility in the Fourier basis, rather than its pixel count, sets the circuit cost. The adjustable truncation order $m \le n-2$ controls the trade-off between fidelity and gate count. The first compression strategy thins the inner cascade of uniformly-controlled rotations by dropping small rotation parameters and cancelling CNOTs through parity; the second splits the image into $2^{n_0} \times 2^{n_0}$ blocks so each block can use a smaller $m$ and fewer qubits.

What would settle it

Run FAQPIE on a 1024x1024 image whose Fourier energy is deliberately spread to high frequencies, such as a checkerboard or a surgical frame with fine instrument texture, and increase $m$ until the fidelity in Eq. (8) meets a chosen threshold. If acceptable fidelity requires $m$ near $n-2=8$, the gate count at that $m$ is close to $4^{10} \approx 1.05 \times 10^6$ gates per circuit, comparable to the pixel count; that would falsify the claimed factor-of-at-least-four reduction for that image class.

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Extended reading notes

Core claim

The central claim is that a quantum image encoding need not touch every pixel. The authors write the image as a 2D DFT, truncate it to a $2^m \times 2^m$ block of low-frequency Fourier coefficients, and load those coefficients with the FSL circuit; they call the resulting state FAQPIE. This state approximates the QPIE state while its circuit has $O(4^{m+2}+n^2)$ gates and $2n$ qubits for a $2^n \times 2^n$ image. In simulations of 1024x1024 frames from a remote-access thyroidectomy video, truncation orders $m=5$ for partitioned blocks and $m=6$ for whole images keep the averaged fidelity above 0.90 while maximal single-qubit and CNOT gate counts drop to roughly 5,741-7,484, reductions of about 82% and 77% against the uncompressed FAQPIE. The paper presents FAQPIE as a candidate input stage for large-scale medical imaging and quantum edge detection.

Load-bearing premise

The load-bearing premise is that medical images are strongly compressible in the two-dimensional Fourier basis, so small truncation orders such as $m=5$ or $m=6$ on a 1024x1024 image still preserve clinically relevant detail; if an image has significant high-frequency content, $m$ must grow and the $O(4^{m+2}+n^2)$ gate count approaches the pixel count.

Editorial extensions

If this is right

  • The exact QPIE of a 1024x1024 image built from uniformly-controlled-rotation gates would need $2^{21}-4$ gates; FAQPIE uses at most $2^{16}-8$ per circuit in the simulations, over 96% fewer, with comparable improvements against improved FRQI.
  • Encoding cost is governed by image compressibility rather than pixel count: images that concentrate their Fourier energy in low frequencies can take smaller $m$, making the circuit cost largely independent of the image resolution.
  • Partitioning a 1024x1024 image into four 512x512 blocks lowers the qubit count from 20 to 18, cuts the truncation order from $m=6$ to $m=5$, and reduces total pre-processing time by about a factor of four in the tests.
  • Combining image partition with UCR compression lowers maximal single-qubit gates by about 82% and maximal CNOT gates by about 77% compared with uncompressed FAQPIE while keeping surgical details visible.
  • FAQPIE is suggested as a missing encoding stage for quantum Hadamard edge detection, which up to now has lacked a general circuit for QPIE.

Reading between the lines

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

  • Beyond the paper's own claims, the same truncate-and-load pipeline could be tested on other structured medical data, such as MRI k-space or angiographic sequences, where Fourier energy concentration is known to be strong; the paper only demonstrates still frames from one surgical video.
  • The fidelity metric in Eq. (8) is evaluated against the original image being encoded, so it is an in-sample measure; a sharper test would feed the reconstructed images into a downstream task such as edge detection or a clinical measurement and check whether the 80% gate reduction survives.
  • One plausible extension is to replace the UCR-based inner unitary with Schmidt-decomposition or matrix-product-state constructions to see whether the $O(4^{m+2}+n^2)$ scaling can be further improved; the paper notes Schmidt decomposition exists but does not benchmark it.
  • The partition strategy implies a parallel-encoding architecture in which many small FAQPIE circuits are prepared separately and later combined through an index register, but the paper does not simulate the combined circuit, only the individual blocks.
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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

4 major / 4 minor

Summary. The paper proposes FAQPIE, a quantum image encoding method that prepares an approximate QPIE state by loading a truncated set of 2D discrete Fourier coefficients with the FSL circuit. For a 2^n x 2^n image the claimed gate complexity is O(4^(m+2)+n^2), where m is a truncation order, which the authors report as a large reduction relative to improved FRQI. Two additional compression strategies are described: thresholding and parity-cancelling the uniformly controlled rotations (CUCR), and partitioning the image into smaller blocks (IP). Numerical simulations on 1024x1024 medical images from BABA robotic thyroidectomy report high fidelities and gate reductions, summarized in Tables II and III.

Significance. If the central claims hold, the method would be practically relevant: it offers a concrete amplitude-encoding circuit whose gate count can scale below the pixel count for compressible images, and it addresses a real bottleneck in quantum image processing. The paper has strengths: it provides an explicit gate-complexity table, uses a standard simulator (Qiskit 1.2.4), and tests multiple surgical images. However, the numerical validation as written is inconsistent with the stated Fourier truncation, so the central trade-off between truncation order, fidelity, and gate count is not established. The compression claims also depend on unspecified thresholds and on a max-per-circuit rather than total-resource metric.

major comments (4)
  1. [Section III, Eqs. (5), (9), Table II] The reported fidelities are impossible for the truncation defined in Eq. (5). For a real image, C_{x,y} = conj(C_{N-x,N-y}), so the first-quadrant mask [0,2^m-1]^2 retains only one member of each non-DC conjugate pair; the squared fidelity in Eq. (9) is therefore at most (1+D)/2, where D is the DC energy fraction. The Fig. 4(b) image is zero-padded from a 636x842 crop to 1024x1024, giving a support fraction of about 0.51; for nonnegative pixel values D is at most this support fraction, so the squared fidelity cannot exceed about 0.755. Table II reports values between 0.9789 and 0.9931 for the uncompressed FAQPIE, which contradicts Eq. (5). The implementation must be using a different coefficient mask (e.g., a centered low-frequency mask after fftshift) or a different fidelity metric; either way, the numerical evidence does not validate the method as written.
  2. [Abstract and Table I] The abstract's claim that the encoding reduces gates below the pixel count 'by a factor at least 4' is not supported for the full parameter range. Table I gives gate complexity O(4^(m+2)+n^2) with m <= n-2; at m = n-2 the leading term is 4^n, equal to the number of pixels, plus n^2. The factor-4 reduction holds only for m <= n-3 and only if the image is compressible enough that this truncation preserves the required fidelity. The claim should be stated as conditional on m and on image compressibility, not as a general property of the method.
  3. [Section II.B.1 and Tables II-III] The UCR compression rule is not specified, which prevents reproduction of the reported gate reductions. The text says the authors 'followed a similar rule in [1] that simply discard a certain percentage of the smallest parameters,' but neither the percentage nor the threshold criterion is given. In addition, the truncation order m in Tables II and III is selected per image to keep fidelity high, with no held-out validation, so the reported quality numbers are in-sample estimates. The compression claims need a concrete rule and an out-of-sample or task-based validation.
  4. [Section II.B.2, Eq. (7), Table II] The image-partition compression is measured by the maximal per-piece gate count, not by the total number of gates needed to encode the full image. For the 1024x1024 image partitioned into four 512x512 blocks, the twelve IP circuits in Table II have total Ry+Rz and CNOT counts very close to the three whole-image circuits (about 98,256 versus 98,292 per gate type). Thus the phrase 'further reduce the number of gates' is only true in the max-per-circuit sense defined in Eq. (7), not for the total encoding resource. The authors should clarify whether the claimed compression is per-circuit resource reduction or total-resource reduction.
minor comments (4)
  1. [Abstract and Introduction] There are several spelling errors, including 'informaiton' in the abstract, 'dicrete' in Section II, 'capcitiy' in Section I, and 'paritition' in Section II.B.2.
  2. [Table II caption] The caption says 'images in Fig. 3', but the table reports simulations of the image in Fig. 4; the reference should be corrected.
  3. [Eq. (6)] The summation range for j appears to be off by one: if there are 4^(n-n0) pieces and j starts at 0, the upper limit should be 4^(n-n0)-1.
  4. [Section III] The abbreviation 'UCR' is used in Figure 2 and throughout without a formal expansion at first use; 'uniformly-controlled rotations' should be introduced with the acronym.

Circularity Check

1 steps flagged · score 2.0 of 10

The structural gate-complexity claim is self-contained and inherited from an external FSL circuit result; only the fidelity validation is mildly circular because the truncation order m is selected on the same images used to report fidelity.

  1. fitted input called prediction [Section III, paragraph following Eq. (9), and Table II caption.]
    "With this figure of merit, Table II shows high fidelity results for all simulated images, which means that the values of m in our simulations are set reasonably."

    The fidelity (8)-(9) is, by construction, the fraction of truncated Fourier energy retained after choosing m. The authors select a reasonable m per image and then report the fidelity on that same image as evidence that the choice was good. This is an in-sample fit: the metric is both the parameter-selection criterion and the validation metric, so the reported high fidelity is partly forced by the way m was chosen. The gate-count reduction itself is not circular, since O(4^{m+2}+n^2) follows from the published FSL circuit complexity and the chosen truncation order, not from the image data.

full rationale

No load-bearing self-citation, imported uniqueness, or ansatz-smuggling is present: the FSL circuit and UCR compression are cited from external works by Moosa et al. and Amankwah et al. The central structural claim, that FAQPIE loads truncated 2D DFT coefficients with O(4^{m+2}+n^2) gates rather than O(4^n) pixel-scale circuits, is a direct consequence of the stated circuit construction and the parameter bound m ≤ n−2, so it is self-contained rather than circular. The only circular element is the mild in-sample validation of image quality: the truncation order m (and effectively the UCR thresholding percentage) are chosen for the same images whose fidelities are then reported, so the 'negligible loss' figures are not independent predictions. Separately, there is a correctness concern that should be weighed in the verdict though it is not itself circularity: the fidelities in Table II (e.g., 0.9931 for the red channel) appear inconsistent with the first-quadrant truncation in Eq. (5). For a real zero-padded image with nonzero support fraction about 0.51, conjugate symmetry and Parseval bound the squared fidelity by at most (1+D)/2 ≤ 0.755, so either the implemented coefficient mask differs from Eq. (5) or the fidelity metric differs from Eq. (9). This undermines the numerical validation, but it does not make the derivation circular, so the circularity score remains low.

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

The central claim rests on the compressibility of medical images in the Fourier basis, the correctness of the FSL circuit from prior work, and the validity of the adopted UCR compression heuristic. The truncation order and compression threshold are manually tuned on the same images used for fidelity reporting.

free parameters (3)
  • Truncation order m = 6 (whole image), 5 (image partition)
    Sets the number of low-frequency Fourier coefficients retained (2^m x 2^m). Chosen per strategy in Table II to keep fidelity high; it directly determines gate count via O(4^(m+2)+n^2).
  • UCR rotation-gate threshold = not specified (described as 'discard a certain percentage' after [1])
    Determines how many small rotation angles are removed in CUCR; the exact percentage is not reported, making the compression result hard to reproduce.
  • Partition size n0 = 9 (512x512 blocks from 1024x1024)
    Image partition divides into 2^n0 x 2^n0 blocks; n0=9 chosen in experiments, affects qubit count (18 vs 20) and truncation order.
assumptions (3)
  • domain assumption The 2D DFT coefficient truncation in Eqs. (4)-(5) preserves the images used in the experiments
    The paper selects truncation order m per color and per compression mode in Table II to keep fidelity high, so the reported numbers apply only to images that are DFT-compressible.
  • domain assumption The FSL circuit of [15] correctly prepares the truncated Fourier coefficients with gate complexity O(4^(m+2)+n^2)
    The authors rely on the FSL construction without rederiving it; this is the backbone of the gate-count reduction.
  • domain assumption The UCR compression rule from [1] removes small rotation gates and cancels CNOTs without materially changing the encoded state
    Adopted from prior work; the exact threshold percentage is not given in this paper.

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Cite this review

Pith. "Pith review of Quantum medical image encoding and compression using Fourier-based methods." pith.science (2026). https://pith.science/paper/FLXKUJBY

@misc{pith2026250506471,
  author       = {Pith},
  title        = {Pith review of: Quantum medical image encoding and compression using Fourier-based methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FLXKUJBY}},
  note         = {Machine review of arXiv:2505.06471}
}
read the original abstract

Quantum image processing (QIMP) has recently emerged as a promising field for modern image processing applications. In QIMP algorithms, encoding classical image informaiton into quantum circuit is important as the first step. However, most of existing encoding methods use gates almost twice the number of pixels in an image, and simulating even a modest sized image is computationally demanding. In this work, we propose a quantum image encoding method that effectively reduces gates than the number of pixels by a factor at least 4. We demonstrate our method for various 1024 by 1024 high-quality medical images captured during the Bilateral Axillo-Breast Approach (BABA) robotic thyroidectomy surgery. Additionally, two compression techniques are proposed to further reduce the number of gates as well as pre-processing time with negligible loss of image quality. We suggest our image encoding strategy as a valuable option for large scale medical imaging.

Figures

Figures reproduced from arXiv: 2505.06471 by the authors.

Figure 1
Figure 1. FIG. 1: Workflow of the FAQPIE of RGB image and two options for reducing circuit resource [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: (a) Illustration of 2d FSL circuit with uniformly-controlled rotations (UCRs) (b): [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Illustration of the change of image quality of FAQPIE for a surgical image with increasing [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: We zero pad the original medical images (a) to one of size 1024 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Comparison of original medical images (left), the corresponding reconstructed images via [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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Reference graph

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