Quantum algorithm for edge detection in digital grayscale images

Alok Shukla; Mohit Rohida; Prakash Vedula

REVIEW 3 major objections 4 minor 24 references

Quantum algorithm for edge detection in digital grayscale images

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims a quantum edge-detection pipeline whose transform-and-filter stage costs O(log2(N1N2)) in gates and depth, improving on the QHED baseline.

desk verdict A correct and honest application of sequency-ordered WHT to quantum edge detection, but the headline O(log N) advantage rests on an unverified filter circuit borrowed from the authors' earlier work. read the letter →

arxiv 2507.06642 v1 pith:URRFDBDW submitted 2025-07-09 quant-ph

Mohit Rohida , Alok Shukla , Prakash Vedula This is my paper

classification quant-ph MSC 68Q1268U1081P68

keywords quantumimageprocessingedgedetectionsequency-orderedWalsh-Hadamardtransformhigh-passfilterQPIEcircuitdepthQHED

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 proposes a quantum edge-detection pipeline for digital grayscale images and claims it runs in $O(\log_2(N_1N_2))$ gate complexity and circuit depth, where $N_1\times N_2$ is the image size. The method replaces Fourier-style filtering with the sequency-ordered Walsh-Hadamard transform, the ordering of Walsh basis functions by number of zero-crossings, so that edges are isolated by passing high-sequency components and suppressing low-sequency ones. The authors argue this improves on the Quantum Hadamard Edge Detection algorithm, whose cost is $O(\mathrm{poly}(\log_2(N_1N_2)))$. Computational examples on synthetic and photographic images are reported, with Structural Similarity Index Measure scores comparable to or slightly better than QHED on the tested cases.

What carries the argument

The central object is the sequency-ordered Walsh-Hadamard transform, built as $H_S^N = U_z H^{\otimes n}$, where $H^{\otimes n}$ is the depth-one natural-ordered transform and $U_z$ maps each basis state $|m\rangle$ to $|g\rangle$ with $g_i = m_0\oplus m_1\oplus\cdots\oplus m_{n-i-1}$. The high-pass filter $U_{HP}$ is a multi-controlled-$X$ operation that flips an ancilla exactly when the sequency index is below a cutoff $c$, tagging low-sequency components for exclusion. The argument's load is carried by the claim that $U_z$ and $U_{HP}$ can each be implemented in $O(n)$ gates and depth, making the transform-and-filter stage logarithmic in the number of pixels.

What would settle it

Decompose the high-pass filter for a $64\times 64$ image at cutoff $N/4$ into elementary gates by expanding every multi-controlled-$X$; if the total grows faster than $O(\log N)$ as the image size doubles, the central complexity claim fails, and the same check applies to an explicit circuit for $U_z$.

Watch

Extended reading notes

Core claim

The central discovery, on the paper's terms, is that edge detection can be completed by the identity $H_S^N = U_z H^{\otimes n}$: a natural-ordered Walsh-Hadamard transform followed by a reordering unitary $U_z$ produces the sequency-ordered transform, and a quantum high-pass filter on the ancilla-tagged state isolates the high-sequency parts that encode edges. Because the natural transform is a single layer of Hadamard gates, and because $U_z$ together with the filter are claimed to cost $O(n)$, the whole transform-filter-inverse pipeline costs $O(\log_2(N_1N_2))$ for an $N_1\times N_2$ image. The paper reports that this beats the QHED baseline's $O(\mathrm{poly}(\log_2(N_1N_2)))$ and shows edge maps with SSIM values slightly above QHED on its test images.

Load-bearing premise

The logarithmic cost claim depends on the unproven assumption that both the sequency-reordering unitary and the high-pass filter can be wired using only $O(n)$ gates and depth; the paper cites another reference for these constructions without giving their circuit decompositions.

Editorial extensions

If this is right

For images of size $N_1\times N_2$ with $N_i=2^{n_i}$, the paper's Algorithm 1 achieves $O(\log_2(N_1N_2))$ gate complexity and circuit depth for the transform-and-filter part, versus $O(\mathrm{poly}(\log_2(N_1N_2)))$ for QHED.
The cutoff sequency $c$ tunes the result: $c=N/2$ extracts strong edges while $c=N/4$ reveals finer detail, as shown in the reported examples.
A complete edge map requires two executions of the algorithm, one on the image and one on its transpose, to capture vertical and horizontal edges.
The complexity comparison is explicitly limited to the transform, filtering, and inverse transform; QPIE state preparation and ancilla measurement are not counted in either algorithm, so the advantage applies when the input already comes from a quantum subroutine or only global features are needed.

Reading between the lines

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

If the cited constructions for $U_z$ and $U_{HP}$ are genuinely $O(n)$ in depth, the same sequency-domain filter-and-invert template should transfer to other Walsh-domain image tasks such as denoising or compression, which the paper does not explore.
The paper's own caveat implies that for standalone classical images the end-to-end cost is dominated by QPIE preparation and readout; the asymptotic improvement is therefore a property of the transform stage, not of a complete quantum image-processing pipeline.
A direct gate count on the displayed 64x64 circuits would settle the complexity question: the number of elementary gates in the filter should be checked to scale linearly in $n$ for fixed cutoff fraction, not in $N$.
The paper's mention of the swap test suggests a testable extension: compare the filtered state against an edge template without full tomography, which would make the logarithmic-depth advantage usable in practice.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the paper has a correct mathematical core, an honest complexity caveat, and a real but narrow application claim. The central O(log N) advantage is not yet established because the filter cost is borrowed from [19] without a circuit decomposition.

What's new: combining sequency-ordered WHT with a quantum high-pass filter for edge detection, benchmarked against QHED. The identity Uz H^⊗n = H_S_N is correctly derived, and the pipeline is internally consistent. The paper also compares against an independent baseline (QHED) and reports SSIM numbers, not just pictures.

Soft spots, in order of importance: 1. The gate/depth claim for UHP is unsupported. The paper says only that UHP is 'a set of multi-controlled X-gates' and defers to [19]. For cutoff c=N/2, the natural implementation could be as cheap as a single 0-controlled X on the most significant qubit, but the text does not say that. If the implementation is one multi-controlled NOT per low-sequency basis state, the cost is O(c), which for c=N/2 is exponential in n. That makes the claimed O(log N) improvement over QHED collapse. A referee needs a concrete circuit decomposition and gate count for Uz and UHP, for the cutoffs actually used. 2. The abstract's 'computational cost O(log2(N1N2))' omits that state preparation and readout are excluded. The authors do acknowledge this in Sec. 5.2, so it is an accuracy issue in the abstract, not a hidden flaw. 3. Minor: the SSIM evaluation compares edge maps against the original image, which is a weak protocol. The scaling factors (3 and 2, or 9 and 6 for the cat) are manually chosen, so the numeric comparison has limited force. But the visual results are consistent across several test patterns.

Overall: the mathematical identity and algorithm pipeline are sound, and the paper is honest about the end-to-end bottleneck. The missing piece is evidence for the circuit cost of the filter, which is the load-bearing part of the complexity claim. This is fixable with a few equations and circuit diagrams. I would send it to review, but instruct the referee to check the filter cost carefully.

Recommendation: engage with it, but do not accept the O(log N) claim on faith. Ask for the circuit decomposition.

Referee Report

3 major / 4 minor

Summary. This paper proposes a quantum edge-detection algorithm for grayscale images. After QPIE state preparation, the algorithm applies a natural-ordered Hadamard transform, a unitary Uz (adapted from ref [19]) that converts natural ordering to sequency ordering, a high-pass filter UHP (also from ref [19]) that tags low-sequency components on an ancilla, the inverse sequency-ordered transform, and a post-selective measurement on the ancilla to extract the edge map. The authors derive the identity HS_N = Uz H^⊗n in Sec. 3 and claim a gate count and circuit depth of O(log2(N1N2)) for an N1×N2 image, improving on QHED's O(poly(log2(N1N2))). The paper reports SSIM comparisons on synthetic and natural images and concludes that the method matches or slightly exceeds QHED at lower cost.

Significance. If the claimed resource counts for Uz and UHP can be substantiated, the algorithm would be a genuinely simpler asymptotic route to quantum edge detection than the amplitude-permutation approach of QHED, and the sequency-ordering identity in Eq. (3.6) is correctly derived and clearly presented. The paper honestly discloses in Sec. 5.2 that state preparation and readout are excluded from the complexity comparison, and the computational examples are described with explicit cutoff and scaling parameters. However, the significance is currently conditional on the missing circuit-level analysis of UHP and on a meaningful SSIM benchmark.

major comments (3)

[Sec. 5.2 and Sec. 4] The claimed O(log2(N1N2)) gate complexity and depth for Algorithm 1 depend on the assertions that the sequency-ordering unitary Uz (Sec. 3) and the high-pass filter UHP (Sec. 4) each require O(n) gates and depth. The paper does not provide a circuit decomposition or gate count for either operator: Uz is introduced abstractly via Eq. (3.4) and UHP is described only as 'a set of multi-controlled X-gates' with construction deferred to ref [19]. This is load-bearing because, for the demonstrated cutoff c = N/2, a naive implementation of UHP with one multi-controlled X per sequency m < c would require O(N) gates, which would eliminate the asymptotic advantage over QHED. Please supply explicit circuits (or a rigorous counting argument) for Uz and UHP and state their gate counts and depths as functions of n and c.
[Sec. 5.1 and Table 1] The SSIM scores are computed between the detected edge map and the original input image (the text says the detected edge maps obtained from both methods are compared against the original image using SSIM). SSIM compares luminance and structure; an edge map and a natural image are not directly commensurable, and a trivial output that approximates the original image would score highly. The reported 'comparable or slightly superior' performance over QHED is therefore not evidence of better edge detection. The evaluation should instead compare detected edge maps with ground-truth edge maps (for synthetic images, the true edge map; for natural images, a classical reference detector) and report SSIM or another edge-focused metric on that basis.
[Abstract and Sec. 5.2] The abstract and conclusion state the full cost of the edge detection algorithm as O(log2(N1N2)) without qualification, while Sec. 5.2 acknowledges only later that QPIE state preparation and the partial measurement of the ancilla are excluded. Since state preparation from classical image data generally costs O(N1N2), the unqualified end-to-end claim is misleading. Please qualify the abstract and conclusion, or provide a count of the state-preparation and readout costs for the cases where the input is produced by a preceding quantum subroutine.

minor comments (4)

[Sec. 3] The comparison with the QFT (O(n^2)) is not directly relevant because the QFT is never used in the pipeline; consider removing it or clarifying that it is motivational.
[Remark 5.0.1(iii) and Fig. 5 caption] The remark states default vertical/horizontal scaling factors of 3 and 2, but the Fig. 5 caption reports scaling factors of 9 and 6; these values should be reconciled.
[Fig. 4] The 64x64 circuits are reproduced too small to read; a legible version or a schematic with explicit gate counts would be helpful for verifying the claimed depth.
[Sec. 4, Eq. (4.3)] The definition of UHP is clear, but the text should specify whether the projectors sum_{m<c}|m><m| act on data qubits only and that the ancilla is flipped by X_a, to avoid ambiguity about the control qubits.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional circularity; the main caveat is that the advertised O(log N) complexity inherits an unverified gate-cost assumption for the high-pass filter from the authors' own ref [19].

full rationale

The paper's derivation chain is mostly self-contained: the sequency-ordered Walsh-Hadamard transform is defined explicitly (Eq. 3.1), the unitary Uz is given by Eq. 3.4 and verified by comparing Eq. 3.5 with Eq. 3.1, and the edge-detection pipeline is concrete and benchmarked against the independent QHED algorithm. There is no fitted parameter disguised as a prediction, no uniqueness theorem imported to forbid alternatives, and no renaming of a known empirical result as organization. The only circularity-adjacent feature is that the claimed gate and depth cost of the high-pass filter UHP is deferred to ref [19], which shares authors with this paper: Sec. 4 says only that UHP is 'implemented using a set of multi-controlled X-gates' and that 'a more detailed discussion ... can be found in [19]', while Sec. 5.2 assumes the sequency-ordered WHT is 'the most computationally significant step.' If [19] genuinely contains a full O(n)-gate construction, this is legitimate external support rather than circularity; if it does not, the complexity claim is unsupported, but that is a correctness/completeness risk rather than a circular derivation. Accordingly, the score reflects a minor self-citation dependency, not a reduction of a prediction to its inputs.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced; the algorithm uses standard qubits, one ancilla qubit, and operators from prior literature. The free parameters are the cutoff sequency and postprocessing scaling factors, all chosen by hand.

free parameters (3)

Cutoff sequency c = N/2 or N/4 in examples
User-selected threshold that determines how many low-sequency components are suppressed; the paper does not provide a principled selection rule.
Vertical pass scaling factor = 3 in Remark 5.0.1(iii), 9 for cat silhouette
Postprocessing multiplier applied to the vertical-pass edge map to enhance visual sharpness; likely influences SSIM scores and is chosen by the authors.
Horizontal pass scaling factor = 2 in Remark 5.0.1(iii), 6 for cat silhouette
Postprocessing multiplier applied to the horizontal-pass edge map; chosen by hand to improve visual output.

assumptions (4)

domain assumption The unitary Uz converts the natural-ordered WHT to sequency-ordered form and can be implemented in O(n) circuit depth (Eq. 3.4, adapted from ref [19]).
Load-bearing borrowed result from the authors' earlier work; not proven in this paper.
domain assumption The high-pass filter UHP can be realized with multi-controlled X gates at no more than O(n) cost (Sec. 4, ref [19]).
This is the key unverified assumption behind the headline complexity.
domain assumption High-sequency Walsh components correspond to image edges, so suppressing low-sequency components and inverse transforming yields a valid edge map.
Domain modeling assumption; no proof or benchmark against true edge maps is provided.
domain assumption For the complexity claim, the quantum image is already available as a quantum state, i.e., QPIE state preparation is excluded (Sec. 5.2).
The authors explicitly state this exclusion; without it the total complexity is O(N1 N2).

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Cite this review

Pith. "Pith review of Quantum algorithm for edge detection in digital grayscale images." pith.science (2026). https://pith.science/paper/URRFDBDW

@misc{pith2026250706642,
  author       = {Pith},
  title        = {Pith review of: Quantum algorithm for edge detection in digital grayscale images},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/URRFDBDW}},
  note         = {Machine review of arXiv:2507.06642}
}

abstract

In this work, we propose a novel quantum algorithm for edge detection in digital grayscale images, based on the sequency-ordered Walsh-Hadamard transform. The proposed method significantly improves upon existing quantum techniques for edge detection by using a quantum algorithm for the sequency-ordered Walsh-Hadamard transform, achieving a circuit depth of $\mathcal{O}(n)$ (where $n$ is the number of qubits). This represents a notable enhancement over the Quantum Fourier Transform (QFT), which has a circuit depth of $\mathcal{O}(n^{2})$. Furthermore, our approach for edge detection has a computational cost (both gate complexity and quantum circuit depth) of $\mathcal{O}(\log_{2}(N_{1}N_{2}))$ for an image of size $N_{1}\times N_{2}$, offering a considerable improvement over the Quantum Hadamard Edge Detection (QHED) algorithm, which incurs a cost of $\mathcal{O}(\text{poly}(\log_{2}(N_{1}N_{2})))$. By integrating a quantum high-pass filter with the sequency-ordered Walsh-Hadamard transform, the algorithm effectively extracts edge information from images. Computational examples are provided to demonstrate the efficacy of the proposed algorithm which provides a better performance in comparison to QHED.

Figures

Figures reproduced from arXiv: 2507.06642 by the authors.

**Figure 2.** Quantum circuit for sequency-ordered Walsh-Hadamard transform. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Block diagram for the proposed quantum edge detection approach. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Quantum circuits for edge detection using Algorithm [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Edge detection results on a 512 × 512 grayscale image of a cat silhouette using Algorithm 1. Both vertical and horizontal passes of the algorithm were applied, and the resulting edge maps were combined to form the final edge-detected image. The next set of computationa…

**Figure 6.** Figure 6: Edge detection comparison on 64 × 64 computer-generated grayscale images. The effectiveness of Algorithm 1 in detecting edges is further illustrated in [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: Edge detection results using Algorithm 1 on two test images: binary blobs (top row) and a checkerboard pattern (bottom row). For each image, results are shown for two cutoff sequency values, N 2 and N 4 . 5.2 Computational complexity For edge detection in a grayscale …

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [19]

A quantum approach for digital signal processing

Alok Shukla and Prakash Vedula. A quantum approach for digital signal processing. The European Physical Journal Plus, 138(12):1–24, 2023

work page 2023

[2] [1]

A fast quantum mechanical algorithm for database search

Lov K Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 212–219, 1996

1996

[3] [2]

An e fficient implementation of a quantum search algorithm for arbitrary N

Alok Shukla and Prakash Vedula. An e fficient implementation of a quantum search algorithm for arbitrary N. The European Physical Journal Plus, 140(6):1–10, 2025

work page 2025

[4] [3]

Algorithms for quantum computation: discrete logarithms and factoring

Peter W Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings 35th annual symposium on foundations of computer science, pages 124–134. Ieee, 1994

1994

[5] [4]

Polynomial-time algorithms for prime factorization and discrete logarithms on a quan- tum computer

Peter W Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quan- tum computer. SIAM Review, 41(2):303–332, 1999

work page 1999

[6] [5]

Quantum algorithm for linear systems of equa- tions

Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equa- tions. Physical Review Letters, 103(15):150502, 2009

work page 2009

[7] [6]

A hybrid classical-quantum algorithm for solution of nonlinear ordinary differential equations

Alok Shukla and Prakash Vedula. A hybrid classical-quantum algorithm for solution of nonlinear ordinary differential equations. Applied Mathematics and Computation, page 127708, 2022

work page 2022

[8] [7]

High-order quantum algorithm for solving linear di fferential equations

Dominic W Berry. High-order quantum algorithm for solving linear di fferential equations. Journal of Physics A: Mathematical and Theoretical, 47(10):105301, 2014

work page 2014