pith. sign in

arxiv: 2604.09869 · v1 · submitted 2026-04-10 · 🪐 quant-ph

Q-PIPE A Practical Quantum Phase Encoding Method

Brian Garc\'ia Sarmina , Emmanuel Mart\'inez-Guerrero , Janeth De Anda Gil , Sun Guo-Hua , Dong Shi-Hai This is my paper

Pith reviewed 2026-05-10 17:11 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum phase encodingquantum image processingGray codephase kickbackquantum edge detectionNISQfinite differencesquantum machine learning
0
0 comments X

The pith

Q-PIPE encodes image intensities into quantum phases with O(qN) gate count using Gray-code traversal and phase kickback.

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

The paper introduces Q-PIPE to transfer classical image data into quantum states more efficiently than existing amplitude or basis encodings. It maps continuous pixel intensities directly to phases via optimized Gray-code spatial traversal and the phase kickback mechanism, reaching an elementary gate count of O(qN) for an O(log N) reduction over standard basis encoding. This phase-domain approach supports native finite-difference calculations without deep arithmetic circuits. Mitigations for readout errors include restricting inputs to [-π, π] and a dimension-dependent probability threshold, with demonstrations on quantum edge detection showing exact matches for quantized data and low reconstruction error for continuous inputs across benchmarks.

Core claim

Q-PIPE exploits the quantum phase kickback mechanism and optimized spatial traversal via a Gray-code sequence to efficiently map continuous intensity values into the computational basis with an elementary gate count of O(qN), a O(log N) improvement over standard basis encoding. Operating directly in the phase domain enables native computation of finite differences without deep arithmetic circuits. Classical readout vulnerabilities, including phase aliasing and spectral leakage, are mitigated by mapping inputs to [-π, π] and introducing a probability threshold equation that scales inversely with the dimension of the spatial register. A proof-of-concept performing Quantum Edge Detection via d

What carries the argument

Phase kickback mechanism combined with Gray-code sequence for spatial traversal, which injects pixel intensities as phases for direct phase-domain operations.

If this is right

  • Finite-difference operations for image processing become native in the phase domain without auxiliary arithmetic circuits.
  • Data-loading gate count for quantum images scales as O(qN) instead of higher costs from basis encoding.
  • Phase aliasing and spectral leakage are controlled via bounded input range and inverse-dimension threshold.
  • Quantum edge detection yields exact pixel reconstructions on quantized inputs and low error on continuous data.
  • Overall input-output overhead decreases for quantum computer vision and machine-learning workflows.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The parallelizable structure may allow Q-PIPE to serve as a building block for other phase-based quantum vision algorithms beyond derivatives.
  • Direct phase arithmetic could simplify implementations of additional image operators that rely on local differences or gradients.
  • The encoding's efficiency profile suggests it could reduce preprocessing costs when embedding classical image datasets into larger quantum models.

Load-bearing premise

That mapping inputs to [-π, π] together with a probability threshold scaling inversely with spatial-register dimension will reliably mitigate phase aliasing and spectral leakage on real hardware without introducing unacceptable reconstruction error.

What would settle it

Executing the Q-PIPE edge-detection circuit on actual NISQ hardware with continuous-intensity benchmark images and checking whether mean absolute error stays low or prominent aliasing artifacts appear in the output.

read the original abstract

A major hurdle in Quantum Image Processing (QIMP) is efficiently transferring classical, high-dimensional image data into quantum states. Current methods face trade-offs: amplitude encoding (FRQI) is computationally expensive in gate complexity and limited arithmetic capabilities, while basis encoding (NEQR) incurs heavy initialization overhead scaling with image resolution and intensity bit-depth. Frequency-domain approaches further demand complex transformations for basic pixel-wise arithmetic and extensive post-processing to reconstruct pixel information. To address the lack of practical phase encodings, we introduce Q-PIPE (Quantum-Gray Phase Injection for Pixel Encoding). Exploiting the quantum phase kickback mechanism and optimized spatial traversal via a Gray-code sequence, Q-PIPE efficiently maps continuous intensity values into the computational basis with an elementary gate count of $O(qN)$ a $O(\text{log}N)$ improvement over standard basis encoding. Operating directly in the phase domain enables native computation of finite differences without deep arithmetic circuits. Classical readout vulnerabilities, including phase aliasing and spectral leakage, are mitigated by mapping inputs to $[-\pi, \pi]$ and introducing a probability threshold equation that scales inversely with the dimension of the spatial register. A proof-of-concept performing Quantum Edge Detection (QED) via directional derivatives demonstrates strong accuracy, yielding exact reconstructions for quantized inputs and low Mean Absolute Error (MAE) for continuous data across multiple benchmark datasets. Ultimately, Q-PIPE establishes a highly parallelizable, NISQ-compatible subroutine that advances quantum computer vision while reducing input/output (I/O) data-loading overhead in broader Quantum Machine Learning (QML) workflows.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces Q-PIPE, a phase-based encoding for quantum image processing that exploits phase kickback and Gray-code spatial traversal to map continuous pixel intensities into the computational basis. It claims an elementary gate complexity of O(qN), representing an O(log N) improvement over standard basis encoding (NEQR), while enabling native finite-difference operations for tasks such as quantum edge detection. Classical readout issues (phase aliasing, spectral leakage) are addressed by restricting inputs to [-π, π] and applying a probability threshold that scales inversely with the spatial-register dimension. A proof-of-concept implementation reports exact reconstruction for quantized inputs and low mean absolute error (MAE) on continuous benchmark data, positioning the method as NISQ-compatible and useful for reducing I/O overhead in quantum machine learning.

Significance. If the efficiency and error-control claims are substantiated, Q-PIPE would constitute a practical advance in quantum image processing by lowering the gate overhead of data loading relative to amplitude or basis encodings and by permitting arithmetic directly in the phase domain. The approach could reduce the barrier to implementing quantum computer-vision primitives on near-term hardware and might generalize to other QML pipelines that require high-dimensional classical inputs.

major comments (3)
  1. [Abstract] Abstract and the description of the encoding procedure: the stated O(qN) gate count and O(log N) improvement over basis encoding are presented without an explicit circuit decomposition, gate-by-gate accounting, or comparison table against NEQR/FRQI. No derivation is supplied showing how the Gray-code traversal plus phase kickback yields this scaling while preserving the ability to perform finite differences.
  2. [Abstract] Abstract, mitigation paragraph: the probability threshold that scales inversely with the dimension of the spatial register is introduced to control aliasing and leakage, yet no bound on the resulting total-variation distance, reconstruction MAE, or post-selection overhead is derived. For register sizes where the threshold drops below typical readout or decoherence error rates, the scheme risks either discarding valid probability mass or incurring exponential overhead that would negate the claimed gate savings.
  3. [Proof-of-concept] Proof-of-concept section on quantum edge detection: the reported low MAE is stated only for the chosen quantized and continuous benchmarks; no experiments or analysis are provided for the regime in which the threshold becomes smaller than hardware noise levels, nor is an error budget relating phase noise, readout fidelity, and reconstruction error supplied.
minor comments (2)
  1. [Abstract] The abstract contains a grammatical error: 'a O(log N) improvement' should read 'an O(log N) improvement'.
  2. [Abstract] Notation for the probability threshold is described only verbally; an explicit equation would improve reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough and constructive review. The comments highlight areas where additional detail and analysis would strengthen the manuscript. We address each major comment point-by-point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the description of the encoding procedure: the stated O(qN) gate count and O(log N) improvement over basis encoding are presented without an explicit circuit decomposition, gate-by-gate accounting, or comparison table against NEQR/FRQI. No derivation is supplied showing how the Gray-code traversal plus phase kickback yields this scaling while preserving the ability to perform finite differences.

    Authors: We agree the abstract is high-level. Section III of the manuscript derives the encoding: Gray-code ordering ensures each successive pixel address differs by a single bit flip, so the phase-kickback circuit applies exactly q controlled-phase gates per pixel (one per intensity bit) for a total of O(qN) elementary gates. This avoids the extra O(log N) factor per pixel present in NEQR's address-controlled rotations. Finite differences remain native because they act directly on the phase register via additional controlled-phase operations without decoding. We will add an explicit gate-count table comparing Q-PIPE to NEQR and FRQI plus a concise derivation paragraph in the revised abstract and main text. revision: yes

  2. Referee: [Abstract] Abstract, mitigation paragraph: the probability threshold that scales inversely with the dimension of the spatial register is introduced to control aliasing and leakage, yet no bound on the resulting total-variation distance, reconstruction MAE, or post-selection overhead is derived. For register sizes where the threshold drops below typical readout or decoherence error rates, the scheme risks either discarding valid probability mass or incurring exponential overhead that would negate the claimed gate savings.

    Authors: The threshold is set to 1/2^s (s = spatial qubits) so that the probability of aliasing-induced invalid outcomes is suppressed below the per-pixel measurement precision. Under this choice the total-variation distance to the ideal distribution is at most O(threshold) and the expected post-selection overhead is polynomial in N. We did not include a formal derivation of these bounds in the original submission. We will add the bound derivation, an explicit expression for reconstruction MAE, and a discussion of overhead versus hardware noise floors in a new subsection of the revised manuscript. revision: yes

  3. Referee: [Proof-of-concept] Proof-of-concept section on quantum edge detection: the reported low MAE is stated only for the chosen quantized and continuous benchmarks; no experiments or analysis are provided for the regime in which the threshold becomes smaller than hardware noise levels, nor is an error budget relating phase noise, readout fidelity, and reconstruction error supplied.

    Authors: The reported MAE values validate the encoding on ideal and low-noise benchmarks. We agree that an explicit error budget and analysis for the regime where the threshold falls below typical hardware noise are missing. We will add a dedicated error-budget paragraph in the discussion section that relates phase noise, readout fidelity, and reconstruction MAE, and we will state the operating regime in which the threshold remains above expected noise. Full noisy-hardware simulations lie beyond the scope of the current proof-of-concept but are identified as future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in the derivation chain

full rationale

The paper introduces Q-PIPE as a phase-encoding construction that exploits phase kickback and Gray-code ordering to achieve an O(qN) gate count. This complexity and the claimed O(log N) improvement over basis encoding are presented as direct consequences of the chosen traversal and kickback mechanism rather than quantities defined in terms of the method's own outputs or fitted parameters. The aliasing mitigation (input restriction to [-π, π] plus an inverse-dimension probability threshold) is introduced as part of the encoding protocol and is validated empirically on benchmarks; no equation reduces the threshold or the reported MAE to a self-referential fit, and no load-bearing self-citation or uniqueness theorem is invoked in the provided text. The central claims therefore remain independent of the validation data and do not collapse to their inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so the ledger is minimal. The method rests on the standard quantum phase-kickback mechanism and the Gray-code ordering (both drawn from prior literature) with no new invented entities or fitted constants described.

pith-pipeline@v0.9.0 · 5599 in / 1170 out tokens · 48862 ms · 2026-05-10T17:11:53.248708+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

79 extracted references · 79 canonical work pages

  1. [1]

    Quantum Information Processing 15(1), 1–35 (2016)

    Yan, F., Iliyasu, A.M., Venegas-Andraca, S.E.: A survey of quantum image representations. Quantum Information Processing 15(1), 1–35 (2016)

    work page 2016

  2. [2]

    IEEE access 12, 46118–46137 (2024)

    Khan, M.A., Aman, M.N., Sikdar, B.: Beyond bits: A review of quantum embed- ding techniques for efficient information processing. IEEE access 12, 46118–46137 (2024)

    work page 2024

  3. [3]

    Computer Science Review 57, 100763 (2025) https://doi.org/10.1016/j.cosrev.2025.100763

    Farooq, U., Singh, P., Kumar, A.: A systematic review of quantum image pro- cessing: Representation, applications and future perspectives. Computer Science Review 57, 100763 (2025) https://doi.org/10.1016/j.cosrev.2025.100763

    work page doi:10.1016/j.cosrev.2025.100763 2025

  4. [4]

    Materials Today Quantum 6, 100044 (2025) https://doi.org/10

    Ryan, D.P., Werner, J.H.: A review of quantum imaging methods and enabling technologies. Materials Today Quantum 6, 100044 (2025) https://doi.org/10. 1016/j.mtquan.2025.100044

    work page arXiv 2025

  5. [5]

    Quan- tum Machine Intelligence 5(1), 2 (2023)

    Lisnichenko, M., Protasov, S.: Quantum image representation: A review. Quan- tum Machine Intelligence 5(1), 2 (2023)

    work page 2023

  6. [6]

    Physical Review A 102(3), 032420 (2020)

    LaRose, R., Coyle, B.: Robust data encodings for quantum classifiers. Physical Review A 102(3), 032420 (2020)

    work page 2020

  7. [7]

    IET Quantum Communication 2(4), 141–152 (2021)

    Weigold, M., Barzen, J., Leymann, F., Salm, M.: Encoding patterns for quantum 27 algorithms. IET Quantum Communication 2(4), 141–152 (2021)

    work page 2021

  8. [8]

    Applied Soft Computing 187, 114273 (2026) https: //doi.org/10.1016/j.asoc.2025.114273

    Zhang, J., Che, X., Peng, S., Chen, G., Ma, Q., Fan, B., Hu, J.: Design and implementation of run-length encoding on quantum computers for resource- efficient data representation. Applied Soft Computing 187, 114273 (2026) https: //doi.org/10.1016/j.asoc.2025.114273

    work page doi:10.1016/j.asoc.2025.114273 2026

  9. [9]

    EPJ Quantum Technology 11(1), 72 (2024)

    Rath, M., Date, H.: Quantum data encoding: A comparative analysis of classical- to-quantum mapping techniques and their impact on machine learning accuracy. EPJ Quantum Technology 11(1), 72 (2024)

    work page 2024

  10. [10]

    Mathematics 12(21), 3318 (2024)

    Ranga, D., Rana, A., Prajapat, S., Kumar, P., Kumar, K., Vasilakos, A.V.: Quan- tum machine learning: Exploring the role of data encoding techniques, challenges, and future directions. Mathematics 12(21), 3318 (2024)

    work page 2024

  11. [11]

    arXiv preprint arXiv:2410.09121 (2024)

    Munikote, N.: Comparing quantum encoding techniques. arXiv preprint arXiv:2410.09121 (2024)

    work page arXiv 2024

  12. [12]

    Quantum Information Processing 10(1), 63–84 (2011)

    Le, P.Q., Dong, F., Hirota, K.: A flexible representation of quantum images for polynomial preparation, image compression, and processing operations. Quantum Information Processing 10(1), 63–84 (2011)

    work page 2011

  13. [13]

    arXiv preprint arXiv:2505.06471 (2025)

    Ko, T., Lee, I., Yu, H.W.: Quantum medical image encoding and compression using fourier-based methods. arXiv preprint arXiv:2505.06471 (2025)

    work page arXiv 2025

  14. [14]

    In: Proceedings of the Great Lakes Symposium on VLSI 2025, pp

    Masum, A.K.M., Shoushtari Moghadam, M., Kouhalvandi, L., Najafi, M.H., Aygun, S.: Quantum image processing: A comparative study of neqr and frqi encoding schemes with hybrid processing. In: Proceedings of the Great Lakes Symposium on VLSI 2025, pp. 575–580 (2025)

    work page 2025

  15. [15]

    Quantum Information Processing 18(7), 201 (2019)

    Khan, R.A.: An improved flexible representation of quantum images: Ra khan. Quantum Information Processing 18(7), 201 (2019)

    work page 2019

  16. [16]

    Le, P.Q., Dong, F., Arai, Y., Hirota, K.: Flexible representation of quantum images and its computational complexity analysis

    work page

  17. [17]

    Multimedia Tools and Applications 78(17), 24067–24082 (2019)

    Lu, Z., Wang, X., Shang, J., Luo, Z., Sun, C., Wu, G.: A multimedia image edge extraction algorithm based on flexible representation of quantum. Multimedia Tools and Applications 78(17), 24067–24082 (2019)

    work page 2019

  18. [18]

    Quantum Information Processing 12(8), 2833–2860 (2013)

    Zhang, Y., Lu, K., Gao, Y., Wang, M.: Neqr: a novel enhanced quantum repre- sentation of digital images. Quantum Information Processing 12(8), 2833–2860 (2013)

    work page 2013

  19. [19]

    Results in Physics 39, 105710 (2022) 28

    Du, S., Luo, K., Zhi, Y., Situ, H., Zhang, J.: Binarization of grayscale quantum image denoted with novel enhanced quantum representations. Results in Physics 39, 105710 (2022) 28

    work page 2022

  20. [20]

    Quantum Information Processing 16(2), 42 (2017)

    Sang, J., Wang, S., Li, Q.: A novel quantum representation of color digital images. Quantum Information Processing 16(2), 42 (2017)

    work page 2017

  21. [21]

    Scientific Reports 11(1), 13879 (2021)

    Su, J., Guo, X., Liu, C., Lu, S., Li, L.: An improved novel quantum image representation and its experimental test on ibm quantum experience. Scientific Reports 11(1), 13879 (2021)

    work page 2021

  22. [22]

    IEEE Sensors Letters 7(2), 1–4 (2023)

    Majji, S.R., Chalumuri, A., Manoj, B.: Quantum approach to image data encoding and compression. IEEE Sensors Letters 7(2), 1–4 (2023)

    work page 2023

  23. [23]

    SoftwareX 31, 102327 (2025)

    Sokol, M., Volf, P., Hejda, J., Kutílek, P.: Qts2d: Quantum-based image encoding of time series. SoftwareX 31, 102327 (2025)

    work page 2025

  24. [24]

    IEEE Transactions on Neural Networks and Learning Systems 35(2), 1472–1486 (2022)

    Easom-McCaldin, P., Bouridane, A., Belatreche, A., Jiang, R., Al-Maadeed, S.: Efficient quantum image classification using single qubit encoding. IEEE Transactions on Neural Networks and Learning Systems 35(2), 1472–1486 (2022)

    work page 2022

  25. [25]

    Internet of Things 31, 101559 (2025) https://doi.org/10.1016/j.iot

    Huang, H., Liu, Z., Wang, Z., Yan, F.: A secure image encryption mechanism using biased fourier quantum walk and addition-crossover structure in the inter- net of things. Internet of Things 31, 101559 (2025) https://doi.org/10.1016/j.iot. 2025.101559

    work page doi:10.1016/j.iot 2025

  26. [26]

    Quantum Information Processing 19(5), 148 (2020)

    Grigoryan, A.M., Agaian, S.S.: New look on quantum representation of images: Fourier transform representation: Am grigoryan, ss agaian. Quantum Information Processing 19(5), 148 (2020)

    work page 2020

  27. [27]

    In: 2023 IEEE International Conference on Quantum Computing and Engineering (QCE), vol

    Lejarza, J.J.M., Grossi, M., Cieri, L., Rodrigo, G.: Quantum fourier iterative amplitude estimation. In: 2023 IEEE International Conference on Quantum Computing and Engineering (QCE), vol. 1, pp. 571–579 (2023). IEEE

    work page 2023

  28. [28]

    Physical Review A 107(1), 012422 (2023)

    Shin, S., Teo, Y.-S., Jeong, H.: Exponential data encoding for quantum supervised learning. Physical Review A 107(1), 012422 (2023)

    work page 2023

  29. [29]

    Scientific Reports 15(1), 40286 (2025)

    Alwan, N.A., Obaiys, S.J., Al-Saidi, N.M., Noor, N.F.B.M., Karaca, Y.: Multi- layered quantum computing and simulation system for enhanced image repre- sentation of hsi based fourier transform and adjacency matrix. Scientific Reports 15(1), 40286 (2025)

    work page 2025

  30. [30]

    Scientific Reports 10(1), 1–15 (2020)

    Wereszczyński, K., Michalczuk, A., Paszkiewicz, A.: Cosine series quantum sam- pling method with applications in signal and image processing. Scientific Reports 10(1), 1–15 (2020)

    work page 2020

  31. [31]

    Grigoryan, A.M., Agaian, S.S.: Image representation on the unit circle and mqftr (2025)

    work page 2025

  32. [32]

    Procedia Computer Science 225, 4354–4363 (2023) 29 https://doi.org/10.1016/j.procs.2023.10.432

    Werner, K., Kordasz, M., Michalczuk, A., Cyran, K.: Data loss in quantum image representation methods. Procedia Computer Science 225, 4354–4363 (2023) 29 https://doi.org/10.1016/j.procs.2023.10.432

    work page doi:10.1016/j.procs.2023.10.432 2023

  33. [33]

    Scientific Reports 13, 4129 (2023) https://doi.org/10.1038/s41598-023-30575-2

    Haque, M.E., Paul, M., Ulhaq, A., Debnath, T.: Advanced quantum image rep- resentation and compression using a dct-efrqi approach. Scientific Reports 13, 4129 (2023) https://doi.org/10.1038/s41598-023-30575-2

    work page doi:10.1038/s41598-023-30575-2 2023

  34. [34]

    & Cai, X

    Liu, H., Zhang, J., Wang, C., Hu, X., Lyu, L., Sun, J., Yang, X., Wang, B., Li, F., Qian, Y., et al.: Scaling embeddings outperforms scaling experts in language models. arXiv preprint arXiv:2601.21204 (2026)

    work page arXiv 2026

  35. [35]

    Cambridge University Press, ??? (2010)

    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, ??? (2010)

    work page 2010

  36. [36]

    Physical Review Letters 130(12), 123603 (2023)

    Nielsen, J.A., Neergaard-Nielsen, J.S., Gehring, T., Andersen, U.L.: Deterministic quantum phase estimation beyond n00n states. Physical Review Letters 130(12), 123603 (2023)

    work page 2023

  37. [37]

    In: Ying, M

    Ying, M.: Chapter 4 - quantum algorithms and communication protocols. In: Ying, M. (ed.) Foundations of Quantum Programming (Second Edition), Second edition edn., pp. 43–59. Morgan Kaufmann, ??? (2024). https://doi.org/10.1016/ B978-0-44-315942-8.00013-7

    work page 2024

  38. [38]

    Cambridge University Press, ??? (2018)

    Griffiths, D.J., Schroeter, D.F.: Introduction to Quantum Mechanics. Cambridge University Press, ??? (2018)

    work page 2018

  39. [39]

    Clinical Biomechanics 29(5), 484–493 (2014)

    Lamb, P.F., Stöckl, M.: On the use of continuous relative phase: Review of current approaches and outline for a new standard. Clinical Biomechanics 29(5), 484–493 (2014)

    work page 2014

  40. [40]

    Foundations of Physics 54(2), 19 (2024)

    Gao, S.: Why the global phase is not real. Foundations of Physics 54(2), 19 (2024)

    work page 2024

  41. [41]

    International Journal of Modern Physics B 22(06), 561–581 (2008)

    Zhu, S.-L.: Geometric phases and quantum phase transitions. International Journal of Modern Physics B 22(06), 561–581 (2008)

    work page 2008

  42. [42]

    Physical Review A 90(6), 062117 (2014)

    Shepard, S.R.: Quantum theory of angle and relative-phase measurement. Physical Review A 90(6), 062117 (2014)

    work page 2014

  43. [43]

    International Journal of Quantum Chemistry 115(19), 1311–1326 (2015)

    Sjöqvist, E.: Geometric phases in quantum information. International Journal of Quantum Chemistry 115(19), 1311–1326 (2015)

    work page 2015

  44. [44]

    Physical review letters 102(4), 040403 (2009)

    Dorner, U., Demkowicz-Dobrzanski, R., Smith, B.J., Lundeen, J.S., Wasilewski, W., Banaszek, K., Walmsley, I.A.: Optimal quantum phase estimation. Physical review letters 102(4), 040403 (2009)

    work page 2009

  45. [45]

    New Journal of Physics 21(2), 023022 (2019) 30

    O’Brien, T.E., Tarasinski, B., Terhal, B.M.: Quantum phase estimation of multi- ple eigenvalues for small-scale (noisy) experiments. New Journal of Physics 21(2), 023022 (2019) 30

    work page 2019

  46. [46]

    Physical review letters 86(9), 1889 (2001)

    Weinstein, Y.S., Pravia, M., Fortunato, E., Lloyd, S., Cory, D.G.: Implementation of the quantum fourier transform. Physical review letters 86(9), 1889 (2001)

    work page 2001

  47. [47]

    IEEE Access 12, 99029–99044 (2024)

    Akinola, T.A., Li, X., Wilkins, R., Obiomon, P.H., Qian, L.: Robust inverse quan- tum fourier transform inspired algorithm for unsupervised image segmentation. IEEE Access 12, 99029–99044 (2024)

    work page 2024

  48. [48]

    Numerical Linear Algebra with Applications 28(1), 2331 (2021)

    Camps, D., Van Beeumen, R., Yang, C.: Quantum fourier transform revisited. Numerical Linear Algebra with Applications 28(1), 2331 (2021)

    work page 2021

  49. [49]

    Geophysics 58(9), 1324–1334 (1993)

    Spagnolini, U.: 2-d phase unwrapping and phase aliasing. Geophysics 58(9), 1324–1334 (1993)

    work page 1993

  50. [50]

    Applied and Computational Harmonic Analysis 35(2), 341–349 (2013)

    Wang, Y., Yılmaz, Ö., Zhou, Z.: Phase aliasing correction for robust blind source separation using duet. Applied and Computational Harmonic Analysis 35(2), 341–349 (2013)

    work page 2013

  51. [51]

    Optics Express 30(3), 3835–3853 (2022)

    Zhu, H., Guo, H.: Anti-aliasing phase reconstruction via a non-uniform phase- shifting technique. Optics Express 30(3), 3835–3853 (2022)

    work page 2022

  52. [52]

    Chinese Physics B 33(5), 050305 (2024)

    Wu, K., Zhou, R., Luo, J.: An anti-aliasing filtering of quantum images in spatial domain using a pyramid structure. Chinese Physics B 33(5), 050305 (2024)

    work page 2024

  53. [53]

    Advanced Quantum Technologies 9(3), 00683 (2026)

    Shukla, A., Vedula, P.: Toward practical quantum phase estimation: A modular, scalable, and adaptive approach. Advanced Quantum Technologies 9(3), 00683 (2026)

    work page 2026

  54. [54]

    In: Proceedings of the Twenty-fifth Annual ACM Symposium on Theory of Computing, pp

    Bernstein, E., Vazirani, U.: Quantum complexity theory. In: Proceedings of the Twenty-fifth Annual ACM Symposium on Theory of Computing, pp. 11–20 (1993)

    work page 1993

  55. [55]

    Kitaev, A.Y., Shen, A., Vyalyi, M.N.: Classical and Quantum Computation vol

    work page

  56. [56]

    American Mathematical Soc., ??? (2002)

    work page 2002

  57. [57]

    In: Proceedings of Symposia in Applied Mathematics, vol

    Vazirani, U.: A survey of quantum complexity theory. In: Proceedings of Symposia in Applied Mathematics, vol. 58, pp. 193–220 (2002)

    work page 2002

  58. [58]

    Carbondale, IL 1 (2014)

    Mohr, A.: Quantum computing in complexity theory and theory of computation. Carbondale, IL 1 (2014)

    work page 2014

  59. [59]

    Quantum Information Processing 16(3), 82 (2017)

    Zhou, S., Loke, T., Izaac, J.A., Wang, J.: Quantum fourier transform in computational basis. Quantum Information Processing 16(3), 82 (2017)

    work page 2017

  60. [60]

    Physical review A 52(5), 3457 (1995)

    Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P., Sleator, T., Smolin, J.A., Weinfurter, H.: Elementary gates for quantum computation. Physical review A 52(5), 3457 (1995)

    work page 1995

  61. [61]

    IEEE Transactions on Quantum Engineering 1, 1–14 (2020)

    Niu, S., Suau, A., Staffelbach, G., Todri-Sanial, A.: A hardware-aware heuristic 31 for the qubit mapping problem in the nisq era. IEEE Transactions on Quantum Engineering 1, 1–14 (2020)

    work page 2020

  62. [62]

    Schillo and A

    Schillo, N., Sturm, A.: Quantum circuit learning on nisq hardware. arXiv preprint arXiv:2405.02069 (2024)

    work page arXiv 2024

  63. [63]

    IEEE Circuits and Systems Magazine 24(1), 14–32 (2024)

    Chen, F., Jiang, L., Müller, H., Richerme, P., Chu, C., Fu, Z., Yang, M.: Nisq quantum computing: A security-centric tutorial and survey [feature]. IEEE Circuits and Systems Magazine 24(1), 14–32 (2024)

    work page 2024

  64. [64]

    In: 2022 Design, Automation & Test in Europe Conference & Exhibition (DATE), pp

    Bandic, M., Feld, S., Almudever, C.G.: Full-stack quantum computing systems in the nisq era: algorithm-driven and hardware-aware compilation techniques. In: 2022 Design, Automation & Test in Europe Conference & Exhibition (DATE), pp. 1–6 (2022). IEEE

    work page 2022

  65. [65]

    Quantum Information Processing 24(4), 118 (2025)

    Joshi, M., Mishra, M.K., Karthikeyan, S.: Impact of hardware connectivity on grover’s algorithm in nisq era. Quantum Information Processing 24(4), 118 (2025)

    work page 2025

  66. [66]

    : Quantum image processing and its application to edge detection: theory and experiment

    Yao, X.-W., Wang, H., Liao, Z., Chen, M.-C., Pan, J., Li, J., Zhang, K., Lin, X., Wang, Z., Luo, Z., et al. : Quantum image processing and its application to edge detection: theory and experiment. Physical Review X 7(3), 031041 (2017)

    work page 2017

  67. [67]

    Quantum 9, 1687 (2025)

    Llorens, S., González, W., Sentís, G., Calsamiglia, J., Munoz-Tapia, R., Bagan, E.: Quantum edge detection. Quantum 9, 1687 (2025)

    work page 2025

  68. [68]

    Science advances 6(51), 4385 (2020)

    Zhou, J., Liu, S., Qian, H., Li, Y., Luo, H., Wen, S., Zhou, Z., Guo, G., Shi, B., Liu, Z.: Metasurface enabled quantum edge detection. Science advances 6(51), 4385 (2020)

    work page 2020

  69. [69]

    Physical Review A 103(4), 042405 (2021)

    Di Matteo, O., McCoy, A., Gysbers, P., Miyagi, T., Woloshyn, R., Navrátil, P.: Improving hamiltonian encodings with the gray code. Physical Review A 103(4), 042405 (2021)

    work page 2021

  70. [70]

    PRX Quantum 3(2), 020356 (2022)

    Chang, C.C., McElvain, K.S., Rrapaj, E., Wu, Y.: Improving schrödinger equation implementations with gray code for adiabatic quantum computers. PRX Quantum 3(2), 020356 (2022)

    work page 2022

  71. [71]

    Complex & Intelligent Systems 9(1), 609–624 (2023)

    Abd-El-Atty, B., ElAffendi, M., El-Latif, A.A.A.: A novel image cryptosystem using gray code, quantum walks, and henon map for cloud applications. Complex & Intelligent Systems 9(1), 609–624 (2023)

    work page 2023

  72. [72]

    Wang, Z., Bovik, A.C.: Modern image quality assessment (2006)

    work page 2006

  73. [73]

    Measurement 25(2), 135–142 (1999)

    Breitenbach, A.: Against spectral leakage. Measurement 25(2), 135–142 (1999)

    work page 1999

  74. [74]

    Journal of object technology 8(7) (2009) 32

    Lyon, D.A.: The discrete fourier transform, part 4: spectral leakage. Journal of object technology 8(7) (2009) 32

    work page 2009

  75. [75]

    Physical review letters 122(4), 040504 (2019)

    Schuld, M., Killoran, N.: Quantum machine learning in feature hilbert spaces. Physical review letters 122(4), 040504 (2019)

    work page 2019

  76. [76]

    Physical Review A 109(3), 032606 (2024)

    Li, J.: Iterative method to improve the precision of the quantum-phase-estimation algorithm. Physical Review A 109(3), 032606 (2024)

    work page 2024

  77. [77]

    Eurasian Mathematical Journal 9(4) (2018)

    Babenko, K.: On the mean convergence of multiple fourier series and the asymp- totics of the dirichlet kernel of spherical means. Eurasian Mathematical Journal 9(4) (2018)

    work page 2018

  78. [78]

    Journal of Multivariate Analysis 187, 104832 (2022)

    Ouimet, F., Tolosana-Delgado, R.: Asymptotic properties of dirichlet kernel density estimators. Journal of Multivariate Analysis 187, 104832 (2022)

    work page 2022

  79. [79]

    Physical Review D 108(12), 125008 (2023) 33

    Thiemann, T.: Renormalization, wavelets, and the dirichlet-shannon kernels. Physical Review D 108(12), 125008 (2023) 33

    work page 2023