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Estimating detector error models from syndrome data

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper establishes that every detector error model event probability is identifiable in closed form from syndrome data, via a scaled Walsh-Hadamard transform of empirical polarizations.

desk verdict Clean Fourier inversion for DEM estimation, with honest prior-work context, plus an error-bar typo and an unaddressed model-violation caveat. read the letter →

arxiv 2504.14643 v1 pith:BL2Y3VA2 submitted 2025-04-20 quant-ph

Robin Blume-Kohout , Kevin Young This is my paper · ORCID
classification quant-ph
keywords detectorerrormodelsyndromedataWalsh-Hadamardtransformpolarizationattenuationquantumcorrectionnoiseestimationdecodergraph
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 show that a detector error model (DEM) describing syndrome extraction noise can be learned directly and exactly from syndrome data, without running a decoder and without iterative fitting. It introduces the attenuation of a DEM event as the key variable and proves that the vector of all attenuations is obtained by a scaled Walsh-Hadamard transform of the vector of empirical depolarizations. If this is correct, every DEM event probability, and every aggregated class of event probabilities, has an explicit closed-form estimator. That matters because accurate DEMs improve decoding and logical error rates, and current methods are limited by degeneracy or require numerical Bayesian inference.

What carries the argument

The central object is the linear relation between depolarizations and attenuations, where the matrix entry records whether a DEM event flips a given parity. Because all DEM transition matrices commute and become diagonal in the Walsh-Hadamard (polarization) basis, this matrix can be inverted by its Moore-Penrose pseudoinverse, which is proportional to the Walsh-Hadamard matrix with its first row and column removed. The same diagonalization shows that attenuations, unlike probabilities, add linearly under aggregation, which is what makes reduced DEMs and the lattice-pruning algorithm work.

What would settle it

Simulate many syndrome histories from a known DEM, estimate all polarizations, and apply the paper's inversion formulas; if the reconstructed event probabilities do not match the input within finite-sample error, the identity itself is wrong. Then repeat with time-correlated or crosstalk errors: mismatches there would show the DEM assumptions, not the algebra, are the limiting factor.

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Extended reading notes

Core claim

The paper's first main result is Eq. 31, which states that the vector of all event attenuations is obtained by a scaled Walsh-Hadamard transform of the vector of depolarizations. Unpacked, this gives explicit formulas for the attenuation, decay factor, and probability of any DEM event in terms of observable polarizations of syndrome data. The paper also shows how to estimate aggregated classes of events using only low-weight depolarizations, and how to recover sparse DEMs by pruning a lattice of event classes. A central conceptual payoff is an answer to what the widely used p_ij coefficients quantify: they are aggregated probabilities of all DEM events that flip both detectors i and j.

Load-bearing premise

The inversion is only as good as the DEM model's assumption that errors are independent, each error always flips the same fixed set of detector bits, and every event's probability is constant across cycles.

Editorial extensions

If this is right

  • Each DEM event probability can be estimated from the full set of empirical polarizations with no decoder and no fitting, in closed form.
  • Aggregated attenuations of event classes can be computed from low-weight depolarizations, so useful DEM estimates remain possible when N is large enough that high-weight polarizations are statistically zero.
  • The total DEM attenuation, a scalar measure of all noise in syndrome extraction, can be estimated efficiently via a Monte Carlo average over depolarizations.
  • If the DEM is sparse and low-weight, lattice pruning can identify which individual events have nonzero probability and estimate them, extending the p_ij method into a data-driven decoder-graph construction with higher-order hyperedges.
  • The derivation of the p_ij coefficients reinterprets existing calibration results and tells experimenters what those numbers actually mean.

Reading between the lines

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

  • The paper leaves implicit that the same closed-form inversion could be applied to time-windowed syndrome data, offering a direct route to detecting time-varying or nonstationary DEMs by comparing attenuation estimates across windows.
  • Because the inversion is exact under the DEM model, a natural benchmark is to compare its estimates against decoder-counting frequencies in regimes where degeneracy is absent; agreement there would validate the algebra, and disagreement would localize the failure to the DEM assumptions.
  • The lattice-pruning algorithm suggests a concrete testable extension: in a surface-code or repetition-code experiment, clutter of nonzero weight-2 classes that cannot be explained by genuine weight-2 events should appear as higher-order hypercliques, and the paper's method would distinguish those cases.
  • The use of sparse Walsh-Hadamard transform ideas points toward a fully blind sparse-DEM estimator that does not require a candidate event list, though the paper only sketches the candidate-based version.
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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 develops a closed-form, decoder-free method for estimating detector error model (DEM) event probabilities from syndrome-extraction data. It defines parities z_y = (-1)^{x·y}, polarizations <z_y>, and depolarizations ω_y = -ln<z_y>; under the DEM axioms of Section 1.2 (independent errors with fixed detector supports, merged identical supports, constant probabilities), the vector of depolarizations is shown to be a linear function of event attenuations a_s = -ln(1-2p_s), namely ω = W a (Eq. 25). The authors derive the Moore-Penrose inverse W^+ and obtain the main inversion (Eqs. 31-34): event attenuations, decay factors, and probabilities are given explicitly by a scaled Walsh-Hadamard transform of the depolarization vector, involving only empirical polarizations. They show that reduced-DEM aggregations reproduce and clarify the p_ij coefficients of Refs. [5,7], give explicit low-weight aggregation formulas (Eqs. 44-48), propose Monte Carlo averaging for high-weight quantities such as total attenuation, and sketch two sparse-DEM algorithms based on submatrix inversion and lattice pruning (Section 4).

Significance. The central derivation is clean and parameter-free: the inversion from empirical polarizations to DEM event probabilities involves no fitted constants and no decoder, and the key identities (Eqs. 25-34) are derived step by step rather than asserted. The paper gives explicit, falsifiable closed-form predictions, and it makes a genuine conceptual contribution by explaining what the widely used p_ij coefficients actually estimate (aggregated probabilities over event classes) and by unifying earlier partial results [5,7,11] in a single algebraic framework. If the results hold up, the formalism is likely to become a standard reference for DEM estimation and calibration. The main weaknesses are equally clear: the paper is purely algebraic, with no numerical experiments, no code, and the sparse algorithms that carry the realistic-N claims are explicitly deferred to future work. The practical claim that Eq. 31 can be used to infer DEMs from data therefore rests on unvalidated finite-sample and model-robustness behavior, and the uncertainty formulas (Eqs. 15-16) contain a mathematical error.

major comments (3)
  1. [Sec. 2.1.1 (Eqs. 15-16)] The standard error formulas are wrong by a square root. Since Eq. 14 correctly gives the variance σ^2(z_y) = 1 - <z_y>^2, the standard error of the mean is sqrt(1-<z_y>^2)/sqrt(K), but Eq. 15 states σ_<z_y> = (1-<z_y>^2)/sqrt(K), and Eq. 16 states σ_ωy = (1-<z_y>^2)/(<z_y> sqrt(K)). The correct form of the latter is sqrt(1-<z_y>^2)/(<z_y> sqrt(K)). As written, the formulas overstate the error bars by a factor that diverges as <z_y> approaches zero, which is exactly the regime the paper later identifies as problematic in Sec. 3.4.1. Because the paper recommends these error bars as the practical uncertainty quantification for the whole estimation pipeline, this needs to be corrected before acceptance.
  2. [Secs. 3.4 and 4; abstract] The abstract promises that the paper shows how to estimate probabilities of individual DEM events from data, but for realistic N the individual-event estimators require the Section 4 algorithms, which the paper explicitly defers: 'We defer the work of demonstrating and analyzing these algorithms to future work' (Sec. 4). There are no numerical experiments anywhere in the manuscript, no finite-sample study, and no discussion of how to choose the threshold wmax that the sparse estimates depend on, even though wmax is a free input to the proposed procedures. Consequently, the central practical claim is not demonstrated in the regime the paper itself identifies as relevant (large N, local low-weight DEMs). I request simulation-based tests, for example sampling from the N=4 DEM of Sec. 3.1.1 and from a local weight-<=2 DEM with N = 20-100, comparing Eq. 31 and the Section 4 algorithms against ground truth with proper error bars.
  3. [Sec. 1.2 axioms; Eqs. 31-34] The inversion is exact only for distributions of the product form (Eq. 23) generated by a DEM; for data outside the DEM family, Eq. 31 does not return a physical DEM, and the paper offers no analysis, warning, or test. A concrete example: the valid N=2 distribution with outcome probabilities (0.525, 0.225, 0.225, 0.025) has <z1>=<z2>=0.5 and <z12>=0.1, yet no DEM generates it (the implied decay factors would require d12^2 = 2.5, impossible since |d12| <= 1), and Eq. 34 returns p_{1,2} = 1/2 - (1/2)sqrt(0.25/0.1) ≈ -0.29. Real syndrome data will contain time-correlated errors, crosstalk, and drift that violate the Section 1.2 axioms, so the statement at the end of Sec. 2.3 that the linear relationship 'can be inverted to learn or infer attenuations from empirical depolarizations' needs either a robustness bound with respect to distance from the DEM family or an explicit demonstration on simulated non-DEM noise that the estimator fails gracefully. As it stands, this is a load-bearing gap for the estimation claim.
minor comments (5)
  1. [Eqs. 9 and 37] Both equations contain the typo '1 - 2<xi> - 2<xi> + 4<xixj>' with the second expectation equal to the first; Eq. 9 should read '1 - 2<xi> - 2<xj> + 4<xixj>' and Eq. 37 should use <x1> and <x2> consistently.
  2. [Sec. 3.4] The Monte Carlo estimator expression 'a_s = 2<[(-1)^{y·s}+1 ω_y]>_y' is ambiguous and, as typeset, incorrect; the intended formula is a_s = 2<(-1)^{y·s+1} ω_y>_y, with the +1 inside the exponent. Please fix the typesetting.
  3. [Sec. 4.1 (footnote 5)] The stated count of additional events, C(N + wmax - 2|E| - 1, wmax - |E|), does not equal the sum over w = 1 to wmax - |E| of C(N - |E|, w) in general. For example, with N=5, |E|=1, and wmax=4, the sum is 4+6+4=14 events, while the formula gives C(6,3)=20. The counting should be corrected or qualified as an upper bound.
  4. [Sec. 3.1.1] The claim that Eq. 36 is 'Equation (11) from the Supplementary Material to Ref. [7], mod a little algebra' would be easier to verify if the intermediate algebra were shown; as written, the reader must obtain the supplement to check the match.
  5. [General] The paper never mentions that Eq. 34 can produce estimates outside [0,1] even for data drawn from a DEM when K is finite, and no guidance is given on how to handle such estimates (e.g., clipping, reframing as a model-violation diagnostic, or reporting the unconstrained attenuation instead). A brief remark would substantially improve practical usability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the DEM inversion is a parameter-free algebraic consequence of the stated model.

full rationale

The derivation is self-contained under the paper's explicitly stated DEM model. Section 1.2 defines a DEM as independent events, each flipping a fixed support of detector bits, with indistinguishable supports merged. Sections 2.2-2.3 derive from this definition that each DEM event s attenuates every polarization ⟨z_y⟩ with y·s=1 by the factor (1-2p_s), which by independence yields Eq. 23, ⟨z_y⟩ = ∏_s d_s^{y·s}, and its logarithmic form Eq. 24. This factorization is a derived consequence of the model, not an assumed conclusion. The Walsh-Hadamard inversion in Eqs. 25-31 is then pure linear algebra: Eq. 31 solves Eq. 24 for the attenuations, and Eq. 34 is the algebraic inverse of Eq. 23 under the stated DEM assumptions. No fitted parameter is later relabeled as a prediction; the empirical polarizations are the data, and the mapping from them to DEM event probabilities is parameter-free. The paper explicitly frames its input as syndrome data generated by an unknown DEM and its goal as estimating that DEM. Reproducing the p_ij formula of Ref. [7] in Section 3.1 is presented as a consistency check and answer to a motivating question, not as a load-bearing input to the derivation. The cited earlier frameworks are acknowledged and re-derived in the present paper rather than used as unverified premises, and no load-bearing self-citation chain appears. The skeptically raised issue of robustness to non-DEM noise is a validity or model-misspecification concern, not evidence of circularity: it asks what Eq. 31 returns when the generating process violates the stated model, which is an evaluation question outside the derivation chain. Because the central claim is an exact algebraic identity within an explicitly stated and independently motivated model class, no circular step is present.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central inversion has no fitted constants; the only hand-chosen numerical input is wmax for the sparse algorithms. The main burden is the DEM modeling assumption: independent, constant-rate, support-identical-merged error events. Attenuation is a mathematical reparametrization, not a new physical entity.

free parameters (1)
  • wmax, maximum event weight for sparse algorithms
    Chosen by hand in Section 4; the sparse DEM estimate depends on this cutoff, and no method for selecting it from data is given.
assumptions (5)
  • standard math Walsh-Hadamard transform and Moore-Penrose pseudoinverse linear algebra are valid and sufficient for the inversion.
    Used in Section 2 to derive Eqs. 25-31; these are standard results in linear algebra and Fourier analysis.
  • domain assumption Syndrome histories are generated by a DEM: independent error events, each flipping a fixed subset of detector bits, with identical-support events merged.
    Section 1.2 defines the model; the central inversion is exact only for data drawn from such a DEM.
  • domain assumption The K syndrome histories are independent and identically distributed samples from one fixed DEM.
    The sample polarization estimator in Eq. 12 and the uncertainty formulas assume i.i.d. sampling.
  • domain assumption Each DEM event probability is below 1/2, so the decay factor d_s = 1-2p_s is positive and attenuation is real.
    Eqs. 20-21 define attenuation as -ln(1-2p); the paper does not state this restriction, but real syndrome extraction error events are rare.
  • domain assumption Realistic DEMs are low-weight, local, and approximately extensive.
    Section 3.4.1 relies on this physical expectation to argue that high-weight depolarizations are unusable and to motivate the sparse algorithms; if false, the practical methods fail.

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Cite this review

Pith. "Pith review of Estimating detector error models from syndrome data." pith.science (2026). https://pith.science/paper/BL2Y3VA2

@misc{pith2026250414643,
  author       = {Pith},
  title        = {Pith review of: Estimating detector error models from syndrome data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BL2Y3VA2}},
  note         = {Machine review of arXiv:2504.14643}
}
read the original abstract

Protecting quantum information using quantum error correction (QEC) requires repeatedly measuring stabilizers to extract error syndromes that are used to identify and correct errors. Syndrome extraction data provides information about the processes that cause errors. The collective effects of these processes can be described by a detector error model (DEM). We show how to estimate probabilities of individual DEM events, and of aggregated classes of DEM events, using data from multiple cycles of syndrome extraction.

Figures

Figures reproduced from arXiv: 2504.14643 by the authors.

Figure 1
Figure 1. Two syndrome extraction cycles (SECs) for a [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. A nontrivial chain of four transformations maps data [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Classes of events whose aggregated attenuation can [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

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Reference graph

Works this paper leans on

31 extracted references · 18 canonical work pages · cited by 3 Pith papers

  1. [1]

    In-situ characterization of quantum devices with error correction

    Joshua Combes, Christopher Ferrie, Chris Cesare, Markus Tiersch, G J Milburn, Hans J Briegel, and Carlton M Caves. “In-situ characterization of quantum devices with error correction” (2014). arXiv:1405.5656

    arXiv 2014

  2. [2]

    Scalable extraction of er- ror models from the output of error detection cir- cuits

    Austin G Fowler, D Sank, J Kelly, R Barends, and John M Martinis. “Scalable extraction of er- ror models from the output of error detection cir- cuits” (2014). arXiv:1405.1454

    arXiv 2014

  3. [3]

    Syndrome-Derived Error Rates as a Benchmark of Quantum Hardware

    James R Wootton. “Syndrome-derived error rates as a benchmark of quantum hardware” (2022). arXiv:2207.00553

    work page Pith review arXiv 2022

  4. [4]

    Learning time- dependent noise to reduce logical errors: real time error rate estimation in quantum error correction

    Ming-Xia Huo and Ying Li. “Learning time- dependent noise to reduce logical errors: real time error rate estimation in quantum error correction”. New J. Phys.19, 123032 (2017)

    work page 2017

  5. [5]

    Adaptive weight estimator for quantum error correction in a time-dependent environment

    Stephen T Spitz, Brian Tarasinski, Carlo W J Beenakker, and Thomas E O’Brien. “Adaptive weight estimator for quantum error correction in a time-dependent environment”. Adv. Quantum Technol.1, 1800012 (2018)

    work page 2018

  6. [6]

    Analysing correlated noise on the surface code using adaptive decoding algorithms

    Naomi H Nickerson and Benjamin J Brown. “Analysing correlated noise on the surface code using adaptive decoding algorithms” (2017). arXiv:1712.00502

    work page Pith review arXiv 2017

  7. [7]

    Exponential suppression of bit or phase errors with cyclic error correction

    Google Quantum AI. “Exponential suppression of bit or phase errors with cyclic error correction”. Nature 595, 383–387 (2021)

    2021

  8. [8]

    Calibrated decoders for experimental quantum error correction

    Edward H Chen, Theodore J Yoder, Youngseok Kim, N Sundaresan, S Srinivasan, Muyuan Li, A C’orcoles, Andrew W Cross, and M Takita. “Cal- ibrated decoders for experimental quantum error correction”. Phys. Rev. Lett.128, 110504 (2021). arXiv:2110.04285

    work page Pith review arXiv 2021

Show all 31 references
  1. [9]

    Optimization of decoder priors for accurate quantum error correction

    Volodymyr Sivak, Michael Newman, and Paul Klimov. “Optimization of decoder priors for accurate quantum error correction” (2024). arXiv:2406.02700

    2024 arXiv

  2. [10]

    Optimal adaptation of surface-code decoders to local noise

    Andrew S Darmawan. “Optimal adaptation of surface-code decoders to local noise” (2024). arXiv:2403.08706

    2024 arXiv

  3. [11]

    Experimentally informed decod- ing of stabilizer codes based on syndrome correla- tions

    Ants Remm, Nathan Lacroix, Lukas Bödeker, Elie Genois, Christoph Hellings, François Swiadek, Graham J Norris, Christopher Eichler, Alexandre Blais, Markus Müller, Sebastian Krinner, and An- dreas Wallraff. “Experimentally informed decod- ing of stabilizer codes based on syndro...

    2025 arXiv

  4. [12]

    Improving error suppression with noise- aware decoding

    Evan T Hockings, Andrew C Doherty, and Robin Harper. “Improving error suppression with noise- aware decoding” (2025). arXiv:2502.21044

    2025 arXiv

  5. [13]

    Optimal noise estimation from syndrome statistics of quantum codes

    Thomas Wagner, Hermann Kampermann, Dag- mar Bruß, and Martin Kliesch. “Optimal noise estimation from syndrome statistics of quantum codes” (2020). arXiv:2010.02243

    2020 arXiv

  6. [14]

    Pauli channels can be estimated from syndrome measurements in quan- tum error correction

    Thomas Wagner, Hermann Kampermann, Dagmar Bruß, and Martin Kliesch. “Pauli channels can be estimated from syndrome measurements in quan- tum error correction”. Quantum 6, 809 (2022). arXiv:2107.14252v2

    2022 arXiv

  7. [15]

    Learning logical pauli noise in quantum error correction

    Thomas Wagner, Hermann Kampermann, Dag- mar Bruß, and Martin Kliesch. “Learning logical pauli noise in quantum error correction” (2022). arXiv:2209.09267

    2022 arXiv

  8. [16]

    Efficient estimation of pauli channels

    Steven T Flammia and Joel J Wallman. “Efficient estimation of pauli channels”. ACM Transactions on Quantum Computing1, 1–32 (2020)

    2020

  9. [17]

    Effi- cientlearningofthestructureandparametersoflo- cal pauli noise channels

    Cambyse Rouzé and Daniel Stilck Franca. “Effi- cientlearningofthestructureandparametersoflo- cal pauli noise channels” (2023). arXiv:2307.02959

    2023 arXiv

  10. [18]

    Learning correlated noiseina39-qubitquantumprocessor

    R Harper and S Flammia. “Learning correlated noiseina39-qubitquantumprocessor”. PRXquan- tum 4, 040311 (2023). arXiv:2303.00780

    2023 arXiv

  11. [19]

    Scalable noise characterization of syndrome-extraction circuits with averaged cir- cuit eigenvalue sampling

    Evan T Hockings, Andrew C Doherty, and Robin Harper. “Scalable noise characterization of syndrome-extraction circuits with averaged cir- cuit eigenvalue sampling”. PRX quantum 6, 010334 (2024). arXiv:2404.06545

    2024 arXiv

  12. [20]

    Bayesian inference of general noise model parameters from surface code’s syndrome statistics

    Takumi Kobori and S Todo. “Bayesian inference of general noise model parameters from surface code’s syndrome statistics” (2024). arXiv:2406.08981

    2024

  13. [21]

    Quantum computer error structure probed by quantum error correction syndrome measure- ments

    S Gicev, L Hollenberg, and Muhammad Us- man. “Quantum computer error structure probed by quantum error correction syndrome measure- ments”. Phys. Rev. Res. 6, 043249 (2023). arXiv:2310.12448

    2023 arXiv

  14. [22]

    Using detector likelihood for benchmarking quan- tum error correction

    Ian Hesner, Bence Hetényi, and James R Wootton. “Using detector likelihood for benchmarking quan- tum error correction” (2024). arXiv:2408.02082

    2024 arXiv

  15. [23]

    Enhanced repetition codes for the cross- platform comparison of progress towards fault- tolerance

    Milan Liepelt, Tommaso Peduzzi, and James R Wootton. “Enhanced repetition codes for the cross- platform comparison of progress towards fault- tolerance” (2023). arXiv:2308.08909

    2023 arXiv

  16. [24]

    Randomized benchmarking as convolution: Fourier analysis of gate dependent errors

    Seth T Merkel, Emily J Pritchett, and Bryan H Fong. “Randomized benchmarking as convolution: Fourier analysis of gate dependent errors”. Quan- tum 5, 581 (2021). arXiv:1804.05951v3

    2021 arXiv

  17. [25]

    The jackknife and bootstrap

    Jun Shao and Dongsheng Tu. “The jackknife and bootstrap”. Springer Science & Business Media. (2012)

    2012

  18. [26]

    Fast walsh–hadamard–fourier transform algorithm

    Monir T Hamood and Said Boussakta. “Fast walsh–hadamard–fourier transform algorithm”. IEEE Transactions on Signal Processing 59, 5627–5631 (2011). 14

    2011

  19. [27]

    The spright algo- rithm for robust sparse hadamard transforms

    Xiao Li, Joseph Kurata Bradley, Sameer Pawar, and Kannan Ramchandran. “The spright algo- rithm for robust sparse hadamard transforms”. In 2014 IEEE International Symposium on Informa- tion Theory. Pages 1857–1861. IEEE (2014)

    2014

  20. [28]

    Robust sublinear complexity walsh-hadamard transform with arbi- trary sparse support

    Xu Chen and Dongning Guo. “Robust sublinear complexity walsh-hadamard transform with arbi- trary sparse support”. In 2015 IEEE International Symposium on Information Theory (ISIT). Pages 2573–2577. IEEE (2015)

    2015

  21. [29]

    Nearly op- timal deterministic algorithm for sparse walsh- hadamard transform

    Mahdi Cheraghchi and Piotr Indyk. “Nearly op- timal deterministic algorithm for sparse walsh- hadamard transform”. ACM Transactions on Al- gorithms (TALG)13, 1–36 (2017)

    2017

  22. [30]

    Compressed sensing

    David L Donoho. “Compressed sensing”. IEEE Transactions on information theory 52, 1289– 1306 (2006)

    2006

  23. [31]

    A structured review of sparse fast fourier trans- form algorithms

    Elias Rajaby and Sayed Masoud Sayedi. “A structured review of sparse fast fourier trans- form algorithms”. Digital Signal Processing 123, 103403 (2022). 15

    2022

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