Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

Out of Tune: Demystifying Noise-Effects on Quantum Fourier Models

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper claims that noise effects on quantum Fourier models—on their Fourier spectrum, expressibility, and entangling capability—are predictable per noise type, uniform across the tested circuits, and shaped more by circuit architecture…

desk verdict A broad, honestly-scoped numerical benchmark of noise effects on quantum Fourier models; the coefficient and training results are credible, but the expressibility metric is applied to mixed states without a stated convention, which is a load-bearing gap. read the letter →

arxiv 2506.09527 v2 pith:4DR4VREQ submitted 2025-06-11 quant-ph

Maja Franz , Melvin Strobl , Leonid Chaichenets , Eileen Kuehn , Achim Streit , Wolfgang Mauerer This is my paper
classification quant-ph
keywords quantummachinelearningFouriermodelsvariationalcircuitsnoisemodellingspectrumexpressibilityentanglingcapabilityerrormitigation
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 the influence of hardware noise on quantum Fourier models can be anticipated: for a given circuit ansatz and input encoding, noise affects the Fourier spectrum, expressibility, and entangling capability in systematic ways, with each noise type acting uniformly across all tested circuits. If true, this means noise-induced degradation of a quantum model can be predicted before running noisy hardware, and architectural choices can be selected for robustness. The authors support the claim with numerical simulations on circuits of three to six qubits, four standard ansätze, seven noise types, and noise levels up to three percent, linking these spectral and entanglement metrics to training performance on synthetic regression problems. A sympathetic reader would take the core message as: the structure of the variational circuit and the way data is encoded matter at least as much as the noise level for what a noisy quantum model can learn.

What carries the argument

The central object is the representation of a variational quantum circuit as a quantum Fourier model: the observable expectation value is written as a truncated Fourier series $f(x,\theta)=\sum_{\omega\in\Omega} c_\omega(\theta)e^{i\omega^T x}$, where the frequency set $\Omega$ is determined by differences of eigenvalues of the Hermitian generators in the input-encoding unitaries, and the coefficients $c_\omega(\theta)$ depend on the trainable parameters and circuit structure. This Fourier lens turns questions about noise into questions about how noise alters coefficients and frequencies. The paper measures coefficients numerically with the fast Fourier transform after cross-validating against analytical expansions, quantifies expressibility as the inverse KL divergence between the model's fidelity distribution and the uniform distribution over quantum states, and quantifies entangling capability with a pure-state global-entanglement measure for noiseless circuits and a mixed-state entanglement measure for noisy ones. Noise is modelled with standard channels: bit flip, phase flip, depolarisation, amplitude damping, phase damping, state-preparation and measurement errors, and coherent gate errors. The mechanism behind the coherent-error result is that an error $\epsilon_x$ on an encoding gate shifts the frequency exponent from $\omega^T x$ to $(\omega+\epsilon_x)^T x$, which produces additional non-integer frequencies and reduces the redundancy of the spectrum.

What would settle it

Compute expressibility for the same noisy circuits using a mixed-state fidelity (for example the Uhlmann fidelity or a purification-based overlap) instead of the pure-state overlap the paper uses, and compare the resulting KL divergences to the uniform-state distribution; if the ranking of ansätze or the direction of noise dependence changes, the expressibility findings are an artefact of the metric rather than a property of the circuits.

Watch

Extended reading notes

Core claim

The central claim, stated as the paper's Result Summary, is that the influence of noise on properties of quantum Fourier models—particularly the Fourier spectrum, expressibility, and entangling capability—can be predicted, that the effects of each noise type are uniform throughout all tested circuits, and that the structure of the variational quantum circuit and the input encoding have a crucial impact on these properties independently of the noise type, also in a noiseless setting. Concretely, decoherent gate errors such as bit flip, phase flip, and depolarisation exponentially damp the mean magnitude of Fourier coefficients across all four ansätze, while SPAM and damping noise produce weaker or no observable contraction. Expressibility, measured by how far the circuit's state-overlap distribution sits from the uniform distribution over quantum states, drops under decoherent gate errors and more mildly under SPAM and damping noise, while coherent gate errors leave it essentially unchanged. Entangling capability falls most clearly under amplitude damping. In training on synthetic Fourier series, decoherent gate errors push all ansätze to a similar elevated mean-squared error, whereas the heavily entangled ansatz that achieves a full noiseless spectrum trains best without noise and the ansätze with incomplete spectra fail to reach the optimum even noiselessly. The paper also finds that coherent gate error acting on the encoding gates changes the frequency set itself, adding new non-integer frequencies whose coefficients are not individually tunable.

Load-bearing premise

The expressibility conclusions for noisy circuits rest on applying a fidelity-based overlap measure designed for pure quantum states to the mixed states that noise actually produces, and the paper supplies no mixed-state generalization; if that step is invalid, the reported noise-induced loss of expressibility is unsupported.

Editorial extensions

If this is right

  • For a chosen ansatz and input encoding, the severity and type of noise-induced degradation can be anticipated without running the noisy circuit: which coefficients decay, how expressibility drops, and how much entanglement is lost are all predictable from the architecture and the noise channel.
  • Decoherent gate errors are the dominant threat to quantum Fourier models, exponentially shrinking Fourier coefficient magnitudes, raising training error, and reducing expressibility, whereas SPAM and damping errors have milder effects.
  • Architecture choice matters even in a noiseless setting: some ansätze have full Fourier spectra and train well but lose expressibility quickly under noise, while others lack full spectra entirely and therefore cannot learn certain functions regardless of noise.
  • Coherent gate errors on encoding gates are qualitatively different from other noise because they alter the frequency set itself; error mitigation must therefore target encoding gates, not only trainable layers, to preserve the intended spectrum.
  • The observed patterns suggest that noise-aware ansatz selection and tailored error mitigation can be informed by spectral properties measured on a few small circuits, since the uniform behaviour across tested circuits indicates broader applicability beyond the limited sizes studied.

Reading between the lines

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

  • If the reported uniformity of noise-type effects holds at larger qubit counts and depths, noise-aware architecture search could rank candidate ansätze by their predicted spectral robustness before any hardware deployment; the paper does not itself propose such a search.
  • The expressibility findings rest on applying a pure-state fidelity metric to mixed states, so a mixed-state formulation of the metric could change quantitative conclusions while perhaps preserving the qualitative ordering of ansätze; this is an open question the paper does not address.
  • Because coherent gate errors add non-integer frequencies that are not individually tunable, a testable extension would be to check whether deliberately introduced coherent shifts on encoding gates can be exploited as a controlled spectral-design tool, for example to reach off-grid frequencies helpful for a specific learning task.
  • A natural next experiment would be to measure the same spectrum, expressibility, and entangling-capability quantities on real quantum hardware under calibrated noise, to see whether the simulated uniform patterns survive device-specific crosstalk and drift.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper presents a numerical study of how various noise models affect three properties of variational quantum circuits used as quantum Fourier models (QFMs): the Fourier spectrum (coefficients), expressibility, and entangling capability. The authors simulate four ansätze (SEA, HEA, C15, C19) with 3–6 qubits, one or two encoding features, and seven noise levels up to 3%, and they also train the QFMs on synthetic regression targets. The central claims are that noise effects are predictable and qualitatively uniform across circuits, and that ansatz structure and input encoding dominate the impact of these properties, including in the noiseless setting. The paper includes a software reproduction package and uses both FFT-based and analytical coefficient computations, with cross-validation.

Significance. If the central claims hold, the work would provide a useful empirical map of how decoherent and coherent noise degrade the Fourier spectrum, expressibility, and entanglement of QFMs, with potential value for ansatz selection and error-mitigation design. The study is broad in its coverage of noise channels and ansätze, and the public code package is a strength. However, several load-bearing methodological gaps currently prevent the results from fully supporting the stated conclusions. In particular, the expressibility metric is applied to mixed states without a defined generalization, the synthetic target definition in Eq. (14) appears internally inconsistent, and the expressibility/entanglement circuits omit the encoding that is central to QFM properties. These issues need to be resolved before the paper's main claims can be accepted.

major comments (4)
  1. [Sec. 4.3, Eq. (11); Sec. 5.2, Fig. 11] The expressibility metric is defined via the pure-state overlap F = |<ψ_φ|ψ_ϕ>|^2 and the KL divergence between the resulting fidelity distribution and that of a Haar t-design. In Sec. 5.2 this same metric is applied to circuits subjected to decoherent noise, whose outputs are mixed states. The manuscript nowhere specifies the mixed-state generalization: it does not state whether noisy states are represented as density matrices (requiring, e.g., Uhlmann fidelity or a purification of the Haar states) or whether each noisy run is stochastically collapsed to a pure state via Kraus-operator sampling. These choices define different distributions, so the KL divergence plotted in Fig. 11 is not well-defined as written. Since Fig. 11 is used to conclude that decoherent noise reduces expressibility, and expressibility is one of the three properties named in the Result Summary, this ambiguity is load-bearing. The authors should either define and justify a mixed-state fidelity distribution or restrict the expressibility claim to a clearly stated operational convention.
  2. [Sec. 4.5, Eq. (14)] The target coefficients for the synthetic regression task are defined as c'_ω = a r_ω |∑_{ω∈Ω} r_ω e^{iω^T x}|^{-1}. Because the denominator depends on the input x, c'_ω is not a constant Fourier coefficient; consequently f'(x) is not a Fourier series with the stated spectrum. In fact, substituting this definition into f'(x) yields f'(x) = a times the phase factor of the random sum, whose Fourier coefficients are not the c'_ω. This makes the target a pathological function rather than a random Fourier series with a controlled spectrum. The coefficient-difference metric Δc_ω in Eq. (15) is therefore ill-defined as written, and the training results in Sec. 5.1.5 and Fig. 10 cannot support conclusions about noise-induced degradation of coefficient learning. The authors should correct Eq. (14) (e.g., normalizing by a constant such as ∑|r_ω|) and verify that the implemented code matches the corrected formula, or clearly report the exact target used in the experiments.
  3. [Sec. 5 (third paragraph); Sec. 5.2] For the expressibility and entanglement measurements, the authors state that they 'discard encoding gates and the second trainable layer to make the results more consistent with Ref. [42]'. Thus the circuits analyzed in Figs. 11–13 do not include the data-encoding unitaries that are essential to the QFM representation. This creates a mismatch: the Result Summary claims that 'the structure of a VQC and the input encoding have a crucial impact on these properties', but the expressibility experiments never vary the input encoding, and the entanglement experiments likewise omit it. At minimum, the claim should be restricted to the trainable ansatz structure, or separate experiments should be performed to test the influence of the encoding on expressibility and entanglement. As written, the connection between these metrics and the QFM properties used in the coefficient and training experiments is indirect.
  4. [Sec. 4.4.2; Fig. 13] The paper reports that the Entanglement of Formation (EF) measure gives different values from the Meyer-Wallach (MW) measure even for noiseless circuits, and attributes this to non-uniqueness of the eigendecomposition used in the EF computation. For a pure state, the eigendecomposition of the density matrix is unique up to a global phase, and the MW measure is invariant under local unitary transformations; the explanation as written is therefore not convincing. The discrepancy might indicate a numerical artifact (e.g., mixing of nearly degenerate eigenvalues) or a different computation than described. Since EF is used to support the central claim that noise reduces entangling capability, the authors should validate the zero-noise limit of EF against the MW measure and clarify the computation, or explicitly restrict EF claims to mixed-state regimes where the non-uniqueness genuinely matters.
minor comments (5)
  1. [Sec. 4.3; Fig. 11] The text says expressibility is the inverse of the KL divergence, but Fig. 11 plots the KL divergence itself with an axis label showing 'more expr.← KL-Divergence [log] → less expr.'; please align the terminology with the plotted quantity.
  2. [Sec. 5.1.3, Fig. 6] The claim that decoherent gate errors cause an 'exponential decay' of the coefficient mean is based on visual inspection across only seven noise levels up to 3%; consider fitting a decay law or stating this as a qualitative trend rather than a functional form.
  3. [Result Summary; Sec. 5.1.3] The statement that 'the effects of each noise type are uniform throughout all tested circuits' is qualified by the reported exception for the SEA with bit-flip noise, where σ_c(ω=0) increases with noise level; please make this exception explicit in the summary.
  4. [Sec. 5.2, final paragraph] The sentence 'In a noiseless environment, the SEA achieves a higher expressibility compared to the other ansätze. However, when coherent noise is applied, the expressibility is quickly equalised' appears to refer to decoherent noise, which contradicts the earlier statement that coherent noise has no measurable effect on expressibility; please correct this apparent wording error.
  5. [Fig. 8 caption] The y-axis label '# Qubits – Limit' is unclear; please define what this limit represents, for instance in relation to the maximum possible number of frequencies for the given encoding.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the numerical results are self-contained and do not reduce to fitted inputs or self-citation chains.

full rationale

The paper's central claims are supported by direct numerical simulation rather than by parameters fitted to the predicted quantities. Fourier coefficients are obtained by FFT from simulated expectation values (Sec. 4.2), expressibility is computed as the KL divergence to a Haar distribution (Sec. 4.3), entangling capability is evaluated with Meyer-Wallach and Entanglement of Formation measures (Sec. 4.4), and training performance is measured against independently generated random Fourier series (Sec. 4.5). The synthetic targets use a chosen normalization a=0.5, but this is a dataset-generation convention, not a fitted parameter that forces the reported MSE or coefficient-difference results. The expressibility metric's application to noisy mixed states is an unstated-convention ambiguity, not a circular reduction: the paper does not derive the noisy expressibility values from the definition of the metric, and the issue would affect correctness rather than indicate that the conclusion is equivalent to its input. Self-citations (e.g., Refs. [13], [27], [43]) appear as reproducibility and framework references and are not load-bearing for the numerical findings; the load-bearing methodological citations, such as Refs. [42], [29], and [12], are external prior work. The Result Summary is an empirical generalization from the systematic experiments, not a claim forced by construction, and the paper explicitly notes that analytical derivations remain future work. No step in the derivation chain equates an output to an input by definition, and no fitted parameter is renamed as a prediction. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

No new physical entities are introduced. The main results rest on standard QFM Fourier theory, simplified noise models, and several hand-chosen thresholds and metrics. The only genuinely tunable constants are the synthetic target normalization and the coefficient cutoff; neither is fitted to make the central noise conclusions true. The expressibility-on-mixed-states and eigendecomposition-EF choices are the largest unproven assumptions.

free parameters (2)
  • Target normalization a = 0.5
    Chosen in Eq. 14 so synthetic target functions lie in the interval [-0.5, 0.5] and are attainable by the QFM; it sets the difficulty of the regression task but does not drive the noise conclusions.
  • Coefficient zero threshold = 1e-14
    Used in Sec. 5.1 to decide which frequencies are absent from the spectrum; directly affects the claims that the hardware-efficient ansatz and C15 lack a full spectrum.
assumptions (6)
  • standard math A VQC with data-encoding unitaries can be written as a finite Fourier series with frequencies given by eigenvalue differences of the encoding generators (Eqs. 4-6).
    Adopted from Refs. [40, 31, 6, 29] and used throughout Sec. 2; not re-derived in this paper.
  • domain assumption The Kraus-operator noise channels and Gaussian coherent gate errors in Sec. 4.1 and Appendix A adequately represent real hardware noise.
    The paper claims hardware-agnostic results but uses fixed probabilities and simplified channels without device calibration or comparison to QPU data.
  • domain assumption The KL divergence to the Haar distribution remains a valid expressibility measure when the circuit output is a mixed state under decoherent noise.
    Sec. 4.3 defines fidelity for pure states; Sec. 5.2 applies it to noisy circuits without specifying a mixed-state generalization.
  • domain assumption Sampling 250 shots per parameter and five seeds adequately characterizes coefficient statistics.
    Sec. 5 reports small seed standard deviations but provides no convergence or resampling analysis.
  • domain assumption Entanglement of Formation computed from one eigen-decomposition of the noisy density matrix is a meaningful upper bound on entangling capability.
    Sec. 4.4.2 and Sec. 5.3 acknowledge the decomposition is not unique and interpret the values as upper bounds.
  • domain assumption Trends observed for 3 to 6 qubits and one to two input features generalize to larger circuits.
    Sec. 6 suggests broader applicability from consistent scaling but provides no analytical or numerical evidence beyond the tested sizes.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Out of Tune: Demystifying Noise-Effects on Quantum Fourier Models." pith.science (2026). https://pith.science/paper/4DR4VREQ

@misc{pith2026250609527,
  author       = {Pith},
  title        = {Pith review of: Out of Tune: Demystifying Noise-Effects on Quantum Fourier Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4DR4VREQ}},
  note         = {Machine review of arXiv:2506.09527}
}
read the original abstract

Variational quantum algorithms have received substantial theoretical and empirical attention. As the underlying variational quantum circuit (VQC) can be represented by Fourier series that contain an exponentially large spectrum in the number of input features, hope for quantum advantage remains. Nevertheless, it remains an open problem if and how quantum Fourier models (QFMs) can concretely outperform classical alternatives, as the eventual sources of non-classical computational power (for instance, the role of entanglement) are far from being fully understood. Likewise, hardware noise continues to pose a challenge that will persist also along the path towards fault tolerant quantum computers. In this work, we study VQCs with Fourier lenses, which provides possibilities to improve their understanding, while also illuminating and quantifying constraints and challenges. We seek to elucidate critical characteristics of QFMs under the influence of noise. Specifically, we undertake a systematic investigation into the impact of noise on the Fourier spectrum, expressibility, and entangling capability of QFMs through extensive numerical simulations and link these properties to training performance. The insights may inform more efficient utilisation of quantum hardware and support the design of tailored error mitigation and correction strategies. Decoherence imparts an expected and broad detrimental influence across all Ans\"atze. Nonetheless, we observe that the severity of these deleterious effects varies among different model architectures, suggesting that certain configurations may exhibit enhanced robustness to noise and show computational utility.

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Fourier Fingerprints of Ansatzes in Quantum Machine Learning

    quant-ph 2025-08 conditional novelty 6.0 of 10

    Variational quantum circuits' Fourier coefficients are correlated in ansatz-specific ways, and the new Fourier coefficient correlation metric predicts their training performance better than expressibility in the tested cases.

Reference graph

Works this paper leans on

51 extracted references · 11 canonical work pages · cited by 1 Pith paper

  1. [42]

    Expressibility and Entangling Capability of Parameterized Quantum Circuits for Hybrid Quantum- Classical Algorithms

    Sukin Sim, Peter D. Johnson, and Alán Aspuru-Guzik. “Expressibility and Entangling Capability of Parameterized Quantum Circuits for Hybrid Quantum- Classical Algorithms”. en. In: Advanced Quantum Technologies2.12 (2019), p. 1900070. doi: 10.1002/qute.201900070

    doi:10.1002/qute.201900070 2019

  2. [1]

    Industry quantum computing applications

    Andreas Bayerstadler, Guillaume Becquin, Julia Binder, et al. “Industry quantum computing applications”. In:EPJ Quantum Technology8.1 (Nov. 2021).doi: 10.1140/epjqt/s40507-021-00114-x

    doi:10.1140/epjqt/s40507-021-00114-x 2021

  3. [2]

    Joseph Bowles, Shahnawaz Ahmed, and Maria Schuld.Better than classical? The subtle art of benchmarking quantum machine learning models. Tech. rep. arXiv: 2403.07059. arXiv, Mar. 2024.doi: 10.48550/arXiv.2403.07059

    arXiv doi:10.48550/arxiv.2403.07059 2024

  4. [3]

    Brennen.An observable measure of entanglement for pure states of multi-qubit systems

    Gavin K. Brennen.An observable measure of entanglement for pure states of multi-qubit systems. Tech. rep. arXiv, Nov. 2003.doi: 10.48550/arXiv.quant- ph/0305094

    doi:10.48550/arxiv.quant- 2003

  5. [4]

    Cecilia Carbonelli, Michael Felderer, Matthias Jung, et al.Challenges for Quan- tum Software Engineering: An Industrial Application Scenario Perspective. Ed. by Iaakov Exman, Ricardo Pérez-Castillo, Mario Piattini, et al. Cham, 2024. doi: 10.1007/978-3-031-64136-7_12

    doi:10.1007/978-3-031-64136-7_12 2024

  6. [5]

    Encoding-dependent generalization bounds for parametrized quantum circuits

    Matthias C. Caro, Elies Gil-Fuster, Johannes Jakob Meyer, et al. “Encoding- dependent generalization bounds for parametrized quantum circuits”. In: Quantum 5 (Nov. 2021). arXiv: 2106.03880, p. 582.doi: 10.22331/q-2021-11-17- 582

    work page Pith review arXiv 2021

  7. [6]

    Multidimensional Fourier series with quantum circuits

    Berta Casas and Alba Cervera-Lierta. “Multidimensional Fourier series with quantum circuits”. In:Physical Review A107.6 (June 2023). arXiv: 2302.03389, p. 062612. doi: 10.1103/PhysRevA.107.062612

    work page Pith review arXiv 2023

  8. [7]

    Cerezo, Martin Larocca, Diego García-Martín, et al.Does provable absence of barren plateaus imply classical simulability? Or, why we need to rethink variational quantum computing

    M. Cerezo, Martin Larocca, Diego García-Martín, et al.Does provable absence of barren plateaus imply classical simulability? Or, why we need to rethink variational quantum computing. Tech. rep. arXiv: 2312.09121. arXiv, Dec. 2023. doi: 10.48550/arXiv.2312.09121

    arXiv doi:10.48550/arxiv.2312.09121 2023

Show all 51 references
  1. [8]

    Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits

    M. Cerezo, Akira Sone, Tyler Volkoff, et al. “Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits”. In:Nature Communica- tions 12.1 (Mar. 2021). arXiv: 2001.00550, p. 1791.doi: 10.1038/s41467-021- 21728-w

    2021 arXiv

  2. [9]

    Random Quantum Circuits Transform Local Noise into Global White Noise

    Alexander M. Dalzell, Nicholas Hunter-Jones, and Fernando G. S. L. Brandão. “Random Quantum Circuits Transform Local Noise into Global White Noise”. In: Communications in Mathematical Physics405.3 (2024). doi: 10.1007/s00220- 024-04958-z

    2024 doi

  3. [10]

    Spectral Bias in Variational Quantum Machine Learning

    Callum Duffy and Marcin Jastrzebski. “Spectral Bias in Variational Quantum Machine Learning”. In:arXiv preprint arXiv:2506.22555(2025)

    2025

  4. [11]

    Rudolph, Ross Duncan, et al.Classical simulations of noisy variational quantum circuits

    Enrico Fontana, Manuel S. Rudolph, Ross Duncan, et al.Classical simulations of noisy variational quantum circuits. Tech. rep. arXiv: 2306.05400. arXiv, June

    arXiv

  5. [12]

    Enrico Fontana, Ivan Rungger, Ross Duncan, et al.Spectral analysis for noise diagnostics and filter-based digital error mitigation. Tech. rep. arXiv: 2206.08811. arXiv, Nov. 2022.doi: 10.48550/arXiv.2206.08811. 22

    2022 doi arXiv

  6. [13]

    Out of Tune: Demys- tifying Noise-Effects on Quantum Fourier Models

    Maja Franz and Melvin Strobl.Reproduction Package for "Out of Tune: Demys- tifying Noise-Effects on Quantum Fourier Models". Apr. 2025. doi: 10.5281/ zenodo.15211317

    2025

  7. [15]

    Uncovering instabilities in variational-quantum deep Q-networks

    Maja Franz, Lucas Wolf, Maniraman Periyasamy, et al. “Uncovering instabilities in variational-quantum deep Q-networks”. In:Journal of the Franklin Institute 360.17 (2023), pp. 13822–13844.doi: https://doi.org/10.1016/j.jfranklin.2022.08. 021

    2023 doi

  8. [16]

    Modeling and sim- ulating the noisy behavior of near-term quantum computers

    Konstantinos Georgopoulos, Clive Emary, and Paolo Zuliani. “Modeling and sim- ulating the noisy behavior of near-term quantum computers”. In:Physical Review A 104.6 (Dec. 2021). Publisher: American Physical Society (APS), p. 062432. doi: 10.1103/physreva.104.062432

    2021 doi

  9. [17]

    Quantum Data Encoding Patterns and their Consequences

    Martin Gogeissl, Hila Safi, and Wolfgang Mauerer. “Quantum Data Encoding Patterns and their Consequences”. In:Proceedings of the 1st Workshop on Quan- tum Computing and Quantum-Inspired Technology for Data-Intensive Systems and Applications. Q-Data ’24. Santiago, AA, Chile: As...

    2024

  10. [18]

    Effects of Imperfections on Quantum Algorithms: A Software Engineering Perspective

    Felix Greiwe, Tom Krüger, and Wolfgang Mauerer. “Effects of Imperfections on Quantum Algorithms: A Software Engineering Perspective”. In:2023 IEEE International Conference on Quantum Software (QSW). 2023, pp. 31–42.doi: 10.1109/QSW59989.2023.00014

    2023

  11. [19]

    arXiv:2209.10345 [quant-ph]

    Dirk Heimann, Gunnar Schönhoff, Elie Mounzer, et al.Learning capability of parametrized quantum circuits. arXiv:2209.10345 [quant-ph]. Mar. 2024.doi: 10.48550/arXiv.2209.10345

    2024 doi arXiv

  12. [20]

    Let Quantum Neural Networks Choose Their Own Frequencies

    Ben Jaderberg, Antonio A. Gentile, Youssef Achari Berrada, et al. “Let Quantum Neural Networks Choose Their Own Frequencies”. In:Physical Review A109.4 (Apr. 2024). arXiv:2309.03279 [quant-ph], p. 042421.doi: 10.1103/PhysRevA. 109.042421

    2024 arXiv

  13. [21]

    Uhrig, and Benedikt Fauseweh.Entanglement-informed Construction of Variational Quantum Circuits

    Alina Joch, Götz S. Uhrig, and Benedikt Fauseweh.Entanglement-informed Construction of Variational Quantum Circuits. arXiv:2501.17533 [quant-ph]. Jan

    arXiv

  14. [22]

    Kingma and Jimmy Ba.Adam: A Method for Stochastic Optimization

    Diederik P. Kingma and Jimmy Ba.Adam: A Method for Stochastic Optimization. 2017

    2017

  15. [23]

    On Information and Sufficiency

    S. Kullback and R. A. Leibler. “On Information and Sufficiency”. In:The Annals of Mathematical Statistics22.1 (Mar. 1951). Publisher: Institute of Mathematical Statistics, pp. 79–86.doi: 10.1214/aoms/1177729694

    1951

  16. [24]

    Jonas Landman, Slimane Thabet, Constantin Dalyac, et al.Classically Approxi- mating Variational Quantum Machine Learning with Random Fourier Features. Tech. rep. arXiv: 2210.13200. arXiv, Oct. 2022.doi: 10.48550/arXiv.2210.13200

    2022 doi arXiv

  17. [25]

    Barren plateaus in variational quantum computing

    Martín Larocca, Supanut Thanasilp, Samson Wang, et al. “Barren plateaus in variational quantum computing”. en. In:Nature Reviews Physics(Mar. 2025). Publisher: Nature Publishing Group, pp. 1–16.doi: 10.1038/s42254-025-00813-9. 23

    2025 doi

  18. [26]

    Make Some Noise! Measuring Noise Model Quality in Real-World Quantum Software

    Stefan Raimund Maschek, Jürgen Schwittalla, Maja Franz, et al. “Make Some Noise! Measuring Noise Model Quality in Real-World Quantum Software”. en. In: Proceedings of the IEEE International Conference on Quantum Software (QSW). July 2025

    2025

  19. [27]

    1-2-3 Reproducibility for Quantum Software Experiments

    Wolfgang Mauerer and Stefanie Scherzinger. “1-2-3 Reproducibility for Quantum Software Experiments”. In:2022 IEEE International Conference on Software Analysis, Evolution and Reengineering (SANER). 2022, pp. 1247–1248. doi: 10.1109/SANER53432.2022.00148

    2022

  20. [28]

    Global entanglement in multiparticle systems

    David A. Meyer and Nolan R. Wallach. “Global entanglement in multiparticle systems”. In:Journal of Mathematical Physics43.9 (Sept. 2002), pp. 4273–4278. doi: 10.1063/1.1497700

    2002 doi

  21. [29]

    arXiv:2403.09417

    Hela Mhiri, Leo Monbroussou, Mario Herrero-Gonzalez, et al.Constrained and Vanishing Expressivity of Quantum Fourier Models. arXiv:2403.09417. Mar. 2024. doi: 10.48550/arXiv.2403.09417

    2024 doi arXiv

  22. [30]

    Fourier expansion in variational quantum algorithms

    Nikita A. Nemkov, Evgeniy O. Kiktenko, and Aleksey K. Fedorov. “Fourier expansion in variational quantum algorithms”. In:Physical Review A108.3 (Sept. 2023). arXiv:2304.03787 [quant-ph], p. 032406.doi: 10.1103/PhysRevA.108. 032406

    2023 arXiv

  23. [31]

    Data re- uploading for a universal quantum classifier

    Adrián Pérez-Salinas, Alba Cervera-Lierta, Elies Gil-Fuster, et al. “Data re- uploading for a universal quantum classifier”. In:Quantum 4 (Feb. 2020). arXiv: 1907.02085, p. 226.doi: 10.22331/q-2020-02-06-226

    2020 arXiv

  24. [33]

    Plenio and S

    Martin B. Plenio and S. Virmani.An introduction to entanglement measures. arXiv:quant-ph/0504163. June 2006.doi: 10.48550/arXiv.quant-ph/0504163

    2006 doi arXiv

  25. [34]

    A Lie Algebraic Theory of Barren Plateaus for Deep Parameterized Quantum Circuits

    Michael Ragone, Bojko N. Bakalov, Frédéric Sauvage, et al. “A Lie Algebraic Theory of Barren Plateaus for Deep Parameterized Quantum Circuits”. In:Nature Communications 15.1 (Aug. 2024). arXiv:2309.09342 [quant-ph], p. 7172.doi: 10.1038/s41467-024-49909-3

    2024 arXiv

  26. [35]

    arXiv:2407.17204 [quant- ph]

    Tobias Rohe, Daniëlle Schuman, Jonas Nüßlein, et al.The Questionable Influence of Entanglement in Quantum Optimisation Algorithms. arXiv:2407.17204 [quant- ph]. July 2024.doi: 10.48550/arXiv.2407.17204

    2024 doi arXiv

  27. [36]

    Schreiber, Jens Eisert, and Johannes Jakob Meyer.Classical surrogates for quantum learning models

    Franz J. Schreiber, Jens Eisert, and Johannes Jakob Meyer.Classical surrogates for quantum learning models. Tech. rep. arXiv: 2206.11740. arXiv, June 2022. doi: 10.48550/arXiv.2206.11740

    2022 doi arXiv

  28. [37]

    Circuit-centric quantum classifiers

    Maria Schuld, Alex Bocharov, Krysta Svore, et al. “Circuit-centric quantum classifiers”. In:Physical Review A101.3 (Mar. 2020). arXiv:1804.00633 [quant-ph], p. 032308. doi: 10.1103/PhysRevA.101.032308

    2020 arXiv

  29. [38]

    Is Quantum Advantage the Right Goal for Quantum Machine Learning?

    Maria Schuld and Nathan Killoran. “Is Quantum Advantage the Right Goal for Quantum Machine Learning?” In:PRX Quantum 3 (3 July 2022), p. 030101. doi: 10.1103/PRXQuantum.3.030101. 24

    2022 doi

  30. [39]

    Maria Schuld and Francesco Petruccione.Supervised Learning with Quantum Computers. en. Quantum Science and Technology. Cham: Springer International Publishing, 2018. doi: 10.1007/978-3-319-96424-9

    2018 doi

  31. [40]

    The effect of data encoding on the expressive power of variational quantum machine learning models

    Maria Schuld, Ryan Sweke, and Johannes Jakob Meyer. “The effect of data encoding on the expressive power of variational quantum machine learning models”. In:Physical Review A103.3 (Mar. 2021). arXiv: 2008.08605, p. 032430. doi: 10.1103/PhysRevA.103.032430

    2021 arXiv

  32. [41]

    Reformulation of the No-Free- Lunch Theorem for Entangled Datasets

    Kunal Sharma, M. Cerezo, Zoë Holmes, et al. “Reformulation of the No-Free- Lunch Theorem for Entangled Datasets”. In:Physical Review Letters 128.7 (Feb. 2022). Publisher: American Physical Society, p. 070501.doi: 10.1103/ PhysRevLett.128.070501

    2022

  33. [43]

    2025.doi: 10.48550/arXiv.2506.06695

    Melvin Strobl, Maja Franz, Eileen Kuehn, et al.QML Essentials – A framework for working with Quantum Fourier Models. 2025.doi: 10.48550/arXiv.2506.06695

    2025 doi

  34. [44]

    Potential and limitations of random Fourier features for dequantizing quantum machine learning

    Ryan Sweke, Erik Recio-Armengol, Sofiene Jerbi, et al. “Potential and limitations of random Fourier features for dequantizing quantum machine learning”. In: Quantum 9 (Feb. 2025). arXiv:2309.11647 [quant-ph], p. 1640.doi: 10.22331/q- 2025-02-20-1640

    2025 arXiv

  35. [45]

    Approximating under the Influence of Quantum Noise and Compute Power

    Simon Thelen, Hila Safi, and Wolfgang Mauerer. “Approximating under the Influence of Quantum Noise and Compute Power”. In:2024 IEEE International Conference on Quantum Computing and Engineering (QCE). Vol. 02. 2024, pp. 274–279. doi: 10.1109/QCE60285.2024.10291

    2024

  36. [46]

    Noise-Induced Barren Plateaus in Variational Quantum Algorithms

    Samson Wang, Enrico Fontana, M. Cerezo, et al. “Noise-Induced Barren Plateaus in Variational Quantum Algorithms”. In:Nature Communications12.1 (Nov. 2021). arXiv: 2007.14384, p. 6961.doi: 10.1038/s41467-021-27045-6

    2021 arXiv

  37. [47]

    Transition role of entangled data in quantum machine learning

    Xinbiao Wang, Yuxuan Du, Zhuozhuo Tu, et al. “Transition role of entangled data in quantum machine learning”. en. In:Nature Communications15.1 (May 2024). Publisher: Nature Publishing Group, p. 3716.doi: 10.1038/s41467-024-47983-1

    2024 doi

  38. [48]

    Scherer.Fourier Anal- ysis of Variational Quantum Circuits for Supervised Learning

    Marco Wiedmann, Maniraman Periyasamy, and Daniel D. Scherer.Fourier Anal- ysis of Variational Quantum Circuits for Supervised Learning. arXiv:2411.03450. Nov. 2024. doi: 10.48550/arXiv.2411.03450

    2024 doi arXiv

  39. [49]

    QPU-System Co-design for Quantum HPC Accelerators

    Karen Wintersperger, Hila Safi, and Wolfgang Mauerer. “QPU-System Co-design for Quantum HPC Accelerators”. In:Architecture of Computing Systems. Ed. by Martin Schulz, Carsten Trinitis, Nikela Papadopoulou, et al. Cham: Springer International Publishing, 2022, pp. 100–114

    2022

  40. [50]

    Entanglement of Formation of an Arbitrary State of Two Qubits

    William K. Wootters. “Entanglement of Formation of an Arbitrary State of Two Qubits”. In:Physical Review Letters80.10 (Mar. 1998). arXiv:quant-ph/9709029, pp. 2245–2248. doi: 10.1103/PhysRevLett.80.2245

    1998 arXiv

  41. [51]

    arXiv:2501.05694 [quant-ph]

    Zoltán Zimborás, Bálint Koczor, Zoë Holmes, et al.Myths around quantum computation before full fault tolerance: What no-go theorems rule out and what they don’t. arXiv:2501.05694 [quant-ph]. Jan. 2025.doi: 10.48550/arXiv.2501. 05694. 25 A Noise Models In the following, we desc...

    2025 doi arXiv

  42. [2023]

    doi: 10.48550/arXiv.2306.05400

    doi arXiv

  43. [2025]

    doi: 10.48550/arXiv.2501.17533

    doi arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.