Pith. sign in

REVIEW 3 major objections 5 minor 54 references

Quantum Spectral Model: Data Reuploading with Input-Conditioned Frequency Support

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read Given matrix input M, the QSM data-encoding unitary is exp(-iH(M)t/2) with H_sym(M)=(M+M^T)/2 or H_block(M)=[[0,M],[M^T,0]]. The paper shows that, for one upload, the output is a truncated Fourier series in t with maximal support consisting

desk verdict Solid new Fourier-support theory for Hamiltonian-based encoders; the empirical leadership claim is honest but confounded by resource differences, so referees should require a same-resource control. read the letter →

arxiv 2607.22516 v1 pith:G6ZYVAJ4 submitted 2026-07-24 quant-ph cs.AIcs.LG

Peiyong Wang , Udaya Parampalli , Casey R. Myers This is my paper
classification quant-phcs.AIcs.LG MSC 81P6815A18 PACS 03.67.-a
keywords quantummachinelearningdatareuploadingHamiltonianembeddingspectralinductivebiasFourierrepresentationsingularvaluedecompositioneigengapsPendigits
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

Quantum data encoding can carry an inductive bias matched to matrix-structured inputs: instead of encoding features through coordinate-wise rotations, a Quantum Spectral Model (QSM) constructs the generator of the data-encoding unitary directly from each input matrix. The paper claims that the observable output of such a model becomes a truncated Fourier series in the upload time whose candidate frequencies are input-dependent spectral gaps, while input-dependent spectral subspaces help set the coefficients. It further claims that at the largest depth tested (32), a QSM variant records the highest mean test accuracy among all tested quantum models on all four benchmarks, and that ablations show a task-dependent reversal between spectral values and spectral subspaces. A sympathetic reader should care because this turns data encoding into a principled choice of representation: the model can expose the spectral structure that the task rewards.

What carries the argument

The central object is the sample-conditioned Hamiltonian H(M), used as the generator of the data-encoding unitary within a standard upload-mixer data-reuploading circuit. Its spectral decomposition supplies the phase carriers: exponentials exp(-i lambda_a t/2) for a symmetric Hamiltonian, and exp(-i sigma_j t/2) for a block Hamiltonian. Taking pairwise differences of the generator eigenvalues yields the maximal one-upload Fourier support—eigengaps for the symmetric variant, signed singular-value gaps for the block variant—while the spectral projectors, together with the initial state, mixer, and observable, determine which of these carriers are actually realized and with what coefficients. T

What would settle it

Take a small matrix with known singular values, train a block-Hamiltonian QSM, and numerically Fourier-transform the observable output in the upload time; if any frequency appears that is not a signed singular-value gap (half-difference, half-sum, or half-singular-value term) of that matrix, Equation (24)'s support characterisation is wrong. Alternatively, re-run the depth-32 comparisons with matched qubit and parameter counts; if a rotation encoder then reaches or exceeds the QSM means, the inductive-bias attribution is unsupported.

Watch

Extended reading notes

Core claim

Given matrix input M, the QSM data-encoding unitary is exp(-iH(M)t/2) with H_sym(M)=(M+M^T)/2 or H_block(M)=[[0,M],[M^T,0]]. The paper shows that, for one upload, the output is a truncated Fourier series in t with maximal support consisting of the half-eigengaps of H_sym(M), or the signed half-differences and half-sums of singular values of M (plus half-singular-value terms if zero modes exist). Coefficients are formed by the spectral projectors, the initial state, the mixer, and the readout observable. Because H(M) depends on M, the candidate frequencies vary per sample; the patch-local variant uses four 2x2 block Hamiltonians and gets a Cartesian-product support.

Load-bearing premise

The empirical conclusion rests on comparing the full encoder-mixer models as implemented, with qubit counts ranging from 2-4 for global QSMs to 16 for rotation encoders and parameter counts differing by an order of magnitude; if unequal resource counts, not the encoder construction, explain the accuracy ordering, the paper's main empirical claim collapses.

Editorial extensions

If this is right

  • If the Fourier support characterisation is right, then no trained symmetric-QSM output can contain a frequency outside the half-eigengaps of the input matrix, and no block-QSM output can contain a frequency outside the signed singular-value gaps; this gives a testable, sample-by-sample spectral fingerprint of the model.
  • Data encoders no longer need to be limited to a globally shared or globally learned frequency grid; each input matrix can determine its own available phase carriers, so the inductive bias is adapted at the sample level.
  • At depth 32 the paper reports a QSM variant as top mean test accuracy among the tested quantum models on all four benchmarks (patch-local block-Hamiltonian on the two Pendigits representations; global block-Hamiltonian on the two synthetic spectral tasks).
  • The ablation result implies that no single spectral component is universally sufficient: for Pendigits, sample-dependent spectral subspaces carry most of the useful information, while for the synthetic eigengap and singular-value tasks the spectral values themselves are the decisive component.

Reading between the lines

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

  • My inference beyond the paper: if this construction is taken as a template, the design lesson is to pick the generator whose spectrum matches the informative statistics of the data family; one could test this by building QSMs from other input-derived operators (covariance matrices, graph Laplacians of patch graphs) and checking whether sample-conditioned carriers continue to beat rotation gates.
  • My inference: the paper leaves the causal role of locality unresolved; a natural follow-up is a resource-matched re-run—same qubit count and comparable parameter count for global vs patch-local QSMs—to see whether the depth-32 ordering survives once width and parameter totals are held fixed.
  • My inference: the value-versus-subspace reversal suggests an adaptive model that chooses the spectral component to preserve based on task statistics; this could be formalised as a meta-learning or routing problem, but the paper does not propose it.
  • Manuscript-disclosed limitation worth flagging: Section 5 explicitly says the comparisons are 'as implemented' rather than isolating the encoder from resource differences, and Appendix F/G disclose that some archived diagnostics came from a dirty worktree (the four depth-32 winner aggregates and global-Hblock maxima are marked clean). A clean-tree, resource-matched reproduction is needed before th
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

3 major / 5 minor

Summary. The paper introduces Quantum Spectral Models (QSMs), a data-reuploading family in which the generator of the encoding unitary is built from each input matrix. Three variants are studied: symmetric Hamiltonian H_sym=(M+M^T)/2, block Hamiltonian H_block=[[0,M],[M^T,0]], and a non-overlapping patch-local block Hamiltonian. The central formal contribution, Eqs. (17)-(25), derives the maximal one-upload candidate Fourier support as eigengaps of H_sym or signed singular-value gaps of H_block, with Fourier coefficients depending on input-derived spectral projectors, the initial state, the trainable mixer, and the readout. Empirically, the paper reports that at reuploading depth 32, a QSM variant attains the largest mean test accuracy among the tested quantum models on all four benchmarks (Pendigits DYN and STA4, and two synthetic spectral tasks), and that value-vs-subspace ablations show a task-dependent reversal: subspace-preserving controls dominate on Pendigits while spectral-value-only controls dominate on the synthetic tasks. The body of the paper is notably candid about the limits of the comparisons, but the abstract and conclusion frame the empirical results as evidence for the input-conditioned spectral inductive bias itself.

Significance. The formal analysis is a genuine contribution: Eqs. (17)-(25) are a clean, self-contained spectral-theorem characterization of a novel encoder family, with no hidden fitted constants in the support derivation. The paper ships code, reports full depth sweeps with 20 seeds and standard deviations, discloses data-provenance issues (git_dirty records, Appendix F), and explicitly labels the latent-state diagnostics as descriptive. The value-versus-subspace ablation template is a useful methodological idea for the QML community. However, the significance is materially tempered by three issues. First, the headline depth-32 empirical claim is confounded with resource count and trainability; the paper itself states in Section 5 that the comparisons are 'as implemented, rather than isolating the encoder from all resource differences.' Second, the Fourier support results, while correct, are expansions in the upload-time variable t for each fixed input, not in the input features; the abstract's phrasing invites the standard QFM input-Fourier reading. Third, on the synthetic benchmarks the value-only ablation success is expected by construction, since the labels are defined by the very spectral

major comments (3)
  1. [§5, Table 1, Appendix E; Figure 4a] The depth-32 leadership of the global QSMs is confounded with circuit width, parameter count, and trainability. At L=32 the global QSMs use 2-4 qubits and 512-1472 trainable scalars, while the rotation-gate baselines use 16 qubits and 7200-7712 scalars (Appendix E). Section 5 explicitly disclaims resource isolation ('The results compare the models as implemented, rather than isolating the encoder from all resource differences'). The paper's own gradient diagnostics (Section 6, Figure 4a) show median log10 mixer-gradient variance of roughly -11 to -20 for the 16-qubit rotation encoders versus roughly -1 to -4 for the QSMs, so a narrower-more-trainable-circuit explanation is fully consistent with the accuracy ordering. No same-width control exists for the 2-4 qubit global QSMs (the 8-qubit patch-SU(4) baseline controls the patch QSM, but not the global variants). Since the abstract credits
  2. [§4, Eqs. (17)-(25); Appendix C.1] The Fourier support results are expansions in the upload-time variable t for each fixed input M, not in the input features. The paper states this correctly in Appendix C.1 ('the matrix M is fixed and the upload time is the Fourier variable'), but the abstract and introduction present 'input-conditioned frequency support' in language that naturally reads as a QFM-style input-Fourier statement (Schuld et al., Eq. (4) of this paper). At inference t is a fixed trained parameter, so the support characterization describes the frequency content of the unitary family along a variational parameter; the input-dependence of the trained output enters through the M-dependence of both the phases and the coefficients, and is not itself a truncated Fourier representation. This is a legitimate analytic framework, but the delimitation should be stated in the abstract-level claims, otherwise the central th
  3. [Table 2, Appendix D.2, Table 6] The 'task-dependent reversal' in the ablations is partially a consequence of benchmark construction. On SYNTHETIC EIGENGAP and SYNTHETIC SINGULAR, the labels are defined by the largest eigengap and the leading singular-value sum respectively - exactly the quantities that the spectral-value-only controls feed directly - and the classical MLP trained on those descriptors reaches 98.01% and 98.50% (Table 6). The value-only controls reaching 97.92% and 98.42% (Table 2) is therefore a consistency check of the data-generation rule rather than an empirical discovery about QSM representations. The paper does acknowledge this ('This reversal agrees with the data-generating rules'), but the abstract presents the reversal as shedding new light. The Pendigits half of the reversal is the informative empirical evidence; the synthetic half should be framed as a calibration check.
minor comments (5)
  1. [Table 1, Figure 3] The depth-32 'leads' claim is based on point estimates. Several QSM-vs-QSM gaps are within one standard deviation (e.g., DYN: 92.75±0.84 vs 92.35±5.95; EIGENGAP: 85.15±2.40 vs 82.81±3.16). The QSM-vs-rotation gaps are large and robust, but pairwise significance tests or bootstrapped confidence intervals over the 20 seeds would substantiate the ranking claims.
  2. [Abstract] Please state explicitly in the abstract that the Fourier representation is in the upload-time variable, e.g., 'for each input matrix, the output as a function of the upload time admits a truncated Fourier representation whose candidate carriers are input-dependent spectral gaps.'
  3. [Appendix C.3] Typo: 'Lloydet al.' should be 'Lloyd et al.'
  4. [§4, Eq. (20)] Consider defining 'maximal candidate support' once near Eq. (7) and distinguishing it consistently from 'realised support' throughout; the distinction is used heavily and the current text introduces it somewhat informally.
  5. [Appendix D.2] The description of the synthetic data generation is clear, but the sentence 'In the clean limit, the input-derived symmetric Hamiltonian has eigenvalues λj(Si)' should note that the model sees the noisy Xi, not Si; the current wording could be misread as claiming the model has access to the clean spectrum.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Fourier-support derivation is self-contained linear algebra; self-citations are attribution only.

full rationale

The formal derivation chain (Eqs. (17)-(25) and Appendix C) is self-contained: substituting the spectral decompositions of Hsym and Hblock into the one-upload expectation is a direct expansion of the standard Fourier representation of data-reuploading circuits, and the candidate supports follow as eigengaps or signed singular-value gaps by construction from the generator eigenvalues. No fitted parameter is relabeled as a prediction, and no load-bearing conclusion is imported from the authors' prior work: the citation [49] only attributes the symmetric Hamiltonian embedding, while Appendix C re-derives the relevant spectral facts, and [33] merely notes the block-Hamiltonian form used elsewhere. The empirical leaderboard is explicitly qualified by the paper's own statement that "The results compare the models as implemented, rather than isolating the encoder from all resource differences," so the resource confound is a disclosed limitation rather than a circular step. Similarly, the synthetic benchmarks are constructed from spectral statistics, so spectral-value-only ablations being strong on them is a benchmark-design property, not evidence that the derivation reduces to its own input. No step in the claimed derivation chain is equivalent by definition or by self-citation to its conclusion.

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

The formal derivation relies on standard spectral theory and the established QFM expansion; no hidden fitted constants are introduced. The synthetic benchmarks are designed so that spectral-value controls should succeed, which makes the reverse ablation result on synthetic tasks expected by construction rather than an empirical surprise. The main premise requiring scrutiny is the resource-mismatched comparison protocol.

assumptions (5)
  • standard math Spectral theorem and eigendecomposition of Hermitian matrices; the block matrix [0 M; M^T 0] has eigenvalues equal to the signed singular values of M.
    Used in Eqs (17), (21), and (108); this is standard linear algebra and does not require independent empirical support.
  • domain assumption Data-reuploading observable outputs admit a truncated Fourier series whose frequencies are differences of data-encoding generator eigenvalues (QFM framework).
    Borrowed from Schuld et al. [46]; relied on for Eqs (4)-(9) and for interpreting spectral gaps as candidate carriers.
  • domain assumption The upload time t is treated as the Fourier variable while M is fixed; trained times t_l do not change the sample-conditioned support claim.
    Section 4 and Appendix C.1 make this modelling choice explicit; it is needed for the Fourier expansion to be meaningful.
  • domain assumption Synthetic-task labels computed from clean latent spectra while models observe noisy matrices; the ε=0.05 noise preserves the label-defining spectral signal.
    Appendix D.2 defines the construction; the classical MLP baselines in Table 6 provide indirect support that the signal survives.
  • domain assumption Accuracy ordering across encoders is primarily attributable to the data-encoding construction despite unequal qubit and parameter counts.
    Section 5 compares complete encoder-mixer configurations without resource normalisation; this premise is load-bearing for the empirical leadership claim.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum Spectral Model: Data Reuploading with Input-Conditioned Frequency Support." pith.science (2026). https://pith.science/paper/G6ZYVAJ4

@misc{pith2026260722516,
  author       = {Pith},
  title        = {Pith review of: Quantum Spectral Model: Data Reuploading with Input-Conditioned Frequency Support},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G6ZYVAJ4}},
  note         = {Machine review of arXiv:2607.22516}
}
read the original abstract

A central design principle in modern machine learning and artificial intelligence is to align a model's inductive bias with the structure of its input data. For matrix-valued inputs, relevant matrix-level relationships can be characterised through spectral values and spectral subspaces; however, common coordinate-wise rotation-gate data-encoding unitaries used in most quantum machine learning models do not explicitly construct such a matrix-level representation. We introduce Quantum Spectral Models (QSMs), in which we construct the generator of the data-encoding unitary directly from each input matrix. We study three QSM variants based on symmetric, global block, and non-overlapping patch-local block Hamiltonians. Their outputs admit truncated Fourier representations in which input-dependent spectral gaps supply candidate phase carriers, while spectral subspaces help determine their coefficients. We evaluate the QSMs and comparison quantum models on two matrix representations of Pendigits and two controlled synthetic tasks defined by spectral statistics. At the largest evaluated circuit depth, QSM variants lead the tested quantum models in mean test accuracy across all four benchmarks. The patch-local QSM leads on Pendigits, whereas the global block-Hamiltonian QSM leads on the controlled spectral tasks. Ablations show a task-dependent reversal: subspace-preserving controls perform better on Pendigits, whereas spectral-value-only controls lead among the tested ablations on the synthetic tasks. Together, these results shed new light on quantum machine-learning model design by showing how input-conditioned spectral representations can provide an analysable inductive bias, while offering a broader perspective on structure-aware model design in machine learning and artificial intelligence.

Figures

Figures reproduced from arXiv: 2607.22516 by the authors.

Figure 1
Figure 1. Synthesiser view of the quantum spectral model. Top, conceptual signal flow in a music synthesiser; bottom, the corresponding flow in a quantum spectral model. Reading the panels from left to right, the control signal corresponds to the input matrix, the oscillator bank to the input-derived Hamiltonian, the available tones to its spectrum and spectral gaps, the mixer and filter to the trainable quantum block, and th… view at source ↗
Figure 2
Figure 2. Representative training samples from the real-world and controlled synthetic bench￾marks. (a) Pendigits. One example from each of the ten original classes is shown using two paired representations. The upper row shows the DYN representation, in which eight ordered two-dimensional pen coordinates are connected in their recorded sequence. The lower row shows the corresponding STA4 representation as a 4 × 4 bitmap-like… view at source ↗
Figure 3
Figure 3. (a) Final test accuracy on the Pendigits benchmark as a function of reuploading depth. The two panels show the DYN 8×2 trajectory representation and the STA4 4×4 matrix representation. Each point is the mean over 20 parameter-initialisation seeds, and error bars show one standard deviation across seeds. QSMs lead the tested depth-32 comparisons. The global block-Hamiltonian QSM has the largest observed mean across t… view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: (a) Gradient variance at initialisation as a function of reuploading depth. For each run configuration, gradients are computed on a fixed diagnostic training batch of size 32 for 20 random initialisations. The plotted quantity is the median log10 variance across parame…
Figure 5
Figure 5. Figure 5: (a) Within-minus-between fidelity gap of the final quantum states. For each trained model, we compute the pure-state fidelity kernel Kij = | ⟨ψi |ψj ⟩ |2 on a fixed 32-example validation diagnostic batch and report the mean same-label off-diagonal fidelity minus the me…
Figure 6
Figure 6. Figure 6: This metric records the largest validation accuracy observed at any evaluation checkpoint [PITH_FULL_IMAGE:figures/full_fig_p036_6.png]
Figure 6
Figure 6. Figure 6: (a) Best validation accuracy on the Pendigits benchmark as a function of reuploading depth. Each point is the mean over 20 seeds, and the error bars show one standard deviation. The metric is the maximum validation accuracy observed over the 2000-step training trajecto…
Figure 7
Figure 7. Figure 7: (a) Area under the validation-accuracy training curve on Pendigits. For each run, AUC is computed by trapezoidal integration of validation accuracy over recorded evaluation steps. The plotted values are means over 20 seeds with one-standard-deviation error bars. The pa…
Figure 8
Figure 8. Figure 8: Initialisation RMS gradient as a function of reuploading depth. For each run configuration, [PITH_FULL_IMAGE:figures/full_fig_p041_8.png]
Figure 9
Figure 9. Figure 9: Empirical-Fisher diagnostics at initialisation. For each diagnostic batch, per-sample [PITH_FULL_IMAGE:figures/full_fig_p041_9.png]
Figure 10
Figure 10. Figure 10: Empirical-Fisher diagnostics at the final trained checkpoint. The trace, effective rank, and [PITH_FULL_IMAGE:figures/full_fig_p042_10.png]
Figure 11
Figure 11. Figure 11: Effective rank of the final quantum-state fidelity kernel. Effective rank is computed as tr(K) 2/tr(K2 ) from the eigenvalues of the final fidelity kernel. A high effective rank indicates that the kernel spectrum is spread across many eigenmodes rather than concentrat…
Figure 12
Figure 12. Figure 12: Layerwise within-minus-between fidelity gap. For each post-mixer layer, we compute the pure-state fidelity kernel on the diagnostic validation batch and report the mean same-label off-diagonal fidelity minus the mean different-label off-diagonal fidelity. The curves s…
Figure 13
Figure 13. Figure 13: Layerwise kernel-target alignment. At each post-mixer layer, the fidelity kernel of the diagnostic validation states is centred and aligned with the label-equality kernel. The resulting curves show how class-aligned quantum-state geometry develops across depth. This d…
Figure 14
Figure 14. Figure 14: Layerwise effective rank of the fidelity kernel. Effective rank is computed from the post-mixer fidelity kernel at each layer. The plot measures spectral spread in the kernel, not label alignment. High effective rank alone is not sufficient for high test accuracy in t…
Figure 15
Figure 15. Figure 15: Training trajectory of kernel-target alignment. The curves show centred alignment between the final-state fidelity kernel and the label-equality kernel at saved checkpoints. The diagnostic is evaluated on the same validation batch used in the final latent-state summar…
Figure 16
Figure 16. Figure 16: Relationship between fidelity gap and diagnostic-batch accuracy. Each point cor￾responds to one encoder-depth configuration on the diagnostic validation batch. The horizontal axis is the final within-minus-between fidelity gap, and the vertical axis is final projector…
Figure 17
Figure 17. Figure 17: Adjacent-layer CKA of post-mixer fidelity kernels. For each trained model, we compute centred kernel alignment between the fidelity kernels of consecutive post-mixer layers. High values indicate that adjacent layers preserve a similar batch geometry. These diagnostic …

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

54 extracted references · 8 canonical work pages

  1. [1]

    Alpaydin and Fevzi

    E. Alpaydin and Fevzi. Alimoglu. Pen-Based Recognition of Handwritten Digits. UCI Machine Learning Repository, 1996. DOI: https://doi.org/10.24432/C5MG6K

    doi:10.24432/c5mg6k 1996

  2. [2]

    Quantum variational algorithms are swamped with traps.Nat

    Eric R Anschuetz and Bobak T Kiani. Quantum variational algorithms are swamped with traps.Nat. Commun., 13(1):7760, 15 December 2022. ISSN 2041-1723. doi: 10.1038/ s41467-022-35364-5. URLhttp://dx.doi.org/10.1038/s41467-022-35364-5

    doi:10.1038/s41467-022-35364-5 2022

  3. [3]

    Relational inductive biases, deep learning, and graph networks.arXiv [cs.LG], 4 June 2018

    Peter W Battaglia, Jessica B Hamrick, Victor Bapst, Alvaro Sanchez-Gonzalez, Vinicius Zam- baldi, Mateusz Malinowski, Andrea Tacchetti, David Raposo, Adam Santoro, Ryan Faulkner, Caglar Gulcehre, Francis Song, Andrew Ballard, Justin Gilmer, George Dahl, Ashish Vaswani, Kelsey Allen, Charles Nash, Victoria Langston, Chris Dyer, Nicolas Heess, Daan Wierstra...

    arXiv doi:10.48550/arxiv.1806.01261 2018

  4. [4]

    Spectral methods: crucial for machine learning, natural for quantum computers?arXiv [quant-ph], 25 March

    Vasilis Belis, Joseph Bowles, Rishabh Gupta, Evan Peters, and Maria Schuld. Spectral methods: crucial for machine learning, natural for quantum computers?arXiv [quant-ph], 25 March

  5. [5]

    Spectral networks and locally connected networks on graphs, 2014

    Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally connected networks on graphs, 2014. URLhttps://arxiv.org/abs/1312.6203

    arXiv 2014

  6. [6]

    Fable: Fast approximate quantum circuits for block- encodings

    Daan Camps and Roel Van Beeumen. Fable: Fast approximate quantum circuits for block- encodings. In2022 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 104–113. IEEE, September 2022. doi: 10.1109/qce53715.2022.00029. URL http://dx.doi.org/10.1109/QCE53715.2022.00029

    arXiv 2022

  7. [7]

    Explicit quantum circuits for block encodings of certain sparse matrices, 2023

    Daan Camps, Lin Lin, Roel Van Beeumen, and Chao Yang. Explicit quantum circuits for block encodings of certain sparse matrices, 2023. URLhttps://arxiv.org/abs/2203.10236

    arXiv 2023

  8. [8]

    Variational quantum algorithms.Nat

    M Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J Coles. Variational quantum algorithms.Nat. Rev. Phys., 3(9):625–644, 12 August 2021. ISSN 2522- 5820,2522-5820. doi: 10.1038/s42254-021-00348-9. URL http://dx.doi.org/10.1038/ s42254-021-00348-9

    doi:10.1038/s42254-021-00348-9 2021

Show all 54 references
  1. [9]

    Algorithms for learning kernels based on centered alignment.J

    Corinna Cortes, Mehryar Mohri, and Afshin Rostamizadeh. Algorithms for learning kernels based on centered alignment.J. Mach. Learn. Res., 13(1):795–828, March 2012. ISSN 1532-4435

    2012

  2. [10]

    On kernel-target align- ment

    Nello Cristianini, John Shawe-Taylor, Andre Elisseeff, and Jaz Kandola. On kernel-target align- ment. InProceedings of the 15th International Conference on Neural Information Processing Systems: Natural and Synthetic, NIPS’01, page 367–373, Cambridge, MA, USA, 2001. MIT Press

    2001

  3. [11]

    Quantum generative adversarial networks

    Pierre-Luc Dallaire-Demers and Nathan Killoran. Quantum generative adversarial networks. Phys. Rev. A, 98(1):012324, 23 July 2018. doi: 10.1103/PhysRevA.98.012324. URL http: //dx.doi.org/10.1103/PhysRevA.98.012324

    2018 doi

  4. [12]

    Convolutional neural networks on graphs with fast localized spectral filtering

    Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. InProceedings of the 30th International Conference on Neural Information Processing Systems, NIPS’16, page 3844–3852, Red Hook, NY , US...

    2016

  5. [13]

    Classification with quantum neural networks on near term processors, 2018

    Edward Farhi and Hartmut Neven. Classification with quantum neural networks on near term processors, 2018. URLhttps://arxiv.org/abs/1802.06002. 19

    2018 arXiv

  6. [14]

    A theory of multineuronal dimensionality, dynamics and measurement

    Peiran Gao, Eric Trautmann, Byron Yu, Gopal Santhanam, Stephen Ryu, Krishna Shenoy, and Surya Ganguli. A theory of multineuronal dimensionality, dynamics and measurement. bioRxiv preprint, November 2017. URL https://www.biorxiv.org/content/10.1101/ 214262v2

    2017

  7. [15]

    Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

    András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, New York, NY , USA, 23 June 2019. ...

    2019

  8. [16]

    Practical hamiltonian learning with uni- tary dynamics and gibbs states.Nat

    Andi Gu, Lukasz Cincio, and Patrick J Coles. Practical hamiltonian learning with uni- tary dynamics and gibbs states.Nat. Commun., 15(1):312, 8 January 2024. ISSN 2041- 1723,2041-1723. doi: 10.1038/s41467-023-44008-1. URL http://dx.doi.org/10.1038/ s41467-023-44008-1

    2024 doi

  9. [17]

    Efficient token mixing for transformers via adaptive fourier neural operators

    John Guibas, Morteza Mardani, Zongyi Li, Andrew Tao, Anima Anandkumar, and Bryan Catanzaro. Efficient token mixing for transformers via adaptive fourier neural operators. In International Conference on Learning Representations, 2022. URL https://openreview. net/forum?id=EXHG-A3jlM

    2022

  10. [18]

    Learning quantum hamiltonians from high- temperature gibbs states and real-time evolutions.Nat

    Jeongwan Haah, Robin Kothari, and Ewin Tang. Learning quantum hamiltonians from high- temperature gibbs states and real-time evolutions.Nat. Phys., 20(6):1027–1031, 6 June 2024. ISSN 1745-2473,1745-2481. doi: 10.1038/s41567-023-02376-x. URL http://dx.doi.org/ 10.1038/s41567-02...

    2024 doi

  11. [19]

    Ansatz-free hamiltonian learning with heisenberg-limited scaling.PRX quantum, 6(4): 040315, 22 October 2025

    Hong-Ye Hu, Muzhou Ma, Weiyuan Gong, Qi Ye, Yu Tong, Steven T Flammia, and Susanne F Yelin. Ansatz-free hamiltonian learning with heisenberg-limited scaling.PRX quantum, 6(4): 040315, 22 October 2025. ISSN 2691-3399. doi: 10.1103/j7b8-pb77. URLhttp://dx.doi. org/10.1103/j7b8-pb77

    2025 doi

  12. [20]

    Experimental quantum generative adversarial networks for image generation.Physical Review Applied, 16(2), August 2021

    He-Liang Huang, Yuxuan Du, Ming Gong, Youwei Zhao, Yulin Wu, Chaoyue Wang, Shaowei Li, Futian Liang, Jin Lin, Yu Xu, Rui Yang, Tongliang Liu, Min-Hsiu Hsieh, Hui Deng, Hao Rong, Cheng-Zhi Peng, Chao-Yang Lu, Yu-Ao Chen, Dacheng Tao, Xiaobo Zhu, and Jian-Wei Pan. Experimental q...

    2021 doi

  13. [21]

    Let quantum neural networks choose their own frequencies.Phys

    Ben Jaderberg, Antonio A Gentile, Youssef Achari Berrada, Elvira Shishenina, and Vincent E Elfving. Let quantum neural networks choose their own frequencies.Phys. Rev. A, 109(4): 042421, 22 April 2024. ISSN 2469-9926,2469-9934. doi: 10.1103/physreva.109.042421. URL http://dx.d...

    2024 doi

  14. [22]

    Adam: A method for stochastic optimization.arXiv [cs.LG], 22 December 2014

    Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization.arXiv [cs.LG], 22 December 2014. doi: 10.48550/arXiv.1412.6980. URL http://arxiv.org/abs/1412. 6980

    2014 doi arXiv

  15. [23]

    Similarity of neural network representations revisited

    Simon Kornblith, Mohammad Norouzi, Honglak Lee, and Geoffrey Hinton. Similarity of neural network representations revisited. InInternational Conference on Machine Learning, pages 3519–3529. PMLR, 24 May 2019. URL https://proceedings.mlr.press/v97/ kornblith19a.html

    2019

  16. [24]

    Neural operator: learning maps between function spaces with applications to pdes.J

    Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: learning maps between function spaces with applications to pdes.J. Mach. Learn. Res., 24(1), January 2023. ISSN 1532-4435

    2023

  17. [25]

    Limitations of the em- pirical fisher approximation for natural gradient descent

    Frederik Kunstner, Lukas Balles, and Philipp Hennig. Limitations of the em- pirical fisher approximation for natural gradient descent. InAdvances in Neu- ral Information Processing Systems, volume 32, pages 4156–4167. Curran Asso- ciates, Inc., 2019. URL https://proceedings.ne...

    2019

  18. [26]

    Barren plateaus in variational quantum computing.Nat

    Martín Larocca, Supanut Thanasilp, Samson Wang, Kunal Sharma, Jacob Biamonte, Patrick J Coles, Lukasz Cincio, Jarrod R McClean, Zoë Holmes, and M Cerezo. Barren plateaus in variational quantum computing.Nat. Rev. Phys., 7(4):174–189, 26 March 2025. ISSN 2522- 5820,2522-5820. d...

    2025 doi

  19. [27]

    Gradient-based learning applied to docu- ment recognition.Proc

    Y Lecun, L Bottou, Y Bengio, and P Haffner. Gradient-based learning applied to docu- ment recognition.Proc. IEEE Inst. Electr. Electron. Eng., 86(11):2278–2324, 1998. ISSN 0018-9219,1558-2256. doi: 10.1109/5.726791. URL https://ieeexplore.ieee.org/ document/726791

    1998 doi

  20. [28]

    FNet: Mixing tokens with Fourier transforms

    James Lee-Thorp, Joshua Ainslie, Ilya Eckstein, and Santiago Ontañón. FNet: Mixing tokens with Fourier transforms. In Marine Carpuat, Marie-Catherine de Marneffe, and Ivan Vladimir Meza Ruiz, editors,Proceedings of the 2022 Conference of the North American Chapter of the Assoc...

    2022 doi

  21. [29]

    Bronstein

    Ron Levie, Federico Monti, Xavier Bresson, and Michael M. Bronstein. Cayleynets: Graph convolutional neural networks with complex rational spectral filters.Trans. Sig. Proc., 67(1): 97–109, January 2019. ISSN 1053-587X. doi: 10.1109/TSP.2018.2879624. URL https: //doi.org/10.11...

    2019

  22. [30]

    Fourier neural operator for parametric partial differential equations

    Zongyi Li, Nikola Borislavov Kovachki, Kamyar Azizzadenesheli, Burigede liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. InInternational Conference on Learning Representations, 2021. URLhttps...

    2021

  23. [31]

    Equivariant machine learning on graphs with nonlinear spectral filters

    Ya-Wei Eileen Lin, Ronen Talmon, and Ron Levie. Equivariant machine learning on graphs with nonlinear spectral filters. InThe Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. URLhttps://openreview.net/forum?id=y8P633E5HQ

    2024

  24. [32]

    The cost of removing tunability in quantum data re-uploading.arXiv [quant-ph], 24 June 2026

    Anthony Yuezhang Liu and Lirandë Pira. The cost of removing tunability in quantum data re-uploading.arXiv [quant-ph], 24 June 2026. doi: 10.48550/ARXIV .2606.25598. URL http://arxiv.org/abs/2606.25598

    2026 doi arXiv

  25. [33]

    Arvidsson-Shukur

    Seth Lloyd, Samuel Bosch, Giacomo De Palma, Bobak Kiani, Zi-Wen Liu, Milad Marvian, Patrick Rebentrost, and David M. Arvidsson-Shukur. Quantum polar decomposition algorithm,

  26. [34]

    Quantum embeddings for machine learning, 2020

    Seth Lloyd, Maria Schuld, Aroosa Ijaz, Josh Izaac, and Nathan Killoran. Quantum embeddings for machine learning, 2020. URLhttps://arxiv.org/abs/2001.03622

    2020 arXiv

  27. [35]

    Guang Hao Low and Isaac L. Chuang. Optimal hamiltonian simulation by quantum signal processing.Physical Review Letters, 118(1), January 2017. ISSN 1079-7114. doi: 10.1103/ physrevlett.118.010501. URLhttp://dx.doi.org/10.1103/PhysRevLett.118.010501

    2017 doi

  28. [36]

    Guang Hao Low and Isaac L. Chuang. Hamiltonian simulation by qubitization.Quantum, 3: 163, July 2019. ISSN 2521-327X. doi: 10.22331/q-2019-07-12-163. URL http://dx.doi. org/10.22331/q-2019-07-12-163

    2019 doi

  29. [37]

    Quantum eigenvalue processing.SIAM J

    Guang Hao Low and Yuan Su. Quantum eigenvalue processing.SIAM J. Comput., 55(1): 135–215, February 2026. ISSN 0097-5397,1095-7111. doi: 10.1137/24m1689363. URL http://dx.doi.org/10.1137/24M1689363

    2026 doi

  30. [38]

    A unified frequency principle for quantum and classical machine learning.arXiv [quant-ph], 6 January 2026

    Rundi Lu, Ruiqi Zhang, Weikang Li, Zhaohui Wei, Dong-Ling Deng, and Zhengwei Liu. A unified frequency principle for quantum and classical machine learning.arXiv [quant-ph], 6 January 2026. doi: 10.48550/arXiv.2601.03169. URL http://arxiv.org/abs/2601. 03169

    2026 doi

  31. [39]

    Barren plateaus in quantum neural network training landscapes.Nat

    Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes.Nat. Commun., 9(1):4812, 16 November 2018. ISSN 2041-1723,2041-1723. doi: 10.1038/s41467-018-07090-4. URL https://www.nature.com/...

    2018 doi

  32. [40]

    Constrained and vanishing expressivity of quantum fourier models.Quantum, 9(1847):1847, 3 September 2025

    Hela Mhiri, Leo Monbroussou, Mario Herrero-Gonzalez, Slimane Thabet, Elham Kashefi, and Jonas Landman. Constrained and vanishing expressivity of quantum fourier models.Quantum, 9(1847):1847, 3 September 2025. ISSN 2521-327X. doi: 10.22331/q-2025-09-03-1847. URL http://dx.doi.o...

    2025 doi

  33. [41]

    Data re- uploading for a universal quantum classifier.Quantum, 4:226, 6 February 2020

    Adrián Pérez-Salinas, Alba Cervera-Lierta, Elies Gil-Fuster, and José I Latorre. Data re- uploading for a universal quantum classifier.Quantum, 4:226, 6 February 2020. ISSN 2521- 327X. doi: 10.22331/q-2020-02-06-226. URL https://quantum-journal.org/papers/ q-2020-02-06-226/

    2020 doi

  34. [42]

    Improving language understanding by generative pre-training

    Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. Improving language understanding by generative pre-training. Online, 2018. URL https://cdn.openai.com/ research-covers/language-unsupervised/language_understanding_paper.pdf

    2018

  35. [43]

    On the spectral bias of neural networks.arXiv [stat.ML], 22 June 2018

    Nasim Rahaman, Aristide Baratin, Devansh Arpit, Felix Draxler, Min Lin, Fred A Hamprecht, Yoshua Bengio, and Aaron Courville. On the spectral bias of neural networks.arXiv [stat.ML], 22 June 2018. doi: 10.48550/arXiv.1806.08734. URL http://arxiv.org/abs/1806.08734

    2018 doi arXiv

  36. [44]

    Spectral representations for convolu- tional neural networks

    Oren Rippel, Jasper Snoek, and Ryan P Adams. Spectral representations for convolu- tional neural networks. InAdvances in Neural Information Processing Systems, volume 28, pages 2449–2457, 2015. URL https://proceedings.neurips.cc/paper/2015/hash/ 536a76f94cf7535158f66cfbd4b113b...

    2015

  37. [45]

    Representing data on a quantum computer

    Maria Schuld and Francesco Petruccione. Representing data on a quantum computer. In Maria Schuld and Francesco Petruccione, editors,Machine Learning with Quantum Computers, pages 147–176. Springer International Publishing, Cham, 2021. ISBN 9783030830984. doi: 10.1007/ 978-3-03...

    2021 doi

  38. [46]

    Effect of data encoding on the expressive power of variational quantum-machine-learning models.Phys

    Maria Schuld, Ryan Sweke, and Johannes Jakob Meyer. Effect of data encoding on the expressive power of variational quantum-machine-learning models.Phys. Rev. A, 103(3): 032430, 24 March 2021. ISSN 1050-2947. doi: 10.1103/PhysRevA.103.032430. URL https://link.aps.org/doi/10.110...

    2021 doi

  39. [47]

    SAQNN: Spectral adaptive quantum neural network as a universal approximator.arXiv [quant-ph], 10 February 2026

    Jialiang Tang, Jialin Zhang, and Xiaoming Sun. SAQNN: Spectral adaptive quantum neural network as a universal approximator.arXiv [quant-ph], 10 February 2026. doi: 10.48550/arXiv. 2602.09718. URLhttp://arxiv.org/abs/2602.09718

    2026 doi

  40. [48]

    Exponential concentra- tion in quantum kernel methods.Nat

    Supanut Thanasilp, Samson Wang, M Cerezo, and Zoë Holmes. Exponential concentra- tion in quantum kernel methods.Nat. Commun., 15(1):5200, 18 June 2024. ISSN 2041- 1723,2041-1723. doi: 10.1038/s41467-024-49287-w. URL http://dx.doi.org/10.1038/ s41467-024-49287-w

    2024 doi

  41. [49]

    Quantum hamiltonian embedding of images for data reuploading classifiers.Quantum Mach

    Peiyong Wang, Casey R Myers, Lloyd C L Hollenberg, and Udaya Parampalli. Quantum hamiltonian embedding of images for data reuploading classifiers.Quantum Mach. Intell., 7 (1):35, June 2025. ISSN 2524-4906,2524-4914. doi: 10.1007/s42484-025-00247-7. URL http://dx.doi.org/10.100...

    2025 doi

  42. [50]

    Predictive performance of deep quantum data re-uploading models

    Xin Wang, Hanxiao Tao, and Rebing Wu. Predictive performance of deep quantum data re-uploading models. In Aarti Singh, Maryam Fazel, Daniel Hsu, Simon Lacoste-Julien, Felix Berkenkamp, Tegan Maharaj, Kiri Wagstaff, and Jerry Zhu, editors,Proceedings of the 42nd International C...

    2025

  43. [51]

    Frequency principle: Fourier analysis sheds light on deep neural networks.arXiv [cs.LG], 19 January 2019

    Zhi-Qin John Xu, Yaoyu Zhang, Tao Luo, Yanyang Xiao, and Zheng Ma. Frequency principle: Fourier analysis sheds light on deep neural networks.arXiv [cs.LG], 19 January 2019. URL http://arxiv.org/abs/1901.06523

    2019 arXiv

  44. [52]

    apply X to the first qubit and do nothing to the second,

    Chengkai Zhu, Shuyu He, Yu-Ao Chen, Lei Zhang, and Xin Wang. Optimal hamiltonian recognition of unknown quantum dynamics.NPJ Quantum Inf., 12(1):36, 21 January 2026. ISSN 2056-6387,2056-6387. doi: 10.1038/s41534-026-01182-6. URL http://dx.doi.org/ 10.1038/s41534-026-01182-6. 2...

    2026 doi

  45. [2020]

    URLhttps://arxiv.org/abs/2006.00841

    2006 arXiv

  46. [2026]

    URLhttp://arxiv.org/abs/2603.24654

    doi: 10.48550/arXiv.2603.24654. URLhttp://arxiv.org/abs/2603.24654

    doi arXiv

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.