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Cited by 1,082Quantum Computing Algorithms and ArchitectureQuantum many-body systemsQuantum Information and Cryptography

The power of quantum neural networks

Open access
Abbas, Amira, Sutter, David, Zoufal, Christa, Lucchi, Aurelien, Figalli, Alessio, Woerner, Stefan·2021·Nature Computational Science
DOI 10.1038/s43588-021-00084-1
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Abstract

It is unknown whether near-term quantum computers are advantageous for machine learning tasks. In this work we address this question by trying to understand how powerful and trainable quantum machine learning models are in relation to popular classical neural networks. We propose the effective dimension-a measure that captures these qualities-and prove that it can be used to assess any statistical model's ability to generalize on new data. Crucially, the effective dimension is a data-dependent measure that depends on the Fisher information, which allows us to gauge the ability of a model to train. We demonstrate numerically that a class of quantum neural networks is able to achieve a considerably better effective dimension than comparable feedforward networks and train faster, suggesting an advantage for quantum machine learning, which we verify on real quantum hardware.

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Written from the paper’s full text.

TL;DR

The paper uses information geometry and the effective dimension to argue that well-designed quantum neural networks can achieve higher capacity and faster training than comparable classical feedforward networks. It links the Fisher information spectrum to barren plateaus and verifies a training advantage on the ibmq montreal 27-qubit device.

Key findings

  • The quantum neural network consistently achieves the highest effective dimension over the finite-data ranges considered, followed by the easy quantum model and then the classical feedforward network; with \(s_{\text{in}}=4\), \(s_{\text{out}}=2\), and \(d=40\), the quantum model’s Fisher information spectrum is more uniform and lacks the classical model’s few very large eigenvalues.
  • In training on the first two classes of the Iris dataset with \(d=8\), 100 iterations, ADAM with initial learning rate \(0.1\), and 100 trials, the quantum neural network reaches the lowest average training loss: \(23.14\%\), versus \(43.05\%\) for the easy quantum model and \(37.90\%\) for the classical network; average Fisher-Rao norms are \(117.84\), \(104.89\), and \(46.45\), respectively.
  • The Fisher information spectrum connects to barren plateaus: the easy, classically simulable feature map becomes more “barren plateau-like” as qubit number increases, while the harder higher-order feature map shows resilience, with eigenvalues not concentrating near zero as system size grows.
  • On real hardware, the quantum neural network is trained on the ibmq montreal 27-qubit device using 4 qubits with linear connectivity; training stabilizes after roughly 33 iterations, and the paper reports performance even better than simulated results.

Methods

  • The paper defines neural networks as statistical models, uses the Fisher information matrix and effective dimension as capacity measures, proves a novel generalisation bound based on effective dimension (Theorem 3.2), and extends it to models without full-rank Fisher information via discretisation.
  • It compares a quantum neural network using a depth-2 higher-order feature map and parameterised RY/CNOT variational form against classical feedforward networks and an “easy” quantum model with classically simulable angle encoding, matching models by number of trainable parameters \(d\), input size \(s_{\text{in}}\), and output size \(s_{\text{out}}\), with \(d \le 100\), \(s_{\text{in}} \in \{4,6,8,10\}\), and \(s_{\text{out}}=2\).
  • Numerical experiments sample 100 parameter sets to estimate Fisher information spectra, train on the first two Iris classes with \(d=8\), 100 iterations, ADAM, learning rate \(0.1\), and 100 trials, and run one hardware training on ibmq montreal with 4 qubits and linear entanglement.

Limitations

  • Stated: the hardware experiment is run only once due to limited hardware availability, and the authors note that analysis of hardware noise and spread over differently sampled initial parameters would make the results more robust.
  • Stated: future work is needed on noise-induced barren plateaus, generalisation on multiple datasets, and larger models; the hard feature map’s classical hardness is conjectured rather than proven.
  • Evident: comparisons are limited to specific feedforward topologies, a small Iris subset, and \(d \le 100\); effective-dimension estimates rely on Monte Carlo sampling and the paper notes that higher dimensions likely require more samples.

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Registry

Crossref · 10.1038/s43588-021-00084-1

Cites

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Claims the paper makes

0 of 30 cited sources checked · 0 supported · 0 flagged

  • Findingcitation
    § 1 Introduction · p. 2

    Quantum neural networks with an easier data encoding strategy have an increased likelihood of encountering a barren plateau.

    “a quantum neural network with an easier data encoding strategy, increases the likelihood of encountering a barren plateau”
    Quote in the paperSubstantial claim
  • Findingown data
    § 3.3 Generalisation error bounds · p. 6

    The effective dimension for a model trained on confusion sets with increasing label corruption accurately captures generalisation behaviour.

    “We find that the effective dimension for a model trained on confusion sets with increasing label corruption, accurately captures generalisation behaviour.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § Abstract · p. 1

    Quantum neural networks are able to achieve a significantly better effective dimension than comparable classical neural networks.

    “We show that quantum neural networks are able to achieve a significantly better effective dimension than comparable classical neural networks.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § 1 Introduction · p. 2

    Quantum neural networks with a harder data encoding strategy show resilience to the barren plateau phenomenon.

    “whilst a harder data encoding strategy shows resilience to the phenomenon.”
    • Also stated in: § Abstract · p. 1 · plain
      “Importantly, certain quantum neural networks can show resilience to this phenomenon and train faster than classical models due to their favourable optimisation landscapes, captured by a more evenly spread Fisher information spectrum.”
    Quote in the paperSubstantial claim
  • Findingown data
    § 4.1 The Fisher information spectrum · p. 8

    The concentration of the classical Fisher information spectrum around zero is observed across all classical network configurations considered.

    “This behaviour is observed across all classical network configurations that we consider.”
    Quote in the paperSubstantial claim
  • May combine two claims. 27
  • Classified as probably not a substantial claim. 18

  • May combine two claims.

    Findingown data
    § 4.1 The Fisher information spectrum · p. 8

    The classical model's Fisher information spectrum is concentrated around zero, with the majority of eigenvalues negligible at order 10^-14 and a few very large eigenvalues.

    “The classical model's Fisher information spectrum is concentrated around zero, where the majority of eigenvalues are negligible 17 , however, there are a few very large eigenvalues.”
    • Also stated in: § 4.1 The Fisher information spectrum · p. 8 · plain
      “Specifically, of the order 10 -14 , i.e., close to machine precision and thus, indistinguishable from zero.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 4.2 Capacity analysis · p. 9

    On average, the classical model trains to a lower loss than the easy quantum model, but with significantly larger spread.

    “Whilst on average, the classical model trains to a lower loss than the easy quantum model, the spread is significantly larger.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data
    § 4.2 Capacity analysis · p. 9

    The quantum neural network's training advantage persists on real quantum hardware, and its performance is even better than simulated results.

    “We further verify the performance of the quantum neural network on real quantum hardware and train the model using the ibmq montreal 27-qubit device, where the training advantage persists.”
    • Also stated in: § 4.2 Capacity analysis · p. 9 · plain
      “We plot the hardware results till they stabilise, at roughly 33 training iterations and find the performance to be even better than the simulated results.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 4.3 Trainability · p. 10

    The quantum neural network trains to a lower loss, faster than the other two models over an average of 100 trials.

    “Using a cross-entropy loss function, optimised with ADAM for a fixed number of training iterations = 100 and an initial learning rate = 0 . 1, the quantum neural network trains to a lower loss, faster than the other two models over an average of 100 trials.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § C.1 Sensitivity analysis for the effective dimension · p. 19

    When higher dimensions are considered, the standard deviation around the results increases, but 100 data and parameter samples are still reasonable given that we consider a maximum of d = 100.

    “When higher dimensions are considered, the standard deviation around the results increases, but 100 data and parameter samples are still reasonable given that we consider a maximum of d = 100.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data
    § 4.2 Capacity analysis · p. 9

    The quantum neural network maintains the lowest loss value on average across all three models, with the lowest spread over all training iterations.

    “The quantum neural network maintains the lowest loss value on average across all three models, with the lowest spread over all training iterations.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data
    § C.4 Training the quantum neural network on real hardware · p. 21

    The hardware training stops after roughly 33 training steps.

    “This occurs after roughly 33 training steps.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § 4.2 Capacity analysis · p. 9

    The quantum neural network has a non-degenerate Fisher information matrix and its effective dimension converges to the maximum effective dimension d.

    “The quantum neural network, on the other hand, has a non-degenerate Fisher information matrix and the effective dimension converges to the maximum effective dimension, d .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § C.2 The Fisher information spectra for varying model size · p. 19

    The easy quantum model has a somewhat uniform spectrum for a smaller input size, but this deteriorates as the input size (also equal to the number of qubits in this particular model) increases.

    “The easy quantum model has a somewhat uniform spectrum for a smaller input size, but this deteriorates as the input size (also equal to the number of qubits in this particular model) increases.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § C.3 Training the models using a simulator · p. 20

    The quantum neural network notably has the highest Fisher-Rao norm and lowest training loss on average.

    “The quantum neural network notably has the highest Fisher-Rao norm and lowest training loss on average.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 4.1 The Fisher information spectrum · p. 8

    The easy quantum model has most of its eigenvalues close to zero, with some large eigenvalues whose magnitudes are not as extreme as the classical model.

    “The easy quantum model also has most of its eigenvalues close to zero, and whilst there are some large eigenvalues, their magnitudes are not as extreme as the classical model.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 4.1 The Fisher information spectrum · p. 8

    The quantum neural network has a more uniform Fisher information spectrum, with no outlying values, and this distribution remains more or less constant as the number of qubits increases.

    “The quantum neural network, on the other hand, has a different Fisher information spectrum. The distribution of eigenvalues is more uniform, with no outlying values and remains more or less constant as the number of qubits increase (see Appendix C.2).”
    • Also stated in: § 4.1 The Fisher information spectrum · p. 8 · plain
      “The distribution of eigenvalues is the most uniform in the quantum neural network, whereas the other models contain mostly small eigenvalues and larger condition numbers.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § B.3 Effective dimension converges to maximal rank of Fisher information matrix · p. 16

    The effective dimension correctly increases as the data becomes 'more random' and is thus able to accurately capture a model's generalisation behaviour.

    “The effective dimension correctly increases as the data becomes 'more random' and is thus, able to accurately capture a model's generalisation behaviour.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Interpretationnone given
    § 3.2 The effective dimension · p. 4

    The number of data observations determines a natural scale or resolution used to observe model space.

    “The number of data observations determines a natural scale or resolution used to observe model space.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning
    § 3.2 The effective dimension · p. 4

    The effective dimension's dependence on data availability helps understand how data availability influences the accurate capture of model complexity.

    “This is beneficial for practical reasons where data is often limited, and can help in understanding how data availability influences the accurate capture of model complexity.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation
    § 3.1 The Fisher information · p. 4

    The Fisher-Rao norm acts as an umbrella for several other existing norm-based measures and has demonstrated desirable properties both theoretically and empirically.

    “Notably, the Fisher-Rao norm acts as an umbrella for several other existing norm-based measures [45-47] and has demonstrated desirable properties both theoretically, and empirically.”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [45] Neyshabur, B., Salakhutdinov, R. R. & Srebro, N. Path-SGD: path-normalized optimization in deep neural networks. In Advances in Neural Information Processing Systems 28, 2422–2430 (NIPS, 2015).

      Not checked in the free run.

    • cites [46] Neyshabur, B., Tomioka, R. & Srebro, N. Norm-based capacity control in neural networks. In Proc. Machine Learning Research Vol. 40, 1376–1401 (PMLR, 2015); http://proceedings.mlr.press/v40/Neyshabur15.html

      Not checked in the free run.

    • cites [47] Bartlett, P. L., Foster, D. J. & Telgarsky, M. J. Spectrally-normalized margin bounds for neural networks. In Advances in Neural Information Processing Systems 30, 6240–6249 (NIPS, 2017); http://papers.nips.cc/paper/7204-spectrally-normalized

      Not checked in the free run.

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    Interpretationreasoning
    § 3.4 The Fisher spectrum and the barren plateau phenomenon · p. 6

    The loss landscape becomes increasingly flat as the size of the model increases, making optimisation more difficult.

    “This implies that the loss landscape becomes increasingly flat as the size of the model increases, making optimisation more difficult.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 4.1 The Fisher information spectrum · p. 8

    The Fisher information matrix of non-linear classical neural networks is known to be highly degenerate, with a few large eigenvalues.

    “This is consistent with results from literature, where the Fisher information matrix of non-linear classical neural networks is known to be highly degenerate, with a few large eigenvalues [34].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [34] Bishop, C. Exact calculation of the Hessian matrix for the multilayer perceptron. Neural Comput. 4, 494–501 (1992). source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § 3.4 The Fisher spectrum and the barren plateau phenomenon · p. 7

    From [23], we are in a barren plateau if, for parameters uniformly sampled from Θ, each element of the gradient of the loss function with respect to θ vanishes exponentially in the number of qubits S.

    “From [23], we know that we are in a barren plateau if, for parameters θ uniformly sampled from Θ, each element of the gradient of the loss function with respect to θ vanishes exponentially in the number of qubits, S .”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [23] Rissanen, J. J. Fisher information and stochastic complexity. IEEE Trans. Inf. Theory 42, 40–47 (1996). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 3.1 The Fisher information · p. 4

    The Fisher-Rao norm is a robust norm-based capacity measure.

    “In [44], the authors leverage geometric invariances associated with the Fisher information, to produce the Fisher-Rao norm-a robust norm-based capacity measure”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [44] Liang, T., Poggio, T., Rakhlin, A. & Stokes, J. Fisher–Rao metric, geometry, and complexity of neural networks. In Proc. Machine Learning Research Vol. 89, 888–896 (PMLR, 2019); http://proceedings.mlr.press/v89/liang19a.html

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § 3.4 The Fisher spectrum and the barren plateau phenomenon · p. 7

    A model with a Fisher information spectrum that is not concentrated around zero, or whose eigenvalues do not vanish as the number of qubits grows, is unlikely to experience a barren plateau.

    “Conversely, a model with a Fisher information spectrum that is not concentrated around zero is unlikely to experience a barren plateau.”
    • Also stated in: § 5 Conclusion · p. 10 · hedged
      “A model with eigenvalues of the Fisher information matrix that do not vanish as the number of qubits grow, is unlikely to suffer from a barren plateau.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § 1 Introduction · p. 2

    The Fisher information describes the geometry of a model's parameter space and is essential in both statistics and machine learning.

    “a quantity that describes the geometry of a model's parameter space and is essential in both statistics and machine learning [20-22].”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [20] Du, Y., Hsieh, M.-H., Liu, T. & Tao, D. Expressive power of parametrized quantum circuits. Phys. Rev. Res. 2, 033125 (2020). source

      Not checked in the free run.

    • cites [21] Huang, H.-Y. et al. Power of data in quantum machine learning. Nat. Commun. 12, 2631 (2021). source

      Not checked in the free run.

    • cites [22] Berezniuk, O., Figalli, A., Ghigliazza, R. & Musaelian, K. A scale-dependent notion of effective dimension. Preprint at https://arxiv.org/abs/2001.10872 (2020).

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § 1 Introduction · p. 2

    Often, quantum neural networks suffer from a barren plateau phenomenon, wherein the loss landscape is perilously flat, and consequently, parameter optimisation is extremely difficult.

    “Often, quantum neural networks suffer from a barren plateau phenomenon, wherein the loss landscape is perilously flat, and consequently, parameter optimisation is extremely difficult [23].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [23] Rissanen, J. J. Fisher information and stochastic complexity. IEEE Trans. Inf. Theory 42, 40–47 (1996). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given
    § 3.3 Generalisation error bounds · p. 6

    The effective dimension does not necessarily increase monotonically with n.

    “it is worthwhile to note that the effective dimension does not necessarily increase monotonically with n”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation
    § 3.4 The Fisher spectrum and the barren plateau phenomenon · p. 6

    These types of spectra slow down a model's training and may render optimisation suboptimal.

    “These types of spectra are known to slow down a model's training and may render optimisation suboptimal [31].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [31] Volkoff, T. & Coles, P. J. Large gradients via correlation in random parameterized quantum circuits. Quant. Sci. Technol. 6, 025008 (2021). source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § 1 Introduction · p. 2

    The effective dimension depends on the Fisher information.

    “measures such as the effective dimension have been motivated from an information-theoretic standpoint and depend on the Fisher information”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [20] Du, Y., Hsieh, M.-H., Liu, T. & Tao, D. Expressive power of parametrized quantum circuits. Phys. Rev. Res. 2, 033125 (2020). source

      Not checked in the free run.

    • cites [21] Huang, H.-Y. et al. Power of data in quantum machine learning. Nat. Commun. 12, 2631 (2021). source

      Not checked in the free run.

    • cites [22] Berezniuk, O., Figalli, A., Ghigliazza, R. & Musaelian, K. A scale-dependent notion of effective dimension. Preprint at https://arxiv.org/abs/2001.10872 (2020).

      Not checked in the free run.

  • May combine two claims.

    Interpretationown data
    § 5 Conclusion · p. 11

    Overall, quantum neural networks can possess a desirable Fisher information spectrum that enables them to train faster and express more functions than comparable classical and quantum models.

    “Overall, we have shown that quantum neural networks can possess a desirable Fisher information spectrum that enables them to train faster and express more functions than comparable classical and quantum models-a promising reveal for quantum machine learning, which we hope leads to further studies on the power of quantum models.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 4.2 Capacity analysis · p. 9

    The effective dimension converges slowest for classical models, followed by the easy quantum model, because classical models contain highly degenerate Fisher information matrices.

    “Since the classical models contain highly degenerate Fisher information matrices, the effective dimension converges the slowest, followed by the easy quantum model.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductown data
    § 4.1 The Fisher information spectrum · p. 8

    A sensitivity analysis verifies that 100 parameter samples are reasonable for the models considered.

    “A sensitivity analysis is included in Appendix C.1 to verify that 100 parameter samples are reasonable for the models we consider.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductnone given
    § B.4 A geometric depiction of the effective dimension · p. 16

    In all experiments conducted in this study, the effective dimension is examined over a wide range of n to ensure it is sufficient in accurately estimating the size of a model.

    “In all experiments conducted in this study, we examine the effective dimension over a wide range of n , to ensure it is sufficient in accurately estimating the size of a model.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § 4.2 Capacity analysis · p. 9

    In the limit n→∞, all models converge to an effective dimension equal to the maximum rank of the Fisher information matrix.

    “In the limit n →∞ , all models will converge to an effective dimension equal to the maximum rank of the Fisher information matrix.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 4.3 Trainability · p. 10

    The observed classical Fisher information spectrum is known to have undesirable optimisation properties where outlying eigenvalues slow down training and loss convergence.

    “Classically, the observed Fisher information spectrum is known to have undesirable optimisation properties where the outlying eigenvalues slow down training and loss convergence [31].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [31] Volkoff, T. & Coles, P. J. Large gradients via correlation in random parameterized quantum circuits. Quant. Sci. Technol. 6, 025008 (2021). source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § C.1 Sensitivity analysis for the effective dimension · p. 19

    The capacity results will be sensitive to the number of data samples used in estimating the Fisher information matrix for a given θ, and to the number of θ samples then used to calculate the effective dimension.

    “The capacity results will, thus, be sensitive to the number of data samples used in estimating the Fisher information matrix for a given θ , and to the number of θ samples then used to calculate the effective dimension.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 5 Conclusion · p. 10

    The quantum neural network with a hard feature map is an example of a model showing resilience to barren plateaus with good trainability, supported by results from real quantum hardware.

    “The quantum neural network with a hard feature map is an example of such a model showing resilience to this phenomenon with good trainability, supported by results from real quantum hardware.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The feature map used in the quantum neural network is hard to simulate classically.

  • May combine two claims.

    Interpretationown data
    § 5 Conclusion · p. 10

    A particular quantum neural network offers advantages from both a capacity and trainability perspective, captured by a high effective dimension and a non-degenerate Fisher information matrix.

    “A particular quantum neural network offers advantages from both a capacity and trainability perspective. These advantages are captured by a high effective dimension and a non-degenerate Fisher information matrix.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The feature map used in the quantum neural network is hard to simulate classically.

  • May combine two claims.

    Interpretationown data
    § 4.3 Trainability · p. 10

    The quantum neural network shows resilience against barren plateaus because its Fisher information eigenvalue distribution remains more or less constant and many eigenvalues are not near zero as system size increases.

    “Upon examining the quantum neural network over an increasing system size (see Appendix C.2), the eigenvalue distribution of the Fisher information matrix remains more or less constant, and a large amount of the eigenvalues are not near zero, thus, the model shows resilience against barren plateaus.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data
    § 5 Conclusion · p. 10

    The effective dimension is a robust capacity measure for quantum and classical models, justified through proof of a novel generalisation bound.

    “In this study, the effective dimension is presented as a robust capacity measure for quantum and classical models, which we justify through proof of a novel generalisation bound.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing). May combine two claims.

    Study design and conductreasoning
    § C.4 Training the quantum neural network on real hardware · p. 21

    Due to limited hardware availability, this experiment is only run once and an analysis of the hardware noise and the spread of the training loss for differently sampled initial parameters would make these results more robust.

    “Due to limited hardware availability, this experiment is only run once and an analysis of the hardware noise and the spread of the training loss for differently sampled initial parameters would make these results more robust.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § B.5 Removing the rank constraint via discretisation · p. 19

    If the model M Θ satisfies an additional regularity assumption ‖∇θ F(θ)‖ ≤ Λ for some Λ ≥ 0 and all θ ∈ Θ, then choosing the discretisation parameter κ ≫ Λ ensures that M Θ ≈ M (κ) Θ and F(θ) ≈ F (κ)(θ); furthermore, √n ≫ κ is required to ensure that the balls used to cover each simplex of the triangulation are smaller than the size of each simplex.

    “Remark B.7 (How to choose the discretisation parameter κ ) . In this remark we discuss conditions such that the generalisation bound of Theorem B.5 for the discretised model M ( k ) Θ is a good approximation to a generalisation bound of the original model M Θ . Assume that the model M Θ satisfies an additional regularity assumption of the form ‖∇ θ ˆ F ( θ ) ‖ ≤ Λ for some Λ ≥ 0 and for all θ ∈ Θ,”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given
    § B.5 Removing the rank constraint via discretisation · p. 17

    The bound ||∇θ log F|| ≤ Λ forces the Fisher information matrix to have constant rank.

    “Indeed, this is a rather strong bound as it forces ˆ F to have constant rank, so it is desirable to find a variant of Lemmas B.1 and B.2 that do not require such an assumption.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationreasoning
    § 5 Conclusion · p. 10

    Replacing the quantum neural network's feature map with one that is easily simulable impairs these advantages.

    “The feature map in the quantum neural network is conjectured to be hard to simulate classically, and replacing it with one that is easily simulable, impairs these advantages.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: The feature map used in the quantum neural network is hard to simulate classically.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning
    § C.1 Sensitivity analysis for the effective dimension · p. 19

    For higher d, it is likely that more samples will be needed.

    “For higher d , it is likely that more samples will be needed.”
    Quote in the paperNot a substantial claim
42 of 42

  • The authors state this as an assumption.

    The statistical model satisfies the regularity assumption (4).

    “We assume the following regularity assumption on the model M Θ := { p ( · , · ; θ ) : θ ∈ Θ } :”

  • The paper relies on this without support.

    The feature map used in the quantum neural network is hard to simulate classically.

    “The feature map in the quantum neural network is conjectured to be hard to simulate classically”
    “The quantum neural network's encoding scheme is conjectured to be difficult.”

  • The paper relies on this without support.

    The capacity of quantum neural networks is not well explored, and no such Fisher information analysis has been done for them.

    “In stark contrast to classical models, understanding the capacity of quantum neural networks is not well explored.”
    “For quantum neural networks, no such analysis has been done.”

  • The authors state this as an assumption.

    The loss function in Theorem 3.2 is bounded.

    “the assumption that the loss function in Theorem 3.2 has to be bounded.”

  • The paper relies on this without support.

    The relative entropy is α-Hölder continuous in its first argument for α∈(0,1) and log-Lipschitz continuous in its first argument.

    “We further note that the relative entropy is α -H¨ older continuous in the first argument for α ∈ (0 , 1). In fact, the relative entropy is even log-Lipschitz continuous in the first argument”

  • The paper relies on this without support.

    In practical applications, the Fisher matrix is evaluated at finitely many points.

    “Our approach to this general problem is based on the idea that, in practical applications, the Fisher matrix is evaluated at finitely many points, so it makes sense to approximate a statistical model with a discretised one where the corresponding Fisher information matrix is piecewise constant.”
6 of 6

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • QNNs exploit quantum superposition and entanglement to process information in ways classical networks cannot, potentially accelerating training times and improving performance on complex data sets.

    “Quantum neural networks (QNNs), for instance, exploit quantum superposition and entanglement to process information in ways that classical networks cannot, potentially accelerating training times and improving performance on complex data sets [18].”

    From Quantum-Trained Convolutional Neural Network for Deepfake Audio Detection (2025) · cites this paper as [18]

    The Introduction (p. 1) says QNNs use superposition and entanglement and describes potential training speed-ups; the Trainability section (p. 10) reports faster training than the classical models tested. But the Introduction says whether QNNs are more powerful than classical networks remains open. The paper does not establish that classical networks cannot process information in those ways or that QNNs improve performance on complex datasets. (LLM, conf=0.87)

    Experimental — accuracy not yet measured

  • Variational approaches have made strides in evaluating or compressing quantum system evolutions.

    “there have been a number of strides in evaluating or compressing these evolutions using variational approaches [90, 91].”

    From Variational quantum algorithms (2021) · cites this paper as [90]

    Section 2 (pp. 2–3) describes variational quantum neural networks and the evolution of an encoded state through a variational circuit. It does not report strides in evaluating or compressing quantum-system evolutions; searches for evolution evaluation and compression found no passage addressing that claim. (LLM, conf=0.75)

    Experimental — accuracy not yet measured

  • Variational methods can be applied incrementally by stepping the electronic wavefunction forward with time-dependent variational principles and sampling forces to move nuclei classically, resulting in Born-Oppenheimer type molecular dynamics.

    “Variational methods can be applied incrementally in these cases, by stepping the electronic wavefunction forward with time-dependent variational principles [90, 91] and sampling the forces [260] to move the nuclei classically, resulting in a Born-Oppenheimer type molecular dynamics.”

    From Variational quantum algorithms (2021) · cites this paper as [90]

    Split into 2 atoms — 0 confirmed, 0 partially supported, 0 disputed, 0 misattributed, 2 unverifiable. Expand the claim to see per-atom verdicts.

    Experimental — accuracy not yet measured

  • Quantum neural networks can be more expressive-even exponentially for specific architectures-than classical counterparts.

    “For a given number of neurons, quantum neural networks can be more expressiveeven exponentially, for specific network architectures -than their classical counterpart.[225][226][227]”

    From Quantum Computing for Molecular Biology** (2023) · cites this paper as [227]

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Conflicting specifications of the easy quantum model's feature map

    Minor

    Section 2 specifies RY-gate angle encoding for the easy quantum model, whereas Appendix A specifies an unrepeated layer of Hadamard and RZ gates. These are different gate sequences for the same feature map.

  • Specified RZZ-gate indices extend beyond the circuit's qubits

    Minor

    The feature map has S qubits, but the stated ranges permit j = S and any i ≥ 1, placing a gate on qubit i + S, beyond qubit S. This conflicts with the description of gates between pairs of the circuit's qubits.

2 of 2

Cites

Flagged first, then by citations

Research works printed but not linked: 13

  1. Goodfellow, I., Bengio, Y. & Courville, A. Deep Learning (MIT Press, 2016); http://www.deeplearningbook.org
  2. Farhi, E. & Neven, H. Classification with quantum neural networks on near term processors. Quant. Rev. Lett. 1, 2 (2020).
  3. Sontag, E. D Neural Networks and Machine Learning 69–95 (Springer, 1998).
  4. Neyshabur, B., Bhojanapalli, S., McAllester, D. & Srebro, N. Exploring generalization in deep learning. In Advances in Neural Information Processing Systems 30, 5947–5956 (NIPS, 2017).
  5. Kunstner, F., Hennig, P. & Balles, L. Limitations of the empirical Fisher approximation for natural gradient descent. In Advances in Neural Information Processing Systems 32 4156–4167 (NIPS, 2019); http://papers.nips.cc/paper/limitations-of-fisher-approximation
  6. Karakida, R., Akaho, S. & Amari, S.-I. Universal statistics of Fisher information in deep neural networks: mean field approach. In Proc. Machine Learning Research Vol. 89, 1032–1041 (PMLR, 2019); http://proceedings.mlr.press/v89/karakida19a.html
  7. Liang, T., Poggio, T., Rakhlin, A. & Stokes, J. Fisher–Rao metric, geometry, and complexity of neural networks. In Proc. Machine Learning Research Vol. 89, 888–896 (PMLR, 2019); http://proceedings.mlr.press/v89/liang19a.html
  8. Neyshabur, B., Salakhutdinov, R. R. & Srebro, N. Path-SGD: path-normalized optimization in deep neural networks. In Advances in Neural Information Processing Systems 28, 2422–2430 (NIPS, 2015).
  9. Neyshabur, B., Tomioka, R. & Srebro, N. Norm-based capacity control in neural networks. In Proc. Machine Learning Research Vol. 40, 1376–1401 (PMLR, 2015); http://proceedings.mlr.press/v40/Neyshabur15.html
  10. Bartlett, P. L., Foster, D. J. & Telgarsky, M. J. Spectrally-normalized margin bounds for neural networks. In Advances in Neural Information Processing Systems 30, 6240–6249 (NIPS, 2017); http://papers.nips.cc/paper/7204-spectrally-normalized
  11. Rissanen, J. J. Fisher information and stochastic complexity. IEEE Trans. Inf. Theory 42, 40–47 (1996). · DOI · Paper page
  12. Marrero, C. O., Kieferová, M. & Wiebe, N. Entanglement induced barren plateaus. Preprint at https://arxiv.org/abs/2010.15968 (2020). · arXiv
  13. Virmaux, A. & Scaman, K. Lipschitz regularity of deep neural networks: analysis and efficient estimation. In Advances in Neural Information Processing Systems 31, 3835–3844 (NIPS, 2018); http://papers.nips.cc/paper/lipschitz-regularity-of-deep-neural-networks

Cited by

Sorted by citations

Related

Authors
Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., & Woerner, S.
Affiliations
  • IBM Research - Zurich (CH)
  • ETH Zurich (CH)
Corresponding author
Woerner, Stefan
ORCID
Venue
Nature Computational Science
Publisher
Springer Science and Business Media LLC
Published
2021-06-24
Volume
1
Issue
6
Pages
403-409
ISSN
2662-8457 (ISSN-L)
DOI
10.1038/s43588-021-00084-1
OpenAlex
W3097610849
Cited by
1,082
References
39
Citations by year
2026: 266 · 2025: 266 · 2024: 216 · 2023: 178 · 2022: 87 · 2021: 69
FWCI
48.59
Type
journal article
Language
en
Open access
Licence
Locations
  • Nature Computational Science (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
International Business Machines Corporation
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
Keywords
quantum neural networks, quantum machine learning, effective dimension, Fisher information, generalization, model capacity, training speed, quantum advantage, feedforward neural networks, quantum computing hardware

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-07.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-07.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

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