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The power of quantum neural networks
Open accessAbstract
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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28 Own interpretation · 23 Own result · 22 Not checked yet · 9 No basis given · 3 Own method
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“a quantum neural network with an easier data encoding strategy, increases the likelihood of encountering a barren plateau”
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“We find that the effective dimension for a model trained on confusion sets with increasing label corruption, accurately captures generalisation behaviour.”
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“We show that quantum neural networks are able to achieve a significantly better effective dimension than comparable classical neural networks.”
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“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.”
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“This behaviour is observed across all classical network configurations that we consider.”
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“We find that well-designed quantum neural networks are able to achieve a higher capacity and faster training ability than comparable classical feedforward neural networks.”
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“which we find adds substantial power to a model (as seen in Section 4.2)”
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“Theorem B.5 (Generalisation bound for effective dimension without rank constraint) . Let Θ = [ -1 , 1] d and consider a statistical model M Θ := { p ( · , · ; θ ) : θ ∈ Θ } satisfying (4) . For κ ∈ N , let M ( κ ) Θ := { p ( κ ) ( · , · ; θ ) : θ ∈ Θ } be the discretised form as described above. Let d ( κ ) γ,n denote the effective dimension of M ( κ ) Θ as defined in (2) . Furthermore, let L : P(”
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“Lemma B.6. Let Θ = [ -1 , 1] d , and let N ( κ ) ( ε ) denote the number of boxes of side length ε required to cover the parameter set Θ , the length being measured with respect to the metric ˆ F ( κ ) ij ( θ ) . Under the assumption of Theorem B.5, there exists a dimensional constant c d < ∞ such that for γ ∈ (0 , 1] and for all n ∈ N , we have”
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“The quantum neural network, however, maintains a more uniform spectrum over increasing s in and d , showing promise in avoiding unfavourable qualities, such as barren plateaus.”
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“Note that, with this definition, the Fisher information matrix of the discretised model F ( κ ) ( θ ) is constant inside each simplex Θ ℓ .”
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“The average Fisher-Rao norm over 100 trials is roughly 250% higher in the quantum neural network than in the classical neural network, after training the models on a simple dataset for a fixed number of iterations (see Appendix C.3 for details).”
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“The classical model never achieves an effective dimension greater or equal to the quantum models for the range of finite data considered in this study.”
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“For networks with less trainable parameters, d , the results stabilise with as little as 40 data and parameter samples.”
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“The quantum neural network consistently achieves the highest effective dimension over all ranges of finite data we consider.”
- Also stated in: § 4.2 Capacity analysis · p. 9 · plain
“Notably, the quantum neural network achieves the highest effective dimension over a wide range of data availability, followed by the easy quantum model.”
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“The results are contained in Figure 3b and the real hardware shows remarkable performance relative to all other models.”
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“This is not the case in the easy quantum model. The Fisher information spectrum becomes more 'barren plateau-like', with the eigenvalues becoming smaller as the number of qubits increase.”
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“In all scenarios, the classical model has a majority of its eigenvalues near or equal to zero, with a few very large eigenvalues.”
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“We note that, by construction, θ ↦→ p ( κ ) ( · , · ; θ ) is still M 1 -Lipschitz continuous.”
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“We can conclude that a model suffering from a barren plateau will have a Fisher information spectrum with an increasing concentration of eigenvalues approaching zero as the number of qubits in the model increase.”
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“The Fisher information spectrum for fully connected feedforward neural networks reveals that the parameter space is flat in most dimensions, and strongly distorted in a few others [34].”
What the sources say
- Not checked yetcites [34] Bishop, C. Exact calculation of the Hessian matrix for the multilayer perceptron. Neural Comput. 4, 494–501 (1992). source
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“train faster than classical models due to their favourable optimisation landscapes, captured by a more evenly spread Fisher information spectrum.”
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“Non-linear feedforward neural networks, which we consider in this study, have particularly degenerate Fisher information matrices [34].”
What the sources say
- Not checked yetcites [34] Bishop, C. Exact calculation of the Hessian matrix for the multilayer perceptron. Neural Comput. 4, 494–501 (1992). source
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“In the quantum realm, the negative effect of barren plateaus on training quantum neural networks has been linked to the Hessian matrix [32].”
What the sources say
- Not checked yetcites [32] Skolik, A., McClean, J. R., Mohseni, M., van der Smagt, P. & Leib, M. Layerwise learning for quantum neural networks. Quant. Mach. Intell. 3, 5 (2021). source
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“The barren plateau result also tells us that Var θ [ ∂ ∂θ j R n ( θ )] ≤ ω S for models in a barren plateau.”
What the sources say
- Not checked yetcites [23] Rissanen, J. J. Fisher information and stochastic complexity. IEEE Trans. Inf. Theory 42, 40–47 (1996). source
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“The effective dimension, which depends on the Fisher information, is used to prove a novel generalisation bound and establish a robust measure of expressibility.”
- Also stated in: § 1 Introduction · p. 2 · plain
“We argue that the effective dimension is a robust capacity measure through proof of a novel generalisation bound with supporting numerical analyses”
- Also stated in: § 3.2 The effective dimension · p. 5 · plain
“the effective dimension can be interpreted as a bounded capacity measure that serves as a useful tool to analyse the power of statistical models.”
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- Also stated in: § 1 Introduction · p. 2 · plain
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In the paper
“Since modern deep neural networks are heavily overparameterised, generalisation bounds based on the VC dimension, and other measures alike, are typically vacuous [17,18].”
What the sources say
- Not checked yetcites [17] 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).
- Not checked yetcites [18] Arora, S., Ge, R., Neyshabur, B. & Zhang, Y. Stronger generalization bounds for deep nets via a compression approach. In Proc. 35th International Conference on Machine Learning Vol. 80, 254–263 (PMLR, 2018); http://proceedings.mlr.press/v80/arora18b.html
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“Theorem 3.2 (Generalisation bound for the effective dimension) . Let Θ = [ -1 , 1] d and consider a statistical model M Θ := { p ( · , · ; θ ) : θ ∈ Θ } satisfying (4) such that the normalised Fisher information matrix ˆ F ( θ ) has full rank for all θ ∈ Θ , and ‖∇ θ log ˆ F ( θ ) ‖ ≤ Λ for some Λ ≥ 0 and all θ ∈ Θ .”
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In the paper
“It was found that the entries of the Hessian vanish exponentially with the size of the system in models that are in a barren plateau.”
What the sources say
- Not checked yetcites [32] Skolik, A., McClean, J. R., Mohseni, M., van der Smagt, P. & Leib, M. Layerwise learning for quantum neural networks. Quant. Mach. Intell. 3, 5 (2021). source
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“Whilst there has been much development in the growing field of quantum machine learning, a systematic study of the trade-offs between quantum and classical models has yet to be conducted [12].”
What the sources say
- Not checked yetcites [12] Aaronson, S. Read the fine print. Nat. Phys. 11, 291–293 (2015). source
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“Minimising the empirical risk with a log-likelihood loss function coincides with the task of minimising the relative entropy D ( ·‖· ) between the distribution induced by applying the neural network to the observed input distribution r and the observed output distribution q .”
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“The generalisation bound in (5) can be modified to hold for a statistical model without a full rank Fisher information matrix.”
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In the paper
“The continuity assumptions of Theorem 3.2 are satisfied for a large class of classical and quantum statistical models [49, 50], as well as many popular loss functions.”
What the sources say
- Not checked yetcites [49] Marrero, C. O., Kieferová, M. & Wiebe, N. Entanglement induced barren plateaus. Preprint at https://arxiv.org/abs/2010.15968 (2020).
- Not checked yetcites [50] Havlíček, V. et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212 (2019). source
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“Further, using the VC dimension to bound generalisation error requires several unrealistic assumptions, including that the model has access to infinite data [15,16].”
What the sources say
- Not checked yetcites [15] Sontag, E. D Neural Networks and Machine Learning 69–95 (Springer, 1998).
- Not checked yetcites [16] Vapnik, V., Levin, E. & Cun, Y. L. Measuring the VC-dimension of a learning machine. Neural Comput. 6, 851–876 (1994). source
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“The full rank assumption on the Fisher information matrix, however, often does not hold in classical models.”
What the sources say
- Not checked yetcites [34] Bishop, C. Exact calculation of the Hessian matrix for the multilayer perceptron. Neural Comput. 4, 494–501 (1992). source
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“For certain loss functions, the Fisher information matrix coincides with the Hessian of the loss function [33].”
What the sources say
- Not checked yetcites [33] Huembeli, P. & Dauphin, A. Characterizing the loss landscape of variational quantum circuits. Quant. Sci. Technol. 6, 025011 (2021). source
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“The effective dimension defined in (2) does not necessarily increase monotonically with the number of data, n .”
- Also stated in: § B.4 A geometric depiction of the effective dimension · p. 17 · plain
“Thus, the effective dimension does not necessarily increase monotonically with the number of data used.”
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- Also stated in: § B.4 A geometric depiction of the effective dimension · p. 17 · plain
In the paper
“Moreover, classical neural networks are known to produce highly degenerate Fisher information matrices, which can significantly slow down training.”
What the sources say
- Not checked yetcites [34] Bishop, C. Exact calculation of the Hessian matrix for the multilayer perceptron. Neural Comput. 4, 494–501 (1992). source
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“It also converges much faster due to its more evenly spread Fisher information spectrum.”
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“This highlights the importance of the feature map which can influence the likelihood of experiencing a barren plateau.”
- Also stated in: § 5 Conclusion · p. 10 · plain
“Changing the feature map, influences the Fisher spectrum and hence, alters the likelihood of encountering a barren plateau.”
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“The higher order feature map used in the quantum neural network seems to structurally change the optimisation landscape and remove the flatness, usually associated with barren plateaus or suboptimal optimisation conditions.”
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“We start be noting that Lemma B.2 remains valid for the discretised setting and under the assumption of Theorem B.5, 27 i.e.,”
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“The quantum neural network, however, maintains a more uniform spectrum over increasing s in and d , showing promise in avoiding unfavourable qualities, such as barren plateaus.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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- It may state two separate claims as one. 27
- Probably not a substantial claim. 18
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Also stated in: § 4.1 The Fisher information spectrum · p. 8 · plain
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In the paper
“Whilst on average, the classical model trains to a lower loss than the easy quantum model, the spread is significantly larger.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Also stated in: § 4.2 Capacity analysis · p. 9 · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
plaindirectn = 100- Asserted: passes — The authors put it forward as true.
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In the paper
“The quantum neural network maintains the lowest loss value on average across all three models, with the lowest spread over all training iterations.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“This occurs after roughly 33 training steps.”
plainSingle case- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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 .”
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The quantum neural network notably has the highest Fisher-Rao norm and lowest training loss on average.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
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In the paper
“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.”
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
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In the paper
“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.”
plainn = 100- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Also stated in: § 4.1 The Fisher information spectrum · p. 8 · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The effective dimension correctly increases as the data becomes 'more random' and is thus, able to accurately capture a model's generalisation behaviour.”
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The number of data observations determines a natural scale or resolution used to observe model space.”
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“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.”
What the sources say
- Not checked yetcites [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 yetcites [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 yetcites [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
boostedcorrelational- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This implies that the loss landscape becomes increasingly flat as the size of the model increases, making optimisation more difficult.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
In the paper
“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].”
What the sources say
- Not checked yetcites [34] Bishop, C. Exact calculation of the Hessian matrix for the multilayer perceptron. Neural Comput. 4, 494–501 (1992). source
boosted- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“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 .”
What the sources say
- Not checked yetcites [23] Rissanen, J. J. Fisher information and stochastic complexity. IEEE Trans. Inf. Theory 42, 40–47 (1996). source
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
In the paper
“In [44], the authors leverage geometric invariances associated with the Fisher information, to produce the Fisher-Rao norm-a robust norm-based capacity measure”
What the sources say
- Not checked yetcites [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
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Also stated in: § 5 Conclusion · p. 10 · hedged
None of the sources cited for this claim have been checked yet.
In the paper
“a quantity that describes the geometry of a model's parameter space and is essential in both statistics and machine learning [20-22].”
What the sources say
- Not checked yetcites [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 yetcites [21] Huang, H.-Y. et al. Power of data in quantum machine learning. Nat. Commun. 12, 2631 (2021). source
- Not checked yetcites [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).
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
In the paper
“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].”
What the sources say
- Not checked yetcites [23] Rissanen, J. J. Fisher information and stochastic complexity. IEEE Trans. Inf. Theory 42, 40–47 (1996). source
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“it is worthwhile to note that the effective dimension does not necessarily increase monotonically with n”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“These types of spectra are known to slow down a model's training and may render optimisation suboptimal [31].”
What the sources say
- Not checked yetcites [31] Volkoff, T. & Coles, P. J. Large gradients via correlation in random parameterized quantum circuits. Quant. Sci. Technol. 6, 025008 (2021). source
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“measures such as the effective dimension have been motivated from an information-theoretic standpoint and depend on the Fisher information”
What the sources say
- Not checked yetcites [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 yetcites [21] Huang, H.-Y. et al. Power of data in quantum machine learning. Nat. Commun. 12, 2631 (2021). source
- Not checked yetcites [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).
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Since the classical models contain highly degenerate Fisher information matrices, the effective dimension converges the slowest, followed by the easy quantum model.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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It may state two separate claims as one.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“A sensitivity analysis is included in Appendix C.1 to verify that 100 parameter samples are reasonable for the models we consider.”
plaindirectn = 100- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the limit n →∞ , all models will converge to an effective dimension equal to the maximum rank of the Fisher information matrix.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
In the paper
“Classically, the observed Fisher information spectrum is known to have undesirable optimisation properties where the outlying eigenvalues slow down training and loss convergence [31].”
What the sources say
- Not checked yetcites [31] Volkoff, T. & Coles, P. J. Large gradients via correlation in random parameterized quantum circuits. Quant. Sci. Technol. 6, 025008 (2021). source
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Rests on an assumption: The feature map used in the quantum neural network is hard to simulate classically.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Rests on an assumption: The feature map used in the quantum neural network is hard to simulate classically.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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It may state two separate claims as one.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
hedgedconditionalSingle case- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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,”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
- Rests on an assumption: The feature map used in the quantum neural network is hard to simulate classically.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For higher d , it is likely that more samples will be needed.”
hedgedconditionaln = 100- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
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.”
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?
The source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“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]
Experimental — accuracy not yet measuredThe check wasn’t confident either way, so it gives no verdict — never disputed.
In the citing paper
“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]
Experimental — accuracy not yet measuredThe check wasn’t confident either way, so it gives no verdict — never disputed.
In the citing paper
“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]
Experimental — accuracy not yet measuredIn the citing paper
“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.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ 3 Information geometry, effective dimension, and trainability of quantum neural networks · p. 3
“We use a straightforward angle encoding scheme, where data points are encoded via RY-gates on each qubit without entangling them, with rotations equal to feature values normalised to [ -1 , 1].”
§ A.1 Specific feature maps · p. 11
“The classically simulable feature map employed in the easy quantum model, is simply the first sets of Hadamard and RZgates, as done in Figure 4 and is not repeated.”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ A.1 Specific feature maps · p. 11
“Then, RZZ-gates are implemented on qubits i, i + j for i ∈ [1 , . . . , S -1] and j ∈ [ i +1 , . . . , S ]”
§ A.1 Specific feature maps · p. 11
“Figure 4 contains a circuit representation of the feature map developed in [39] and used in this study in the quantum neural network model. First, the feature map applies Hadamard gates on each of the S := s in qubits”
Citations
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Cited by
Sorted by citations
Variational quantum algorithms
Cerezo, M., Arrasmith, A., Babbush, R., Benjamin, S. C., Endo, S., Fujii, K., McClean, J. R., Mitarai, K., Yuan, X., Cincio, L., et al. (2021) · Nature Reviews Physics · doi:10.1038/s42254-021-00348-9 · Cited by 3,385
“For instance, it has been shown that quantum neural networks can achieve a significantly better effective dimension than comparable classical neural networks [258].”
BackgroundNoisy intermediate-scale quantum algorithms
Bharti, K., Cervera-Lierta, A., Kyaw, T. H., Haug, T., Alperin-Lea, S., Anand, A., Degroote, M., Heimonen, H., Kottmann, J. S., Menke, T., et al. (2022) · Reviews of Modern Physics · doi:10.1103/revmodphys.94.015004 · Cited by 2,231
The Variational Quantum Eigensolver: A review of methods and best practices
Tilly, J., Chen, H., Cao, S., Picozzi, D., Setia, K., Li, Y., Grant, E., Wossnig, L., Rungger, I., Booth, G. H., et al. (2022) · Physics Reports · doi:10.1016/j.physrep.2022.08.003 · Cited by 1,031
Noise-induced barren plateaus in variational quantum algorithms
Wang, S., Fontana, E., Cerezo, M., Sharma, K., Sone, A., Cincio, L., & Coles, P. J. (2021) · Nature Communications · doi:10.1038/s41467-021-27045-6 · Cited by 773
Challenges and opportunities in quantum machine learning
Cerezo, M., Verdon, G., Huang, H.-Y., Cincio, Ł., & Coles, P. J. (2022) · Nature Computational Science · doi:10.1038/s43588-022-00311-3 · Cited by 862
Metadata
- 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
- Venue
- Nature Computational Science
- Published
- 2021-06-24
- Volume
- 1
- Issue
- 6
- Pages
- 403-409
- ISSN
- 2662-8457 (ISSN-L)
- 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
- green
- arxiv.org
- 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 › Physics and Astronomy › Atomic and Molecular Physics, and Optics › 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
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