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On the learnability of quantum neural networks

Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao

arXiv:2007.12369v1quant-ph

TL;DR

The learnability of hybrid quantum neural networks remains theoretically unclear amid non-convex optimization, finite-measurement error, and gate noise. The paper derives ERM utility bounds and analyzes noisy-QNN learnability, reporting polynomial-sample results for noisy-gate learning and QSQ simulation.

  • Problem

    QNN learnability remains largely unknown because non-convex optimization, finite-measurement estimation error, and unavoidable gate noise complicate its analysis.

    Source: §1 · p. 2

  • Method

    The paper analyzes QNN utility under empirical risk minimization and studies noisy-gate learning through differential privacy and quantum statistical query simulation.

    Source: §1 · pp. 4–5

  • Results

    The paper derives ERM utility bounds and reports that noiseless-QNN PAC learnability transfers to modified noisy QNNs with polynomial samples, while noisy QNNs can simulate QSQ efficiently.

    Source: §1 · pp. 4–5, §6 · p. 11

  • Takeaways & Limitations

    The results support modified noisy QNN implementations on NISQ devices for learning tasks with quantum advantages, including parity, juntas, and DNF.

    Source: §6 · p. 11

    The gate-noise analysis models noise at each circuit depth as depolarization noise, while noting that results extend to more general noisy channels.

    Source: §1 · p. 3

Abstract

from arXiv · show

We consider the learnability of the quantum neural network (QNN) built on the variational hybrid quantum-classical scheme, which remains largely unknown due to the non-convex optimization landscape, the measurement error, and the unavoidable gate errors introduced by noisy intermediate-scale quantum (NISQ) machines. Our contributions in this paper are multi-fold. First, we derive the utility bounds of QNN towards empirical risk minimization, and show that large gate noise, few quantum measurements, and deep circuit depth will lead to the poor utility bounds. This result also applies to the variational quantum circuits with gradient-based classical optimization, and can be of independent interest. We then prove that QNN can be treated as a differentially private (DP) model. Thirdly, we show that if a concept class can be efficiently learned by QNN, then it can also be effectively learned by QNN even with gate noise. This result implies the same learnability of QNN whether it is implemented on noiseless or noisy quantum machines. We last exhibit that the quantum statistical query (QSQ) model can be effectively simulated by noisy QNN. Since the QSQ model can tackle certain tasks with runtime speedup, our result suggests that the modified QNN implemented on NISQ devices will retain the quantum advantage. Numerical simulations support the theoretical results.

1 Introduction

The paper studies learnability of hybrid quantum neural networks, where quantum circuits encode inputs and trainable circuits are optimized using measurement-derived information. It addresses open questions about empirical-risk utility and learnability under finite measurements and gate noise.

1 Introduction

  • QNN learnability remains theoretically open amid non-convex optimization, finite-measurement estimation error, and gate noise; the paper studies utility under ERM.The setup uses square loss with an L2 regularizer and models gate noise at each circuit depth as depolarization noise.Source: §1 · pp. 2–3
  • More measurements, lower gate noise, shallower circuits, and fewer trainable parameters yield better QNN utility bounds for both R1 and R2.The bounds cover typical encoding circuits and trainable circuits composed of parameterized single- and two-qubit gates; they may also inform variational quantum-circuit applications to physical-system properties.Source: §1 · p. 4
  • If noiseless QNNs PAC-learn a concept, a modified QNN with certain noisy gates can also learn it using polynomial samples.The paper relates this result to equal sample complexity for noiseless and noisy QNNs and to preserving quantum advantages on NISQ machines with a modification.Source: §1 · p. 4
  • QNNs with gate noise can be treated as (ε, δ)-differentially private models, supporting the paper’s noisy-QNN learnability result.The stated proof strategy uses the DP property together with shared sample-complexity learnability of private and non-private algorithms.Source: §1 · pp. 4–5
  • A quantum statistical query model can be efficiently simulated by a QNN with noisy gates using polynomial samples.The paper motivates this result by noting that QSQ can efficiently tackle tasks with quantum advantages that are computationally hard for classical SQ models.Source: §1 · p. 5

1.1 Related work

  • The related-work review says earlier quantum learning-theory and QNN studies had not addressed QNN learnability, while other QNN platforms use frameworks incomparable with this hybrid scheme.The review discusses noiseless quantum machines, quantum reservoirs, and quantum annealers as distinct implementation approaches.Source: §1.1 · p. 5

2 Preliminary

The preliminaries establish notation for data dimensions and complexity, then introduce quantum states, operations, measurements, depolarization noise, and differential privacy.

2 Preliminary

  • D denotes the feature dimension, d the number of training parameters, and n the number of training examples; O(·) and Õ(·) denote complexity bounds.The complexity notation hides poly-logarithmic factors.Source: §2 · pp. 5–6
  • Quantum states are represented in Hilbert space, while quantum channels act on states and measurements extract classical outcomes with probabilities Tr(Π_bρ).A qubit has a two-dimensional Hilbert space; measurements are represented by positive semidefinite operators.Source: §2 · p. 6
  • The depolarization channel preserves state ρ with probability 1 − p and otherwise replaces it with the maximally mixed state; differential privacy bounds output-probability changes between neighboring datasets.The DP definition uses datasets at Hamming distance at most one.Source: §2 · p. 6

3 Utility bounds of quantum neural network towards ERM

The section relates noisy, finite-measurement gradients to analytic gradients and uses that relation to derive QNN utility bounds for empirical risk minimization. The bounds worsen with higher gate noise, deeper circuits, fewer measurements, and more trainable parameters.

3 Utility bounds of quantum neural network towards ERM

  • QNN estimates gradients from finite measurement statistics on noisy circuits, while the parameter-shift rule evaluates shifted parameters because unitary gradients generally are not quantum operators.Finite measurements introduce sample error, and gate noise changes the pre-measurement state.Source: §3 · p. 7
  • The estimated gradient is centered around a gate-noise-scaled analytic gradient plus a zero-mean term, making its deviation quantitatively analyzable.The result indicates gate error can be more harmful than finite measurements and may lead to divergence; with maximal effective noise, analytic gradient information is lost.Source: §3 · pp. 7–8
  • Larger measurement count K and batch size B, lower depolarizing error p, fewer parameters d, and shallower circuit depth LQ yield better utility bounds R1 and R2.R1 measures expected distance from a stationary point; R2 measures expected excess empirical risk.Source: §3 · p. 8
  • Smoothness supports analysis of convergence to a stationary point, while the PL condition connects that point to excess empirical risk and thus supports R2.The paper combines these objective-function properties with the gradient relation to obtain the utility bounds.Source: §3 · p. 8
  • The gate-noise analysis is stated for depolarization and is said to extend to a more general noisy channel.The extension is given in Appendix I.Source: §3 · p. 8

4 The learnability of quantum neural networks with noisy gates

The section asks whether noisy QNNs retain the learnability of noiseless QNNs and can capture QSQ learning capabilities. It connects noisy QNNs to differential privacy and establishes polynomial-sample learnability and QSQ simulation results.

4 The learnability of quantum neural networks with noisy gates

  • Noisy QNNs can be treated as differentially private models, providing the bridge used to transfer learnability guarantees from noiseless QNNs.The argument combines this connection with equivalence results for PAC and private PAC sample complexities.Source: §4 · p. 9
  • If noiseless QNNs PAC-learn a concept, modified QNNs with certain noisy gates can also learn it using polynomial samples.The result is presented as ensuring there is no concept class efficiently learnable by noiseless QNNs but computationally hard for noisy QNNs.Source: §4 · p. 9
  • Noisy QNNs can efficiently simulate QSQ learning using polynomial samples, extending to parity, juntas, and DNF tasks identified as hard for classical SQ models.The paper states these tasks can therefore be accomplished by QNN with quantum advantage.Source: §4 · p. 9
  • The section’s results assume depolarization gate noise but are stated to extend to a more general gate-noise model.The extension is referred to Appendix I.Source: §4 · p. 9

5 Numerical simulations

The simulations test QNN learnability on preprocessed handwritten digits under noisy and noiseless circuit settings. The achieved loss falls farther from the noiseless optimum as circuit depth increases, matching the stated utility-bound trend.

5 Numerical simulations

  • The study uses a binary handwritten-digit task, reducing 64 image features to 3 with PCA before splitting 360 examples into 280 training and 80 test examples.The selected labels are 0 and 1.Source: §5 · p. 10
  • The QNN uses three qubits, kernel encoding, and a layered trainable circuit following the cited proposal.Figure 2 depicts the encoding and trainable circuits; its x3 and xL labels denote repeated gate blocks.Source: §5 · p. 10, Figure 2 · p. 10
  • The noisy simulations set depolarization rate p = 0.0025, use depths L = 5 and L = 20 with 15 and 60 trainable parameters, and estimate expectations from K = 20 measurements.Noiseless runs at both depths estimate the optimal parameter and minimized objective; all simulations run for T = 400 iterations.Source: §5 · p. 10
  • The gap between optimal loss L* and achieved loss L(θ(T)) grows with circuit depth, consistent with poorer utility bounds at larger parameter count and effective noise.The simulations compare depths L = 5 and L = 20.Source: §5 · p. 10, Figure 3 · p. 11
  • Figure 3 reports training loss across hyperparameter settings and training and test accuracy for quantum classifiers.The loss baselines use p = 0 and K →∞ at depths L = 5 and L = 20.Source: §5 · p. 11, Figure 3 · p. 11

6 Conclusion

The conclusion summarizes the paper’s ERM and sample-complexity analysis, including utility bounds, noisy-gate learnability, and quantum-advantage results for selected concept classes. The implementation discussion describes common encoding and trainable-circuit designs, while the appendix notation section states its conventions.

6 Conclusion

  • The conclusion reports that more measurements, lower noise, and shallower circuits improve QNN utility bounds, and that noisy QNNs retain learnability for parity, juntas, and DNF with quantum advantages.The authors present the noiseless/noisy learnability result as theoretical support for implementing QNNs on NISQ chips.Source: §6 · p. 11

A The summary of notations

  • The notation appendix uses Del(x0) for a point-mass random variable, U(a, b) for a uniform random variable, and ∥v∥ for the ℓ2 norm.It also denotes the ℓp norm of v as ∥v∥p.Source: A The summary of notations · p. 18

B Implementation details of encoding circuit and trainable circuit of QNN

  • The encoding discussion covers amplitude encoding, kernel mapping, dimension reduction, and basis encoding, with encoding methods described as having low circuit depth.Amplitude encoding requires low feature dimension; dimension reduction encodes projected features with D′ ≪ D.Source: B Implementation details of encoding… · p. 18
  • Trainable circuits are products of parameterized layers with single-qubit trainable gates and fixed two-qubit gates, organized as hardware-efficient, tensor-network-inspired, or Hamiltonian-based architectures.Hardware-efficient layouts follow the target NISQ machine’s connectivity and gate set.Source: B Implementation details of encoding… · pp. 18–19

C Proof of Lemma 2

The proof establishes regularity properties of the QNN objective under the stated parameter setting. It shows smoothness and Lipschitz bounds and establishes a Polyak-Lojasiewicz condition when λ ≥ 1/π.

C Proof of Lemma 2

  • Lemma 2’s setup uses single-example batches and a QNN prediction defined by a trainable circuit and a two-outcome POVM, with rotation parameters constrained to [π, 3π].The objective is stated to be S-smooth, G-Lipschitz, and PL under λ ≥ 1/π.Source: C Proof of Lemma 2 · p. 19

C.1 Proof of S-smooth, Lemma 3

C.2 Proof of G-Lipschitz, Lemma 4

C.3 Proof of PL condition, Lemma 5

  • Under λ ≥ 1/π, the objective satisfies the Polyak-Lojasiewicz condition.The condition is stated relative to the minimum objective value L* over the constraint set.Source: §C.3 · pp. 21–22

D Proof of Theorem 4

The proof relates gradients estimated from noisy, finite-measurement QNN runs to analytic gradients. It first combines the depth-wise depolarization channels into one effective channel, then uses that simplification to characterize measurement statistics.

D Proof of Theorem 4

  • Theorem 4 addresses the relation between estimated and analytic QNN gradients, with the estimated gradients affected by depolarization noise and finite measurements.The proof treats the noisy channel as shifting expectation values and finite measurements as making circuit outputs random.Source: D Proof of Theorem 4 · p. 22
  • Theorem 5 expresses the estimated gradient in terms of the analytic gradient and noise-related random variables.The proof represents gradient-estimation noise using zero-mean random variables and their variances.Source: D Proof of Theorem 4 · pp. 22–23

D.1 Proof of Lemma 6

  • A sequence of depth-wise depolarization channels is equivalent to one depolarization channel at the final circuit depth, with effective rate p̃ = 1 − (1 − p)^LQ.Source: §D.1 · pp. 24–25
  • Lemma 6 models repeated measurements as sample statistics from a mixture determined by the effective depolarization channel and the circuit’s expected output.The proof then uses the sample mean’s moments to relate measured statistics to expectation values.Source: §D.1 · p. 25

E Proof of Theorem 1

Theorem 1 bounds QNN utility for empirical risk minimization through R1 and R2. Its proof derives R1 from estimated-gradient convergence and uses the PL condition to obtain R2.

E Proof of Theorem 1

  • Theorem 1 gives utility bounds R1 and R2 for QNN after T iterations, with the stated bounds depending on measurements, circuit depth, gate noise, and batches.The passages identify K as the number of quantum measurements, LQ as circuit depth, p as gate noise, and B as the number of batches.Source: E Proof of Theorem 1 · p. 26

E.1 Proof of Theorem 7: The utility bound R1

  • R1 measures distance from a stationary point; smoothness and the estimated-to-analytic gradient relation support its convergence bound.The proof applies the S-smooth objective with step size η = 1/S and uses the estimated-gradient analysis.Source: E Proof of Theorem 1 · p. 26, §E.1 · p. 27

E.2 Proof of Theorem 8: The utility bound R2

  • R2 bounds the disparity between expected empirical risk and optimal risk, using the PL condition to relate stationary points to the global minimum.The proof applies the PL condition alongside the R1 result.Source: E Proof of Theorem 1 · p. 26, §E.2 · p. 29

E.3 Proof of Lemma 8

  • Lemma 8 supports the R1 derivation by bounding terms in the estimated-gradient analysis.The proof invokes the explicit estimated-gradient formula and uses bounds involving C(i,t) and the G-Lipschitz property.Source: §E.3 · p. 30

F Proof of Lemma 1

The proof treats noisy QNN gradient generation as a differentially private mechanism. It establishes privacy for that mechanism and composes the result across parameters and iterations.

F Proof of Lemma 1

  • Gate noise and a finite number of measurements make the estimated gradient random, motivating treatment of noisy QNN as a DP learning model.For K measurements, the sample mean takes discrete values in {0, 1/K, ..., 1}.Source: F Proof of Lemma 1 · p. 31
  • The proof establishes that the per-parameter mechanism producing the measurement sample mean satisfies differential privacy.It compares outputs on neighboring datasets and uses the distribution of the sample mean.Source: F Proof of Lemma 1 · pp. 31–32
  • Privacy for the full QNN follows by composing the private mechanisms across trainable parameters and iterations.The proof applies the composition property first to parameter mechanisms and then to T iterations.Source: F Proof of Lemma 1 · pp. 31–32

G Proof of Theorem 2

The proof connects QNN learnability to quantum PAC and private PAC results. It concludes that a concept class PAC-learnable by QNN remains PAC-learnable with noisy gates.

G Proof of Theorem 2

  • Quantum examples provide the setting for comparing noiseless and noisy QNN learnability, and the encoding circuit Ux can prepare them efficiently.The proof invokes results establishing quantum PAC = PAC and quantum PPAC = PPAC, together with classical PAC = PPAC.Source: G Proof of Theorem 2 · p. 33
  • A concept class PAC-learnable by QNN is also PAC-learnable by a DP model, identified here with QNN using noisy gates.The conclusion follows from the equivalence of learnable classes for non-private and DP learning models and the established DP property of noisy QNN.Source: G Proof of Theorem 2 · p. 33

H Proof of Theorem 3

The proof analyzes noisy QNN as a simulator for the quantum statistical query model. It uses concentration of measurement outcomes to bound the sample-mean deviation and states conditions for simulation.

H Proof of Theorem 3

  • The proof uses concentration inequalities to compare the noisy QNN sample mean with the QSQ target expectation within tolerance τ.It models the QSQ output α as the QNN sample mean; under depolarization, the measurement expectation is shifted toward 1/2^(N+1).Source: H Proof of Theorem 3 · p. 34
  • The theorem’s setup uses an identity trainable unitary and the quantum example prepared by the encoding circuit.The QSQ definition specifies an oracle receiving tolerance and an observable and returning a number satisfying its query condition.Source: H Proof of Theorem 3 · pp. 33–34
  • Theorem 3 quantifies the query complexity required for noisy QNN to simulate one QSQ query.The stated simulation condition is accompanied by a probability bound and a relation between measurement count K and success probability b, but the supplied passages do not expose their full expressions.Source: H Proof of Theorem 3 · pp. 33–34

I Generalization the results to more general quantum channels

The appendix extends the QNN analysis to a general noisy channel and treats utility bounds, differential privacy, and learnability under that setting. It states that the general-channel QNN remains differentially private and can learn concepts that noiseless QNNs PAC-learn, using polynomial samples.

I Generalization the results to more general quantum channels

I.1 Utility bounds of QNN

  • The utility-bound analysis simplifies the noisy QNN model and relates estimated gradients to analytic gradients before deriving bounds.The section treats separate utility bounds R1 and R2 under E_p1.Source: §I.1 · pp. 35–37

I.2 Differential privacy of QNN

  • Under E_p1, the QNN remains a differentially private learning model.The section attributes this to the sample-mean mechanism being differentially private, with composition supporting the QNN's DP property.Source: §I.2 · p. 37

I.3 Learnability of QNN

  • If a concept is PAC-learnable by a noiseless QNN, it is also learnable by an E_p1-noisy QNN using polynomial samples.Source: §I.3 · p. 38

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