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Entanglement Induced Barren Plateaus

Carlos Ortiz Marrero, Mária Kieferová, Nathan Wiebe

arXiv:2010.15968v2quant-phcs.LG

TL;DR

The paper asks how excess entanglement between visible and hidden QNN units affects learnability. It combines entanglement and thermalization arguments to show that typical volume-law states produce entanglement-induced barren plateaus, while generative pretraining is proposed as a possible workaround.

  • Problem

    Excess entanglement between visible and hidden units may undermine the learning benefits of hidden units in QNNs.

  • Method

    The paper analyzes unitary and thermal QNN states using entanglement, thermalization, and typicality arguments.

  • Results

    Volume-law entanglement makes visible outputs approach the maximally mixed state and produces exponentially vanishing gradients, hindering local optimization.

  • Takeaways & Limitations

    Generative pretraining with a nonlinear objective such as quantum relative entropy is proposed to mitigate these barren plateaus.

  • Takeaways & Limitations

    The analysis assumes typical unitary 2-design states and observables supported only on the visible system.

Abstract

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We argue that an excess in entanglement between the visible and hidden units in a Quantum Neural Network can hinder learning. In particular, we show that quantum neural networks that satisfy a volume-law in the entanglement entropy will give rise to models not suitable for learning with high probability. Using arguments from quantum thermodynamics, we then show that this volume law is typical and that there exists a barren plateau in the optimization landscape due to entanglement. More precisely, we show that for any bounded objective function on the visible layers, the Lipshitz constants of the expectation value of that objective function will scale inversely with the dimension of the hidden-subsystem with high probability. We show how this can cause both gradient descent and gradient-free methods to fail. We note that similar problems can happen with quantum Boltzmann machines, although stronger assumptions on the coupling between the hidden/visible subspaces are necessary. We highlight how pretraining such generative models may provide a way to navigate these barren plateaus.

I. INTRODUCTION

The paper studies feed-forward unitary QNNs and quantum Boltzmann machines, arguing that excessive visible–hidden entanglement can create barren plateaus and hinder learning. It connects this mechanism to thermalization and motivates generative pretraining as a possible way around it.

  • Motivation: Excess entanglement between visible and hidden units can cause barren plateaus in QNN optimization.The paper identifies entanglement, in addition to concentration of measure and hardware noise, as a source of vanishing gradients.
  • Mechanism: Random initial states are typically close to maximally mixed on the visible subsystem, making escape from this state difficult.The paper uses thermalization results for random states and contrasts them with polynomial bounds for states drawn from a k-design.
  • Mechanism: Information can become stored non-locally in correlations between layers rather than locally in the layers themselves.Removing hidden units in this setting leaves a state close to maximally mixed, and low-cost gradient descent is unlikely to escape the resulting plateau.
  • Contribution: The paper links quantum machine learning with the thermalization literature.This connection is presented as previously absent from the literature surveyed by the authors.
  • Models: The analysis focuses on feed-forward unitary QNNs and quantum Boltzmann machines.Quantum Boltzmann machines model data as thermal states and can use objectives such as quantum relative entropy for generative training.
  • Possible route: Quantum generative pretraining using a nonlinear objective such as quantum relative entropy is proposed as a route to mitigate these training challenges.The proposed strategy begins discriminative learning after generative training of the quantum model.

II. THE IMPACT OF ENTANGLEMENT ON DEEP MODELS

The paper formalizes how visible–hidden entanglement affects QNN outputs and distinguishes harmful volume-law scaling from more tolerable area-law scaling. Its central conclusion is that uncontrolled volume-law entanglement can make deep QNN predictions ineffective, especially when the hidden subsystem is much larger.

  • Impact: Visible–hidden entanglement causes thermalization on the visible subsystem and can harm QNN models unless carefully controlled.The paper frames this as the central question of its analysis.
  • Volume law: For volume-law states, the visible entropy approaches log(Dv) with deviation Θ(Dv/Dh).This scaling is stated for states whose hidden dimension satisfies Dh ≥ Dv.
  • Consequences: Volume-law outputs can make QNN predictions asymptotically no better than random guessing.The paper identifies this as especially catastrophic for deep models with Dh ≫ Dv.
  • Area law: Area-law entanglement may be tolerated unless the first hidden layer becomes much larger than the visible layer.The authors contrast this limited-entanglement regime with the catastrophic behavior associated with volume laws.
  • Design implication: The paper concludes that QNN design should target sub-volume-law scaling, although such states may admit concise matrix-product-state representations.The authors also state that sub-volume-law scaling is not typical in the ensembles they consider.

III. TYPICALITY OF VOLUME-LAW SCALING

The paper argues that volume-law entanglement is typical for sufficiently scrambling quantum models and makes visible subsystems close to maximally mixed, undermining observable-based learning.

  • III. TYPICALITY OF VOLUME-LAW SCALING: Volume-law entanglement scalings are expected to be more common than area laws under appropriate interaction assumptions.The analysis models sufficiently scrambling joint states with unitary 2-designs rather than the stronger Haar-random assumption.
  • III. TYPICALITY OF VOLUME-LAW SCALING: For unitary-network and Boltzmann-machine states generated from unitary 2-designs, bounded visible observables are indistinguishable in expectation from measurements on the maximally mixed state with high probability.The result applies to both state-generated unitary networks and Gibbs states of conjugated Hamiltonians.
  • III. TYPICALITY OF VOLUME-LAW SCALING: The visible subsystem of a random initial state is exponentially close to maximally mixed, whereas k-design assumptions provide polynomial bounds in k.The argument uses concentration results and Markov’s inequality to extend expectation statements to high-probability claims.
  • III. TYPICALITY OF VOLUME-LAW SCALING: As the hidden dimension grows relative to the visible dimension, entanglement weakens rather than strengthens the analogous classical model.For deep networks, the paper anticipates many more hidden than visible neurons, making this regime generic for deep QNNs.

IV. ENTANGLEMENT INDUCED BARREN PLATEAUS

The paper develops an entanglement-based explanation for barren plateaus, requiring more nuanced assumptions than the preceding visible-state typicality argument because parameter perturbations must be controlled.

  • IV. ENTANGLEMENT INDUCED BARREN PLATEAUS: Analyzing gradient-descent failure requires assumptions about how parameter perturbations affect the resulting quantum state.The authors note that these assumptions parallel those used in earlier barren-plateau analyses.

A. Plateaus for Unitary networks

For sufficiently deep scrambling unitary networks, changing a parameter barely changes visible observables: the objective is Lipschitz continuous with a constant inversely scaling in hidden dimension, affecting gradient and gradient-free optimization.

  • A. Plateaus for Unitary networks: U(θ1, . . . , θn) := e^-iHnθn . . . e^-iH1θ1 defines the layered unitary ansatz analyzed for parameter perturbations.The study shifts one parameter by δk and bounds the resulting change in a visible-only observable.
  • A. Plateaus for Unitary networks: A sufficiently deep random circuit is assumed to scramble almost all relevant subsequences into unitary 2-designs.This assumption is used to analyze the sensitivity of the objective to parameter changes.
  • A. Plateaus for Unitary networks: The visible objective is Lipschitz continuous with a constant scaling inversely with the hidden dimension Dh = 2^nh.The paper concludes that both gradient descent and gradient-free methods encounter the resulting plateau.
  • A. Plateaus for Unitary networks: The result extends barren-plateau implications beyond gradient-based optimization to gradient-free methods.The paper relates this implication to earlier work that implicitly suggested the same possibility for unitary networks.

B. Plateaus for Boltzmann Machines

Quantum Boltzmann machines exhibit entanglement-induced plateaus under stronger coupling and spectral assumptions, with hidden-layer gradients becoming exponentially small while visible-only coefficients need not share that suppression.

  • B. Plateaus for Boltzmann Machines: Boltzmann-machine plateaus arise under assumptions involving the hidden-visible coupling, including a condition on Tr(h_h^2)/Tr(h^2).The analysis uses random Hermitian Hamiltonians whose eigenbasis is generated by a unitary 2-design.
  • B. Plateaus for Boltzmann Machines: Theorem 4 analyzes differentiable objectives under random-Hamiltonian and unitary-2-design assumptions, with high-probability bounds obtained through perturbation theory.The inverse minimal spectral gap enters through the parameter Γ.
  • B. Plateaus for Boltzmann Machines: Adding hidden-unit bias terms may restore predictive power only by disentangling hidden and visible units and effectively reverting the model to a shallow one.This identifies a concrete trade-off between escaping the plateau and retaining hidden-layer structure.
  • B. Plateaus for Boltzmann Machines: Gradients for terms acting nontrivially on hidden layers are exponentially small in the number of hidden qubits, whereas visible Hamiltonian coefficients need not be.The contrast follows from taking the hidden operator’s trace to be zero for hidden-active terms.
  • B. Plateaus for Boltzmann Machines: Typical Gibbs states explain why adding hidden units did not improve Quantum Boltzmann Machine performance in earlier observations.For typical Hamiltonians, the resulting thermal states are close to maximally mixed, so hidden units are not generally expected to improve performance.

V. HAAR RANDOM UNITARIES

Under Haar-random assumptions, concentration strengthens beyond the unitary 2-design analysis: most networks have vanishing gradients, while higher-order designs interpolate between the bounds.

  • Haar-random networks have vanishing gradients with high probability because large deviations from the Haar expectation are exponentially unlikely.Levy’s lemma provides tighter concentration than Markov’s inequality.
  • Unitary k-designs can interpolate between the 2-design and Haar-random concentration results.The resulting bounds under only a 2-design assumption are not superior to the Markov-based analysis.
  • The 2-design analysis remains the relevant baseline when stronger scrambling assumptions are unavailable.The paper explicitly notes that the improved bounds from higher-order designs do not improve the 2-design result under the weaker assumption.

VI. NUMERICAL RESULTS

Numerical experiments test the asymptotic predictions on small quantum networks, showing concentration toward the maximally mixed state and declining gradient norms as hidden-system size grows.

  • The ansatz uses terms from a random two-local Hamiltonian, with unitary and Boltzmann models constructed from sampled local terms and a Hamiltonian, respectively.The implementation distinguishes onsite coefficients from offsite couplings.
  • Increasing hidden units drives the trace distance of the GUE, unitary QNN, and Quantum Boltzmann Machine models toward zero.Figure 4 shows the corresponding trace-distance histograms concentrating around zero.
  • The unitary QNN’s gradient ∞-norm decreases as the hidden-system size increases, with an exponential decay rate estimated by least-squares fitting.This behavior is predicted by Theorem 3.
  • The Boltzmann Machine experiments amplify offsite relative to onsite couplings to encourage volume-law entanglement and expose gradient decay.Perturbation theory motivates small onsite coefficients and larger offsite coefficients because energy gaps suppress entanglement generation.

VII. CONCLUSION

The paper concludes that entanglement can create barren plateaus and make hidden units harmful, while identifying assumption violations and generative pretraining as possible ways to mitigate the problem.

  • For Haar-random pure states and thermal states of random Hamiltonians, observable-objective gradients vanish exponentially with the number of hidden units.Thus, adding hidden units does not necessarily increase QNN power.
  • The barren plateaus can be avoided by violating proof assumptions, including using atypical initial states or objectives independent of the density operator.The paper is skeptical that gradient-free methods alone will succeed without knowledge of the objective’s global properties.
  • Generative pretraining with a nonlinear objective such as quantum relative entropy is advocated as a possible way to mitigate training difficulties.The paper proposes beginning discriminative learning with generative training.
  • Entanglement should be deployed selectively because uncontrolled entanglement can harm models despite its potential usefulness.The paper frames this as a design requirement for leveraging quantum effects successfully.

Appendix A: Proof of Unitary Network Gradient

The appendix proves a Lipschitz bound for visible-layer objectives in unitary networks by analyzing parameter perturbations, commutator expansions, and unitary 2-design concentration.

  • Theorem 3 assumes a unitary 2-design and a hidden-visible Hilbert space with dimensions Dh and Dv.The theorem concerns gradients generated through a unitary ansatz.
  • The objective’s parameter dependence is analyzed through the visible expectation of the reduced-state difference under a parameter perturbation.Hadamard’s lemma expresses the perturbed-state difference in a form suitable for bounding.
  • The proof expands commutator products into partial-trace terms and uses two copies of the state linked by a visible-system flip operator.Symmetry and unitary invariance simplify the resulting expectations.
  • The Schatten infinity-norm’s unitary invariance yields the final operator-norm bound used in the Lipschitz estimate.The argument assumes ∥Hk∥∞|δk| ∈ O(1).
  • Positive and negative terms cancel up to small errors because their expected values are independent of the partition indices.The cancellation is established across the expanded adjoint terms.

Appendix B: Proof of Quantum Boltzmann Machine Gradient

The appendix proves a high-probability gradient bound for quantum Boltzmann machines under random-Hamiltonian and coupling assumptions. It uses perturbation theory, unitary 2-design averaging, and Markov’s inequality to control visible-observable changes.

  • Proof setup: The Hamiltonian may be taken traceless because subtracting a scalar multiple of the identity preserves its thermal state.This removes an irrelevant energy offset before analyzing the gradient.
  • Theorem assumptions: The theorem assumes a random Hermitian Hamiltonian whose eigenbasis is generated by a unitary 2-design and whose energy gaps satisfy a bounded inverse-square condition.The trainable coupling is required to factor as H_k = h_v ⊗ h_h.
  • Perturbative analysis: Perturbation theory expands the eigenvectors of H + θ_kH_k when no level crossings occur on the parameter interval.The expansion is controlled to order O(θ_k^2).
  • Proof setup: The proof chooses product eigenbases that diagonalize the visible and hidden factors of H_k, yielding eigenvectors |pq⟩ with eigenvalues λ_pq.Expectation values are then expressed through coefficients in this visible-hidden product basis.
  • Gradient bound: Unitary 2-design averaging bounds the expanded terms, with the displayed term scaling as O(Γ^2∥H_k∥∞^2/D_h), and the same bound applies to the remaining products.The argument then uses the preceding bounds to obtain the derivative estimate.
  • Gradient bound: Markov’s inequality converts the expectation bound into a statement that the derivative estimate holds with high probability over the Hamiltonian ensemble.The proof first relates visible-observable differences to density-operator differences under imaginary-time evolution.
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