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Quantum Entanglement in Neural Network States

Dong-Ling Deng, Xiaopeng Li, S. Das Sarma

arXiv:1701.04844v3cond-mat.dis-nncond-mat.quant-gasquant-ph

TL;DR

The paper asks how much entanglement restricted Boltzmann machines can represent and whether their entanglement structure is characterized by locality or randomness. It studies RBM entanglement analytically and numerically, finding area laws for short-range states, volume laws for suitable long-range states, and non-Page, Poisson-type statistics for random-parameter states. These results show that RBMs can efficiently represent quantum states with massive entanglement while also supporting selected ground-state and entanglement-spectrum calculations.

  • Problem

    The entanglement properties connecting neural-network quantum-state representations to many-body-state structure were unknown.

  • Method

    The paper combines analytical proofs and constructions with numerical studies, reinforcement learning, and an RBM-based analytical calculation for selected states.

  • Results

    Short-range RBMs obey area laws, constructed long-range RBMs achieve maximal volume-law entanglement, and random-parameter RBMs show sub-Page entropy with Poisson-type entanglement-spectrum statistics.

  • Takeaways & Limitations

    RBM representations can efficiently describe quantum states with massive entanglement, unlike MPS or tensor-network representations whose tractable efficiency is limited by such entanglement.

  • Takeaways & Limitations

    The analytical RBM method for entanglement calculations applies only in specific circumstances, requiring simplifying symmetries and lacking a systematic RBM construction for arbitrary systems.

Abstract

from arXiv · show

Machine learning, one of today's most rapidly growing interdisciplinary fields, promises an unprecedented perspective for solving intricate quantum many-body problems. Understanding the physical aspects of the representative artificial neural-network states is recently becoming highly desirable in the applications of machine learning techniques to quantum many-body physics. Here, we study the quantum entanglement properties of neural-network states, with a focus on the restricted-Boltzmann-machine (RBM) architecture. We prove that the entanglement of all short-range RBM states satisfies an area law for arbitrary dimensions and bipartition geometry. For long-range RBM states we show by using an exact construction that such states could exhibit volume-law entanglement, implying a notable capability of RBM in representing efficiently quantum states with massive entanglement. We further examine generic RBM states with random weight parameters. We find that their averaged entanglement entropy obeys volume-law scaling and meantime strongly deviates from the Page-entropy of the completely random pure states. We show that their entanglement spectrum has no universal part associated with random matrix theory and bears a Poisson-type level statistics. Using reinforcement learning, we demonstrate that RBM is capable of finding the ground state (with power-law entanglement) of a model Hamiltonian with long-range interaction. In addition, we show, through a concrete example of the one-dimensional symmetry-protected topological cluster states, that the RBM representation may also be used as a tool to analytically compute the entanglement spectrum. Our results uncover the unparalleled power of artificial neural networks in representing quantum many-body states, which paves a novel way to bridge computer science based machine learning techniques to outstanding quantum condensed matter physics problems.

I. INTRODUCTION

The paper addresses the unknown entanglement structure of neural-network quantum states by studying restricted Boltzmann machines analytically and numerically. It establishes area-law and volume-law regimes and connects RBM structure to many-body-state representation.

  • Motivation: Entanglement properties of neural-network states remained unknown despite their growing use in many-body calculations.The paper frames this as a gap because entanglement underlies the usefulness of MPS and tensor-network representations.
  • Approach: The study focuses on restricted Boltzmann machines and examines entanglement entropy, entanglement spectra, and representative RBM constructions.The analysis combines analytical proofs, exact constructions, numerical studies, and reinforcement learning.
  • Main findings: Short-range RBM states obey an entanglement area law, whereas analytically constructed long-range states can exhibit maximal volume-law entanglement.The short-range result holds generally, while the volume-law result follows from an exact construction.
  • Main findings: Random-parameter RBM states show volume-law averaged entropy but differ from completely random pure states in entropy and entanglement-spectrum statistics.Their entropy is below the Page entropy and their spectrum has Poisson-type rather than random-matrix-associated statistics.
  • Applications: RBMs also represent long-range-interaction ground states and permit analytical entanglement-spectrum calculations for one-dimensional SPT cluster states.These examples extend the study beyond general scaling results to optimization and exact spectral analysis.

II. NEURAL-NETWORK REPRESENTATION AND QUANTUM ENTANGLEMENT: CONCEPTS AND NOTATIONS

This section introduces RBMs as variational quantum-state representations and defines the entanglement quantities used throughout the paper. It also explains why these quantities matter for characterizing many-body states.

  • RBM representation: An RBM has visible neurons representing physical spins and hidden neurons representing auxiliary variables, with connections only between the two layers.There are no intra-layer connections, and the hidden variables are traced out to obtain the quantum-state representation.
  • RBM representation: The RBM wavefunction is a variational state whose amplitude and phase are specified by trainable weights and biases.The parameters are collected as Ω=(a_r,b_r′,W_rr′), and the physical state is formed from the amplitudes over visible configurations.
  • Representational scope: RBMs are expressive in principle, but practical representations may require exponentially many neurons and parameters.Representability theorems guarantee approximation capability without guaranteeing an efficient numerical representation.
  • Entanglement quantities: The paper studies reduced-density-matrix-based entanglement entropy, Rényi entropy, and entanglement spectrum for bipartitions into subsystems A and B.The reduced density matrix is obtained by tracing out subsystem B, while the entanglement spectrum is defined through the entanglement Hamiltonian.
  • Entanglement quantities: Rényi entropy is defined from Tr(ρ_A^α), with α→1 recovering the von Neumann entropy.The zeroth-order Rényi entropy is related to the rank of ρ_A.

III. AREA-LAW ENTANGLEMENT FOR SHORT-RANGE NEURAL-NETWORK STATES

The paper proves that every finite-range RBM obeys an entanglement area law, independent of spatial dimension and bipartition geometry. The proof follows from the locality of hidden-neuron factors near the subsystem boundary.

  • Theorem: An R-range RBM has hidden neurons connected only to visible neurons within distance R.This finite-range condition defines the short-range class analyzed by the theorem.
  • Theorem: For every Rényi order, the entropy of an R-range RBM is bounded by a constant times the surface area of subsystem A.The bound applies in any dimension and for arbitrary bipartition geometry.
  • Proof mechanism: Local Γ-factors depend only on visible-spin configurations within an R-neighborhood, which supplies the locality underlying the area law.This dependence limits entanglement-relevant degrees of freedom to regions near the boundary.
  • Proof mechanism: Partitioning A and B into three regions isolates boundary neighborhoods so the reduced density matrix depends only on boundary-scale degrees of freedom.The regions A2, A3, B2, and B3 form hypersurfaces of thickness R in higher dimensions.
  • Consequences: The result applies to exactly represented one-dimensional SPT cluster states and two- and three-dimensional toric-code states.These states use short-range RBMs with R=1, so their entanglement satisfies the area law.
  • Consequences: In one dimension, bounded Rényi entropies of all orders imply that short-range RBM states also admit efficient MPS descriptions.The converse implication—from efficient MPS descriptions to short-range RBMs—remains unknown.
  • Practical implication: Short-range RBMs can reduce parameters and computational cost when the target problem is known to involve only area-law entanglement.For problems involving large entanglement, the paper states that long-range RBMs should instead be used.

IV. VOLUME-LAW ENTANGLEMENT IN LONG-RANGE NEURAL-NETWORK STATES

The paper shows that long-range RBMs can support volume-law entanglement, complementing the area-law theorem for finite-range architectures. It establishes this through an exact construction and a numerical benchmark.

  • Volume-law regime: Long-range RBMs are explicitly shown to exhibit volume-law entanglement through a rigorous exact construction and a numerical benchmark.This extends the finite-range area-law analysis to neural networks with nonlocal connections.

A. Exact construction of maximal volume-law entangled neural-network states

The paper constructs exact long-range RBM families whose entanglement is maximal and volume-law while their representations use only linearly many parameters. The construction extends to higher dimensions and outperforms tractable tensor-network descriptions in representational efficiency.

  • Representational efficiency: Equivalent MPS or tensor-network descriptions require exponentially growing bond dimension, making them computationally intractable as system size increases.The RBM instead remains efficient with only linearly many parameters.
  • 1D construction: The 1D construction gives maximal volume-law entanglement for every contiguous region no larger than half the system.The reduced density matrix is maximally mixed, ρA = I/2^l, for a region of length l.
  • 1D construction: In the thermodynamic limit, any contiguous 1D region has entanglement entropy proportional to its size.The result holds regardless of the region’s position, not only for a boundary segment at the left end.
  • Construction mechanism: The construction uses local and nonlocal hidden-neuron connections to generate volume-law entanglement while retaining an efficient RBM representation.The 2D architecture separates nearest-neighbor X and Y groups from a nonlocal Z group.
  • Higher-dimensional construction: The 2D construction yields maximal entropy SA = NA log 2 for any small regular contiguous region.NA is the number of qubits inside region A.
  • Higher-dimensional construction: In d dimensions, the construction requires hidden neurons and nonzero parameters that both scale linearly with system size.It uses M = 2d+1 2 L^d − dL^{d−1} hidden neurons and 3M nonzero weight parameters.

B. Entanglement benchmarking

The benchmarking study evaluates random-parameter RBM states through entropy, spectrum, and level statistics. These states show volume-law entanglement but remain distinct from completely random pure states and random-matrix predictions.

  • Entropy scaling: For small γ, averaged von Neumann entropy scales linearly with system size, indicating volume-law entanglement.Here γ = M/N is the hidden-to-visible neuron ratio.
  • Entropy scaling: γ*≈0.7 is the entropy maximum, after which averaged entropy decreases as γ increases.At large γ, the wavefunction becomes less dependent on spin configurations and approaches a product state.
  • Entropy benchmarking: The entropy is always below the Page entropy, showing that random-parameter RBM states differ from random pure states.The paper attributes this to RBM states occupying a restricted subspace of the full Hilbert space.
  • Rényi entropies: Second-order Rényi entropy shows similar behavior, while half-order Rényi entropy retains volume-law scaling with finite-size effects.The bending feature at γ = 4 is absent for S1/2 because of finite-size effects.
  • Entanglement spectrum: The entanglement spectrum differs completely from the Marchenko–Pastur distribution associated with random-matrix Wishart ensembles.Increasing γ broadens the entanglement-Hamiltonian eigenvalue distribution and shifts its peak rightward.
  • Level statistics: Consecutive entanglement-spectrum spacings follow Poisson statistics rather than GOE, GUE, or GSE predictions.For N = 20 and γ = 3, the averaged adjacent gap ratio is 0.378 over 10^4 samples.

C. Reinforcement learning of ground states with power-law entanglement

The paper uses reinforcement learning to represent a long-range model’s ground state with power-law entanglement, finding accurate correlations and scalable energy estimates with an RBM.

  • Model and entanglement: The modified Haldane-Shastry model has long-range interactions whose strength decays as the inverse square of chord distance, and its ground state has power-law entanglement.The power-law entanglement is numerically verified through exact diagonalization rather than rigorously proven.
  • RBM reinforcement learning: RBM reinforcement learning efficiently represents the highly entangled ground state, using translation symmetry to reduce variational parameters.The approach is motivated by the difficulty of MPS/DMRG simulations when large entanglement and long-range interactions are present.
  • Small-system validation: The RBM spin correlations match exact-diagonalization results well for small systems, with accuracy improving through larger hidden-unit density and more training iterations.For the displayed parameters, the relative error is approximately 10^-5.
  • Large-system results: For N = 100, reinforcement learning produces spin correlations and a smoothly converging ground-state energy density near E0/N ≈ −2.12.The larger-system calculation is beyond exact diagonalization’s capability.

V. AN ANALYTICAL RBM RECIPE FOR CALCULATING ENTANGLEMENT

The paper develops an RBM-based analytical route to the entanglement entropy and spectrum of a one-dimensional SPT cluster state. Short-range factorization reduces the calculation to a compact reduced density matrix, yielding an exactly characterized entanglement spectrum while limiting general applicability.

  • Analytical RBM method: The RBM representation provides an alternative analytical method for calculating entanglement entropy and spectrum in finite systems, illustrated using the 1D SPT cluster state.The cluster state is a topological state protected by Z2 × Z2 symmetry and has an exact, efficient short-range RBM representation.
  • Analytical RBM method: The short-range RBM is partitioned into A1, A2, B1, and B2 so factors independent of the opposite subsystem can be summed and factorized explicitly.This factorization is the key simplification in constructing the reduced density matrix.
  • Reduced density matrix: The reduced density matrix can be rotated into a basis where the subsystem-dependent states become Hilbert-space basis vectors, without changing entanglement entropy or spectrum.The resulting expression contains only 2^8, rather than 2^N, terms in the relevant summation.
  • Entanglement entropy: The reduced density matrix has four nonzero degenerate eigenvalues, each equal to 1/4, producing the reported Rényi entropies.The eigenvalue structure follows from the analytically simplified reduced density matrix.
  • Entanglement spectrum: The entanglement Hamiltonian has a four-fold-degenerate lowest level at 2 log 2, with all remaining levels infinite, signaling the SPT phase.The four-fold degeneracy is identified as a signature of SPT phases.
  • Limitations: The analytical approach is restricted to specific cases because no systematic RBM wave-function construction exists for arbitrary systems and simplifying symmetries are required.The authors expect extensions to toric-code states, but note that those calculations would be more technically involved.

VI. CONCLUSION AND OUTLOOK

The paper establishes distinct entanglement regimes for RBM states and demonstrates that neural-network representations can efficiently capture highly entangled quantum states. It also identifies open questions about the conditions governing efficient RBM representations and their relation to tensor-network states.

  • Main conclusions: Short-range RBM states obey an entanglement area law for arbitrary dimensions and bipartition geometries.This includes one-dimensional SPT cluster states and two- and three-dimensional toric-code states, with or without anyonic excitations.
  • Main conclusions: Random-parameter long-range RBM states have volume-law averaged entanglement entropy below the Page entropy and Poisson-type entanglement-spectrum statistics.Their spectra lack a universal component associated with random matrix theory.
  • Main conclusions: Analytically constructed one- and two-dimensional RBM families achieve maximal volume-law entanglement while using parameters that scale linearly with system size.These states cannot be efficiently represented by matrix-product or tensor-network states with computationally tractable bond dimension.
  • Main conclusions: Reinforcement learning enables RBM to calculate a long-range-interaction model’s ground state with power-law entanglement, including its energy and correlations.The example uses a modified Haldane–Shastry model.
  • Outlook: Open questions concern the necessary and sufficient conditions for efficient neural-network representations and their relationship to MPS and tensor-network representations.The paper also proposes studying other artificial-neural-network architectures and higher-dimensional extensions.
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