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Machine Learning Topological States

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

arXiv:1609.09060v2cond-mat.dis-nncond-mat.stat-mechcond-mat.str-elquant-ph

TL;DR

The paper asks whether symmetry-protected and intrinsically ordered topological states admit efficient neural-network representations. It constructs short-range RBMs for representative states and applies reinforcement learning to an interacting model, finding exact linear-resource representations and successful phase-transition studies.

  • Problem

    It is unknown whether topological states can be represented by neural networks with efficient classical-resource scaling.

  • Method

    The paper constructs short-range FRRBM representations for the 1D cluster state and 2D and 3D toric-code states, and uses reinforcement learning for a non-integrable 1D Hamiltonian.

  • Results

    The three topological ground states use hidden-layer size equal to the number of physical spins and a number of nonzero weights scaling linearly with system size; 2D networks also describe abelian anyons and mutual statistics.

  • Takeaways & Limitations

    The exact constructions provide guidance, training data, and initial parameters for machine-learning studies of topological phases and many-body problems.

Abstract

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Artificial neural networks and machine learning have now reached a new era after several decades of improvement where applications are to explode in many fields of science, industry, and technology. Here, we use artificial neural networks to study an intriguing phenomenon in quantum physics--- the topological phases of matter. We find that certain topological states, either symmetry-protected or with intrinsic topological order, can be represented with classical artificial neural networks. This is demonstrated by using three concrete spin systems, the one-dimensional (1D) symmetry-protected topological cluster state and the 2D and 3D toric code states with intrinsic topological orders. For all three cases we show rigorously that the topological ground states can be represented by short-range neural networks in an \textit{exact} and \textit{efficient} fashion---the required number of hidden neurons is as small as the number of physical spins and the number of parameters scales only \textit{linearly} with the system size. For the 2D toric-code model, we find that the proposed short-range neural networks can describe the excited states with abelain anyons and their nontrivial mutual statistics as well. In addition, by using reinforcement learning we show that neural networks are capable of finding the topological ground states of non-integrable Hamiltonians with strong interactions and studying their topological phase transitions. Our results demonstrate explicitly the exceptional power of neural networks in describing topological quantum states, and at the same time provide valuable guidance to machine learning of topological phases in generic lattice models.

I. INTRODUCTION

The paper asks whether topological quantum states can be represented by neural networks with practical classical-resource scaling. It addresses this through exact constructions spanning one to three dimensions and numerical study of a non-integrable system.

  • The unknown resource scaling of neural-network representations for topological states limits practical machine-learning applications to topological quantum phases.
  • Exact constructions cover the 1D symmetry-protected cluster state and 2D and 3D toric-code states with intrinsic topological order.
  • Short-range neural networks precisely represent toric-code states despite their long-range entanglement, and describe abelian anyon braiding and mutual statistics.
  • Reinforcement learning finds topological ground states and studies the transition from a symmetry-protected topological phase to a ferromagnetic phase in a 1D interacting Hamiltonian.

II. ARTIFICIAL-NEURAL-NETWORK REPRESENTATION

The paper uses restricted Boltzmann machines to represent many-body wave functions, with visible physical-spin nodes and hidden auxiliary nodes. A further restriction limits hidden-to-visible connections to local neighborhoods for efficient representations.

  • An RBM represents a spin-1/2 many-body wave function using N visible nodes for physical spins and M hidden nodes for auxiliary classical variables.
  • The ANNQS is a variational quantum state whose wave-function amplitudes are determined by trainable RBM parameters.
  • The further restricted RBM connects hidden neurons only locally to visible neurons, avoiding the impractical exponential resource requirements of unrestricted representations.
  • Figure 1 illustrates the short-range network for the 1D cluster state, distinguishing visible physical-spin neurons from hidden artificial neurons.

III. 1D SPT CLUSTER STATE

The 1D cluster state is a Z2 × Z2 symmetry-protected topological ground state represented by a locally connected RBM. The construction converts exact stabilizer constraints into solvable nonlinear parameter equations.

  • The periodic cluster Hamiltonian has Z2 × Z2 symmetry, and its stabilizer ground state is a symmetry-protected topological state analogous to the Haldane phase.
  • The construction exploits short-range connections, factors hidden variables into a product, and imposes the ground-state constraints for every spin configuration and stabilizer site.
  • ∼N2^N highly nonlinear equations arise for a general network, motivating the restricted-network approach rather than direct general-network solving.
  • 25 equations are recast as an optimization problem whose zero minimum yields parameters (b, ω1, ω0, ω−1) = π/4 (1, 2, 3, 1).
  • ∼4N nonzero parameters give a compact representation that is exact because the resulting FRRBM satisfies the cluster-state equation analytically.

IV. 2D KITAEV TORIC CODE STATES

The 2D Kitaev toric code has intrinsically topologically ordered, four-fold-degenerate ground states and abelian anyonic excitations. The paper constructs exact neural-network representations for its ground and excited states, including nontrivial mutual statistics.

  • Model and topological structure: N = 2L^2 qubits occupy the edges of an L × L square lattice with periodic boundary conditions, forming a 2D torus.Vertex and face operators act on the lattice degrees of freedom.
  • Model and topological structure: The toric code is an exactly solvable Z2 gauge theory whose four-fold-degenerate ground state signals intrinsic topological order.Its low-energy excitations are abelian anyons with nontrivial mutual statistics.
  • Ground-state representation: The toric-code ground state has an exact and efficient neural-network representation constructed and verified in the paper.The representation is illustrated using visible neurons for physical spins and hidden neurons associated with vertices and faces.
  • Ground-state representation: The neural network remains short-range: hidden vertex and face neurons connect locally to the corresponding visible spins.The exactness follows by factoring out hidden neurons and analyzing how spin flips affect neighboring cosine factors.
  • Excitations and braiding: Excited states with pairs of x-type magnetic vortices or z-type electric charges are generated by string operators acting along paths.The quasiparticles reside at path endpoints and can be moved by extending or shortening the paths.
  • Excitations and braiding: Linked x- and z-type loops acquire a global phase −1, and the corresponding excited states and braiding processes are represented precisely and efficiently by FRRBMs.Same-type quasiparticles fuse to the vacuum, while linked loops realize their nontrivial mutual statistics.

V. 3D TORIC CODE STATES

The 3D toric-code ground states admit exact and efficient short-range FRRBM representations, extending the neural-network treatment to a paradigmatic three-dimensional topological model. The FRRBM also describes selected low-energy excitations and their mutual statistics, while generic applicability remains unresolved.

  • 3D toric-code model: The 3D model is an exactly solvable paradigmatic model of topological order featuring closed-string and closed-membrane condensation.Its finite-temperature behavior can exhibit classical topological order up to a transition temperature T_c.
  • 3D toric-code representation: The 3D toric-code ground states have an exact and efficient FRRBM representation.The representation is verified by showing that the relevant vertex and face equations are satisfied.
  • 3D toric-code representation: The same weight parameters used in the 2D construction can represent the 3D toric-code ground states.This gives a direct extension of the short-range construction to the three-dimensional model.
  • Excited states: The FRRBM can also describe low-energy 3D excited states generated by string or membrane operators and their mutual statistics.This parallels the corresponding excited-state description in the 2D toric-code case.
  • Open scope: Necessary and sufficient conditions for generic intrinsically topological states to admit FRRBM representations remain to be determined.The authors identify this as requiring substantial future effort.

VI. REINFORCEMENT LEARNING OF SPT PHASES AND PHASE TRANSITIONS

The paper uses numerical reinforcement learning with RBMs to study non-integrable interacting one-dimensional SPT systems. The trained networks reproduce ground-state observables and identify the transition from the SPT phase to a ferromagnetic phase near V_c ∼0.9.

  • Reinforcement-learning approach: Numerical reinforcement learning shows that RBMs can find topological ground states of non-integrable interacting Hamiltonians and study their phase transitions.The method is applied after adding an interaction term and magnetic field to the 1D SPT cluster Hamiltonian.
  • Model and diagnostics: The modified 1D Hamiltonian retains Z2 × Z2 symmetry and approaches the cluster-state SPT phase when h_x and V tend to zero.The SPT character is monitored with a string-order parameter.
  • Phase transition: V_c ∼0.9 marks sharp jumps in magnetization and string-order parameter, indicating a transition from the SPT phase to the ferromagnetic phase.The larger-system result uses N = 50, beyond the capability of exact diagonalization.
  • Accuracy and outlook: Increasing the hidden-unit density γ and the number of iterations can systematically improve the trained-RBM accuracy.The paper also notes that reinforcement learning may be extended to topological states and phase transitions in higher dimensions.

VII. CONCLUSION AND DISCUSSION

The paper demonstrates efficient neural-network representations of symmetry-protected and intrinsically topologically ordered quantum states, while outlining practical uses and directions for generalization.

  • VII. CONCLUSION AND DISCUSSION: Exact FRRBM representations cover 1D cluster states and 2D and 3D toric-code states, with hidden neurons matching physical spins and linearly scaling nonzero weights.The construction applies to both symmetry-protected and intrinsic topological states.
  • VII. CONCLUSION AND DISCUSSION: FRRBMs also describe toric-code excitations containing abelian anyons and their nontrivial mutual statistics.The paper additionally reports analytical and numerical demonstrations of these capabilities.
  • VII. CONCLUSION AND DISCUSSION: Exact results can serve as supervised-learning training data or initialize RBM reinforcement learning for quantum many-body problems.The authors present these uses as practical applications of the exact parameter values and representations.
  • VII. CONCLUSION AND DISCUSSION: The study focuses on single-layer networks, while deep-network generalizations are left for future investigation.The paper states that deeper networks are expected to improve representation power, but does not develop that extension here.

APPENDIX A: CONSTRUCTING FRRBM FOR THE 2D TORIC CODE STATES

The appendix constructs an exact FRRBM for the 2D toric-code ground state by restricting hidden-to-visible connections locally and solving the resulting operator constraints.

  • APPENDIX A: CONSTRUCTING FRRBM FOR THE 2D TORIC CODE STATES: The 2D toric code uses visible neurons on lattice edges and hidden neurons associated with vertices and faces.The hidden variables and weights are introduced as the parameters of the RBM representation.
  • APPENDIX A: CONSTRUCTING FRRBM FOR THE 2D TORIC CODE STATES: The FRRBM restriction connects each hidden vertex or face neuron only to visible neurons belonging to the corresponding vertex or face.This locality constraint defines the network architecture used for the construction.
  • APPENDIX A: CONSTRUCTING FRRBM FOR THE 2D TORIC CODE STATES: The weight parameters are chosen so the proposed neural-network state satisfies the commuting vertex and face-operator ground-state equations.The appendix first exploits the simpler face-operator constraints before solving the vertex-operator conditions.
  • APPENDIX A: CONSTRUCTING FRRBM FOR THE 2D TORIC CODE STATES: 65,536 local configuration equations arise in the vertex-operator construction, which the authors recast as a numerical optimization problem.The equations involve the spins in the affected local region around a vertex.
  • APPENDIX A: CONSTRUCTING FRRBM FOR THE 2D TORIC CODE STATES: The resulting construction gives an exact ANNQS representation of the 2D toric-code state.The appendix identifies this as the exact representation used in the main text.

APPENDIX B: EXCITED STATES WITH ABELIAN ANYONS

The appendix shows how local FRRBM parameter changes represent toric-code excitations, move anyons along paths, preserve contractible loops, and reproduce their mutual-statistics phase.

  • APPENDIX B: EXCITED STATES WITH ABELIAN ANYONS: A Hopf link between different string types braids an x-type anyon with a z-type anyon and produces an overall phase −1.The FRRBM realizes this through hidden-neuron additions and path-dependent weight-sign flips.
  • APPENDIX B: EXCITED STATES WITH ABELIAN ANYONS: Flipping signs of weights associated with a path represents moving an x-type anyon between faces.The construction exactly reproduces the excited state after the path operation.
  • APPENDIX B: EXCITED STATES WITH ABELIAN ANYONS: Adding hidden neurons along a path represents creating and moving z-type quasiparticles between vertices.The appendix gives a path with five hidden neurons for the illustrated process.
  • APPENDIX B: EXCITED STATES WITH ABELIAN ANYONS: Contractible x- and z-type loops leave the toric-code ground state unchanged under the corresponding FRRBM parameter transformations.The x-loop operation flips path-associated weights, whereas the z-loop adds hidden neurons along its path.
  • APPENDIX B: EXCITED STATES WITH ABELIAN ANYONS: The final RBM state equals the ground state apart from the −1 phase generated during the mixed-string process.The phase is attributed to a sign flip of a specific weight parameter in the construction.
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