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Constructing neural stationary states for open quantum many-body systems

Nobuyuki Yoshioka, Ryusuke Hamazaki

arXiv:1902.07006v3cond-mat.dis-nncond-mat.stat-mechquant-ph

TL;DR

The paper addresses the computational difficulty of finding stationary states in open quantum many-body systems governed by Lindblad equations. It introduces an RBM-based neural stationary state ansatz and maps the search to a zero-energy ground-state problem optimized by variational Monte Carlo. The method represents stationary states in one- and two-dimensional transverse-field Ising models and a one-dimensional XYZ model, with high fidelity demonstrated for a one-dimensional case.

  • Problem

    Finding stationary states of open quantum many-body systems is difficult because density matrices require resources scaling with the square of the Hilbert-space dimension.

  • Method

    The neural stationary state uses an RBM ansatz and maps stationary-state search to finding the zero-energy ground state of L†L with variational Monte Carlo.

  • Results

    The NSS represents stationary states of dissipative transverse-field Ising models in one and two dimensions and the one-dimensional XYZ model; for one case, fidelity exceeds 0.999.

  • Takeaways & Limitations

    Neural-network quantum states provide a variational route for simulating stationary states of open quantum many-body systems beyond the demonstrated one-dimensional and two-dimensional spin models.

Abstract

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We propose a new variational scheme based on the neural-network quantum states to simulate the stationary states of open quantum many-body systems. Using the high expressive power of the variational ansatz described by the restricted Boltzmann machines, which we dub as the neural stationary state ansatz, we compute the stationary states of quantum dynamics obeying the Lindblad master equations. The mapping of the stationary-state search problem into finding a zero-energy ground state of an appropriate Hermitian operator allows us to apply the conventional variational Monte Carlo method for the optimization. Our method is shown to simulate various spin systems efficiently, i.e., the transverse-field Ising models in both one and two dimensions and the XYZ model in one dimension.

I. INTRODUCTON

The paper introduces neural stationary states (NSS), an RBM-based variational scheme for stationary states of open quantum many-body systems. It maps Lindblad dynamics into a doubled-space problem and demonstrates expressive power for dissipative spin models.

  • NSS uses a restricted Boltzmann machine ansatz to represent stationary states of open quantum many-body systems.
  • The method duplicates the Hilbert space, forms a Lindblad operator on the doubled space, and optimizes the NSS with variational Monte Carlo.
  • A generic NSS exhibits volume-law operator space entanglement entropy, indicating expressive capacity for complex mixed states.
  • The ansatz represents stationary states of dissipative transverse-field Ising models in one and two dimensions and the one-dimensional XYZ model.
  • Compared with tensor-network approaches limited by small operator-space entanglement, the neural-network representation is intended to capture correlations beyond one dimension.

II. OPEN QUANTUM SYSTEMS IN THE LINDBLAD FORM

The Lindblad master equation describes non-unitary, open-system dynamics through Hamiltonian evolution and dissipative jump processes. Its stationary states provide the target for the paper’s variational simulation scheme.

  • A. Lindblad master equation: Open quantum systems interact with external environments and, under conditions such as Markovianity, are described by completely positive, trace-preserving Lindblad dynamics.
  • A. Lindblad master equation: The Lindblad equation evolves a mixed density matrix through a time derivative generated by the Lindblad superoperator.
  • A. Lindblad master equation: The Hamiltonian commutator generates unitary evolution, while dissipative superoperators generate non-unitary dynamics with strengths γ_i.
  • A. Lindblad master equation: Each jump operator Γ_i specifies the details of the corresponding dissipation.

B. Vector representation of the Lindblad equation

The vector representation converts a density matrix into a vector in a doubled Hilbert space, allowing Lindblad evolution to be treated with standard linear algebra. Physical and fictitious spins encode the two operator-space indices.

  • A stationary density matrix is mapped into an operator-space vector |ρ⟩⟩ in the doubled Hilbert space H⊗H.
  • The vector representation expands ρ into basis states |σ,τ⟩⟩, with σ and τ interpreted as physical and fictitious spin configurations.
  • The matrix representation uses unit trace, whereas the vector representation uses unit L2 norm.
  • Left and right actions on the density matrix become operators acting on the doubled-space vector.
  • The Lindblad dynamics becomes a non-Hermitian operator equation in the doubled Hilbert space.
  • Nonstationary eigenmodes have eigenvalues with negative real parts and therefore decay over time.

C. Stationary state as a “ground state” of ˆL† ˆL

The stationary-state search is converted into a zero-energy ground-state problem for the Hermitian positive-semidefinite operator L†L. Its expectation value provides a direct variational cost function.

  • The zero-eigenvalue states of the Hermitian operator L†L correspond to stationary states of the Lindblad dynamics.
  • Because L†L is positive semidefinite, its stationary states can be sought with conventional ground-state variational methods.
  • The approach assumes or benefits from settings where the stationary state is unique, although multiple stationary states are possible in general.
  • The expectation value of L†L is exactly zero for a stationary state, making it a direct indicator of optimization quality.

III. NEURAL STATIONARY STATES

The NSS uses a complex-valued RBM ansatz optimized by variational Monte Carlo to approximate stationary states. Its optimization targets physical density matrices, although positivity and Hermiticity are only approximate before sufficient optimization.

  • The complex-valued RBM introduces auxiliary binary spins and complex parameters to represent the vectorized density matrix.
  • The number ratio α = M/(N + N̄) compares NSS performance across system sizes and hidden-spin counts.
  • Sufficient optimization is expected to make the generally non-positive-semidefinite, non-Hermitian NSS approximately physical.Negative eigenvalues and the deviation from the Hermitian density matrix are reported to be of order 10^-3 and can be reduced by increasing α.
  • The NSS parameters are updated with VMC sampling and stochastic reconfiguration until the cost function reaches approximately 10^-3 or less.

IV. MODEL AND RESULT

The paper evaluates the NSS method on dissipative spin models, using one- and two-dimensional transverse-field Ising models and a one-dimensional XYZ model.

  • The NSS method is tested on transverse-field Ising models in 1D and 2D and on the XYZ model in 1D.These models are described as experimentally realizable using cold atoms or trapped ions.

A. Random-valued NSS

Random-valued NSS states exhibit volume-law operator space entanglement entropy, indicating that the ansatz can represent states with large operator-space complexity. The authors caution that efficient representation is not guaranteed for every volume-law state.

  • Random-valued NSS states show volume-law scaling of operator space entanglement entropy as system size increases.The entropy is defined in the vector representation of the density matrix, where the mixed state becomes a pure state on a doubled Hilbert space.
  • The NSS ansatz can accommodate large operator space entanglement without the tensor-network restriction associated with small operator space entanglement.
  • The authors caution that not all volume-law states can be represented efficiently by NSS.
  • The random-valued construction connects hidden spins to physical and fictitious spins under conjugation constraints that ensure positive semidefiniteness.The total hidden-spin ratio is α = α1 + 2α2, with random real and imaginary parameters drawn from [-r, r].
  • Figure 2 averages random-state results over 10 independently generated states for hidden-spin ratios (α1, α2) = (2, 8), (2, 6), (2, 4), and (2, 2).The finite α1 values show no evident difference, so α1 is fixed at 2.

B. Transverse-field Ising model in one dimension

For a one-dimensional dissipative transverse-field Ising model, the NSS closely reproduces a volume-law stationary state obtained with Lanczos. It achieves fidelity above 0.999, optimizes the residual to order 10^-3, and reproduces entropy accurately.

  • Fidelity exceeds 0.999 between the NSS stationary state and the Lanczos result for the tested volume-law state.The model uses V = 0.3, g = 1, γ = 0.5, and α = 1 under periodic boundary conditions with translation symmetry imposed.
  • The optimized expectation value ⟨⟨L† L⟩⟩ reaches order 10^-3, providing a small residual for the stationary-state approximation.
  • The relative error of the total entropy is of order 10^-3, and individual eigenvalue entropy contributions closely match the exact results.
  • NSS optimization uses substantially less numerical cost and memory than methods operating on the whole Hilbert space.The supplied passage states this comparison qualitatively and points to an appendix for wall-time measurements against Lanczos.

C. Transverse-field Ising model in two dimension

For the 2D transverse-field Ising model, the NSS optimization works well and reproduces stationary-state entropy contributions with high accuracy. The reported fidelity is 0.9996, with total-entropy relative error of order 10^-5.

  • C. Transverse-field Ising model in two dimension: The 2D model is defined on a square lattice with periodic boundary conditions and neighboring-site interactions.
  • C. Transverse-field Ising model in two dimension: The NSS reproduces the entropy contribution for each stationary-state eigenvalue with high accuracy in two dimensions.The comparison is presented alongside the cost-function optimization in Fig. 4.
  • C. Transverse-field Ising model in two dimension: The cost function optimization works well for the 2D transverse-field Ising model.
  • C. Transverse-field Ising model in two dimension: Fidelity reaches 0.9996 for the 2D transverse-field Ising model, while the total-entropy relative error is of order 10^-5.The calculation uses Lx = Ly = 3, α = 4, and γ = 1.
  • D. XYZ model in one dimension: The study also examines a 1D XYZ model, where the NSS describes exact results well despite a non-simple stationary state.The chosen parameters are Jx = 0.9, Jy = 0.4, Jz = 1, and γ = 1.

V. CONCLUSIONS AND OUTLOOKS

The paper proposes the neural stationary state, an RBM-based variational ansatz for open-system stationary states. It demonstrates accurate representations across dissipative Ising and XYZ models, while leaving larger systems and broader model classes for future work.

  • V. CONCLUSIONS AND OUTLOOKS: The method maps the Lindblad stationary-state problem to finding the zero-energy ground state of an appropriate Hermitian operator using an RBM variational ansatz.
  • V. CONCLUSIONS AND OUTLOOKS: The NSS ansatz expresses stationary states of dissipative transverse-field Ising models in one and two dimensions and the one-dimensional XYZ model.
  • V. CONCLUSIONS AND OUTLOOKS: The XYZ benchmark reports fidelity over 0.998 and total-entropy relative error of order 10^-2.
  • V. CONCLUSIONS AND OUTLOOKS: Whether NSS can simulate larger systems and more diverse open quantum many-body models remains an open question.The outlook specifically mentions long-range interacting systems with dissipations.
  • V. CONCLUSIONS AND OUTLOOKS: Related work has also applied complex RBMs and RBM ansätze to time evolution and stationary states of open quantum many-body systems.

Appendix A: Approximating Random Density Matrices by NSS

The appendix tests NSS expressivity on random density matrices and finds a trade-off: larger spin ratios improve approximation but rapidly increase parameter count and numerical cost.

  • Random density matrices generated from Gaussian-unitary-ensemble Hermitian matrices exhibit volume-law operator space entanglement entropy.The matrices have size 2^L × 2^L, and the entropy scales proportionally to L.
  • Larger NSS spin ratios α approximate random density matrices better, but reaching fixed fidelity requires rapidly increasing parameters and numerical cost.
  • The stationary-state benchmark compares NSS optimization with Lanczos scaling for the one-dimensional transverse-field Ising model.The comparison uses V = 2, g = 1, γ = 1, α = 4, Ns = 2000, and Nit = 1500.
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