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arXiv:2212.13453 · Physical review. B./Physical review. B

Quantum transport in open spin chains using neural-network quantum states

Johannes Mellak, Enrico Arrigoni, Thomas Pock, Wolfgang von der Linden

Published May 1, 2023• 7 citations
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Abstract

In this work we study the treatment of asymmetric open quantum systems with neural networks based on the restricted Boltzmann machine. In particular, we are interested in the nonequilibrium steady-state current in the boundary-driven (anisotropic) Heisenberg spin chain. We address previously published difficulties in treating asymmetric dissipative systems with neural-network quantum states and Monte Carlo sampling and present an optimization method and a sampling technique that can be used to obtain high-fidelity steady-state approximations of such systems. We point out some inherent symmetries of the Lindblad operator under consideration and exploit them during sampling. We show that local observables are not always a good indicator of the quality of the approximation and finally present results for the spin current that are in agreement with known results of simple open Heisenberg chains.

DOI
10.1103/physrevb.107.205102

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In this work we study the treatment of asymmetric open quantum systems with neural networks based on the restricted Boltzmann machine. In particular, we are interested in the nonequilibrium steady-state current in the boundary-driven (anisotropic) Heisenberg spin chain. We address previously published difficulties in treating asymmetric dissipative systems with neural-network quantum states and Monte Carlo sampling and present an optimization method and a sampling technique that can be used to obtain high-fidelity steady-state approximations of such systems. We point out some inherent symmetries of the Lindblad operator under consideration and exploit them during sampling. We show that local observables are not always a good indicator of the quality of the approximation and finally present results for the spin current that are in agreement with known results of simple open Heisenberg chains.
Variational Monte Carlo with neural network quantum states has proven to be a promising avenue for evaluating the ground state energy of spin Hamiltonians. Based on anecdotal evidence, it has been claimed repeatedly in the literature that neural network quantum state simulations are insensitive to the sign problem. We present a detailed and systematic study of restricted Boltzmann machine (RBM) based variational Monte Carlo for quantum spin chains, resolving exactly how relevant stoquasticity is in this setting. We show that in most cases, when the Hamiltonian is phase connected with a stoquastic point, the complex RBM state can faithfully represent the ground state, and local quantities can be evaluated efficiently by sampling. On the other hand, we identify a number of new phases that are challenging for the RBM Ansatz, including non-topological robust non-stoquastic phases as well as stoquastic phases where sampling is nevertheless inefficient. Our results suggest that great care needs to be taken with neural network quantum state based variational Monte Carlo when the system under study is highly frustrated.
Abstract In this work we apply deep neural networks to find the non-equilibrium steady state solution to correlated open quantum many-body systems. Motivated by the ongoing search to find more powerful representations of (mixed) quantum states, we design a simple prototypical convolutional neural network and show that parametrizing the density matrix directly with more powerful models can yield better variational ansatz functions and improve upon results reached by neural density operator based on the restricted Boltzmann machine. Hereby we give up the explicit restriction to positive semi-definite density matrices. However, this is fulfilled again to good approximation by optimizing the parameters. The great advantage of this approach is that it opens up the possibility of exploring more complex network architectures that can be tailored to specific physical properties. We show how translation invariance can be enforced effortlessly and reach better results with fewer parameters. We present results for the dissipative one-dimensional transverse-field Ising model and a two-dimensional dissipative Heisenberg model compared to exact values.
While established neural network approaches based on restricted Boltzmann machine architectures and Metropolis sampling methods are well suited for symmetric open quantum systems, they result in poor scalability and systematic errors for setups without symmetries of translational invariance, independent of training parameters such as the sample size. To overcome this representational limit, we present a hybrid sampling strategy which takes asymmetric properties explicitly into account, achieving fast convergence times and high scalability for asymmetric open systems, underlining the universal applicability of artificial neural networks.
We consider an open quantum system of NNN interacting spin-1/2 particles on a ddd-dimensional lattice whose time evolution is (weakly) symmetric under the action of a finite group GGG that is a subgroup of all the possible permutations of the lattice sites. We define a neural network representation for the system density operator that explicitly accounts for the system symmetries, and we show that its properties are preserved when using variational techniques such those recently used in the optimization of restricted-Boltzmann machines (RBMs). As a proof of principle, we demonstrate the validity of our approach by determining the steady state structure of the one-dimensional dissipative XYZ model in the presence of a uniform magnetic field and uniform dissipation. We first show a direct comparison between the results obtained by optimizing a RBM representation for the density operator and those obtained by means of the invariant neural network defined in the present paper, and then we show results concerning the convergence to the unique steady state configuration when using our approach only. Numerical results support the conjecture that the invariant Ansatz defined in this paper is more expressive that a standard RBM and can be optimized with lesser numerical efforts.
An exact and explicit ladder-tensor-network ansatz is presented for the nonequilibrium steady state of an anisotropic Heisenberg XXZ spin-1/2 chain which is driven far from equilibrium with a pair of Lindblad operators acting on the edges of the chain only. We show that the steady-state density operator of a finite system of size n is-apart from a normalization constant-a polynomial of degree 2n - 2 in the coupling constant. Efficient computation of physical observables is facilitated in terms of a transfer operator reminiscent of a classical Markov process. In the isotropic case we find cosine spin profiles, 1/n(2) scaling of the spin current, and long-range correlations in the steady state. This is a fully nonperturbative extension of a recent result [Phys. Rev. Lett. 106, 217206 (2011)].
Artificial neural networks have been recently introduced as a general ansatz to represent many-body wave functions. In conjunction with variational Monte Carlo calculations, this ansatz has been applied to find Hamiltonian ground states and their energies. Here, we provide extensions of this method to study excited states, a central task in several many-body quantum calculations. First, we give a prescription that allows us to target eigenstates of a (nonlocal) symmetry of the Hamiltonian. Second, we give an algorithm to compute low-lying excited states without symmetries. We demonstrate our approach with both restricted Boltzmann machines and feed-forward neural networks. Results are shown for the one-dimensional spin-1/2 Heisenberg model, and for the one-dimensional Bose-Hubbard model. When comparing to exact results, we obtain good agreement for a large range of excited-states energies. Interestingly, we find that deep networks typically outperform shallow architectures for high-energy states.