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arXiv:2012.10990 · Physical review. B./Physical review. B

Sampling asymmetric open quantum systems for artificial neural networks

Oliver Kaestle, Alexander Carmele

Published May 17, 2021• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.103.195420

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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.
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.
Although artificial neural networks have recently been proven to provide a promising new framework for constructing quantum many-body wave functions, the parametrization of a quantum wave function with non-abelian symmetries in terms of a Boltzmann machine inherently leads to biased results due to the basis dependence. We demonstrate that this problem can be overcome by sampling in the basis of irreducible representations instead of spins, for which the corresponding ansatz respects the non-abelian symmetries of the system. We apply our methodology to find the ground states of the one-dimensional antiferromagnetic Heisenberg (AFH) model with spin-1/2 and spin-1 degrees of freedom, and obtain a substantially higher accuracy than when using the s_{z} basis as an input to the neural network. The proposed ansatz can target excited states, which is illustrated by calculating the energy gap of the AFH model. We also generalize the framework to the case of anyonic spin chains.
Artificial neural networks have been established as a universally applicable tool in both science and industry. In the context of symmetric open quantum systems, specifically the restricted Boltzmann machine architecture has emerged as a natural and highly compressed representation of the density matrix for open spin-1/21/21/2 quantum systems via a direct mapping of spins to the visible neuron layer. Here we present a bit encoding scheme for a highly efficient and scalable representation of bosonic Fock number states in the restricted Boltzmann machine neural network architecture, extending its applicability beyond pure spin-1/21/21/2 open quantum systems. In contrast to common density matrix implementations, the complexity of the neural network scales only with the number of bit-encoded neurons rather than the maximum boson number. Crucially, in the high occupation regime its information compression efficiency is shown to surpass even maximally optimized density matrix implementations, where a projector method is used to access the sparsest Hilbert space representation available.
Artificial neural networks were recently shown to be an efficient representation of highly entangled many-body quantum states. In practical applications, neural-network states inherit numerical schemes used in variational Monte Carlo method, most notably the use of Markov-chain Monte Carlo (MCMC) sampling to estimate quantum expectations. The local stochastic sampling in MCMC caps the potential advantages of neural networks in two ways: (i) Its intrinsic computational cost sets stringent practical limits on the width and depth of the networks, and therefore limits their expressive capacity; (ii) its difficulty in generating precise and uncorrelated samples can result in estimations of observables that are very far from their true value. Inspired by the state-of-the-art generative models used in machine learning, we propose a specialized neural-network architecture that supports efficient and exact sampling, completely circumventing the need for Markov-chain sampling. We demonstrate our approach for two-dimensional interacting spin models, showcasing the ability to obtain accurate results on larger system sizes than those currently accessible to neural-network quantum states.
Monte Carlo simulations of hybrid quantum-classical models such as the double exchange Hamiltonian require calculating the density of states of the quantum degrees of freedom at every step. Unfortunately, the computational complexity of exact diagonalization grows as a function of the system's size NNN, making it prohibitively expensive for any realistic system. We consider leveraging data-driven methods, namely, neural networks, to replace the exact diagonalization step in order to speed up sample generation. We explore a model that learns the free energy for each spin configuration and a second one that learns the Hamiltonian's eigenvalues. We implement data augmentation by taking advantage of the Hamiltonian's symmetries to artificially enlarge our training set and benchmark the different models by evaluating several thermodynamic quantities. While all models considered here perform exceedingly well in the one-dimensional case, only the neural network that outputs the eigenvalues is able to capture the right behavior in two dimensions. The simplicity of the architecture we use in conjunction with the model agnostic form of the neural networks can enable fast sample generation without the need of a researcher's intervention.
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.