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arXiv:2407.01451 · Physical review. B./Physical review. B

Representing arbitrary ground states of the toric code by a restricted Boltzmann machine

Penghua Chen, Bowen Yan, Shawn Xingshan Cui

Published Jan 2, 2025• 1 citation
arXivHTMLPDF

Abstract

We systematically analyze the representability of toric code ground states by a restricted Boltzmann machine with only local connections between hidden and visible neurons. This analysis is pivotal for evaluating the model's capability to represent diverse ground states, thus enhancing our understanding of its strengths and weaknesses. Subsequently, we modify the restricted Boltzmann machine to accommodate arbitrary ground states by introducing essential nonlocal connections efficiently. This model is not only analytically solvable but also demonstrates efficient and accurate performance when solved using machine learning techniques. Then we generalize our the model from ${Z}_{2}$ to ${Z}_{n}$ toric code and discuss future directions.

DOI
10.1103/physrevb.111.045101

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We systematically analyze the representability of toric code ground states by a restricted Boltzmann machine with only local connections between hidden and visible neurons. This analysis is pivotal for evaluating the model's capability to represent diverse ground states, thus enhancing our understanding of its strengths and weaknesses. Subsequently, we modify the restricted Boltzmann machine to accommodate arbitrary ground states by introducing essential nonlocal connections efficiently. This model is not only analytically solvable but also demonstrates efficient and accurate performance when solved using machine learning techniques. Then we generalize our the model from Z2{Z}_{2}Z2​ to Zn{Z}_{n}Zn​ toric code and discuss future directions.
Machine-learning representations of many-body quantum states have recently been introduced as an ansatz to describe the ground states and unitary evolutions of many-body quantum systems. We investigate one of the most important representations, the restricted Boltzmann machine (RBM), in the stabilizer formalism. A general method to construct RBM representations for stabilizer code states is given, and exact RBM representations for several types of stabilizer groups with the number of hidden neurons equal to or less than the number of visible neurons are presented. The result indicates that the representation is extremely efficient. Then we analyze a surface code with boundaries, defects, domain walls, and twists in full detail and find that almost all the models can be efficiently represented via the RBM ansatz: the RBM parameters of the perfect case, boundary case, and defect case are constructed analytically using the method we provide in the stabilizer formalism, and the domain wall and twist case is studied numerically. In addition, the case for Kitaev's D(Zd)D({\mathbb{Z}}_{d})D(Zd​) model, which is a generalized model of the surface code, is also investigated.
Restricted Boltzmann machines (RBMs) have demonstrated considerable success as variational quantum states; however, their representational power remains incompletely understood. In this work, we present an analytical proof that RBMs can exactly and efficiently represent stabilizer code states-a class of highly entangled quantum states that are central to quantum error correction. Given a set of stabilizer generators, we develop an efficient algorithm to determine both the RBM architecture and the exact values of its parameters. Our findings provide new insights into the expressive power of RBMs, highlighting their capability to encode highly entangled states, and may serve as a useful tool for the classical simulation of quantum error-correcting codes.
Abstract In this work, the capability of restricted Boltzmann machines (RBMs) to find solutions for the Kitaev honeycomb model with periodic boundary conditions is investigated. The measured groundstate energy of the system is compared and, for small lattice sizes (e.g. 3 × 3 with 18 spinors), shown to agree with the analytically derived value of the energy up to a deviation of 0.09 % . Moreover, the wave-functions we find have 99.89 % overlap with the exact ground state wave-functions. Furthermore, the possibility of realizing anyons in the RBM is discussed and an algorithm is given to build these anyonic excitations and braid them for possible future applications in quantum computation. Using the correspondence between topological field theories in (2 + 1)d and 2d conformal field theories, we propose an identification between our RBM states with the Moore-Read state and conformal blocks of the 2d Ising model.
Obtaining accurate properties of many-body interacting quantum matter is a long-standing challenge in theoretical physics and chemistry, rooting into the complexity of the many-body wave-function. Classical representations of many-body states constitute a key tool for both analytical and numerical approaches to interacting quantum problems. Here, we introduce a technique to construct classical representations of many-body quantum systems based on artificial neural networks. Our constructions are based on the deep Boltzmann machine architecture, in which two layers of hidden neurons mediate quantum correlations. The approach reproduces the exact imaginary-time evolution for many-body lattice Hamiltonians, is completely deterministic, and yields networks with a polynomially-scaling number of neurons. We provide examples where physical properties of spin Hamiltonians can be efficiently obtained. Also, we show how systematic improvements upon existing restricted Boltzmann machines ansatze can be obtained. Our method is an alternative to the standard path integral and opens new routes in representing quantum many-body states.
Representation by neural networks, in particular by restricted Boltzmann machines (RBMs), has provided a powerful computational tool to solve quantum many-body problems. An important open question is how to characterize which class of quantum states can be efficiently represented with RBMs. Here, we show that RBMs can efficiently represent a wide class of many-body entangled states with rich exotic topological orders. This includes (1) ground states of double semion and twisted quantum double models with intrinsic topological orders, (2) states of the AKLT model and two-dimensional CZX model with symmetry protected topological orders, (3) states of stabilizer Fracton models with fracton topological order, and (4) (generalized) stabilizer states and hypergraph states that are important for quantum information protocols. One twisted quantum double model state considered here harbors non-Abelian anyon excitations. Our result shows that it is possible to study a variety of quantum models with exotic topological orders and rich physics using the RBM computational toolbox.
Abstract Obtaining accurate properties of many-body interacting quantum matter is a long-standing challenge in theoretical physics and chemistry, rooting into the complexity of the many-body wave-function. Classical representations of many-body states constitute a key tool for both analytical and numerical approaches to interacting quantum problems. Here, we introduce a technique to construct classical representations of many-body quantum systems based on artificial neural networks. Our constructions are based on the deep Boltzmann machine architecture, in which two layers of hidden neurons mediate quantum correlations. The approach reproduces the exact imaginary-time evolution for many-body lattice Hamiltonians, is completely deterministic, and yields networks with a polynomially-scaling number of neurons. We provide examples where physical properties of spin Hamiltonians can be efficiently obtained. Also, we show how systematic improvements upon existing restricted Boltzmann machines ansatze can be obtained. Our method is an alternative to the standard path integral and opens new routes in representing quantum many-body states.