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Efficiency of neural-network state representations of one-dimensional quantum spin systems

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Explain the main idea of the paper?

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Explain the main idea of the paper?

AI: The paper analyzes the computational complexity of 1D quantum spin systems represented by restricted Boltzmann machines (RBM) and proposes a class of long-range-fast-decay (LRFD) RBM states that may approximate a wide range of quantum systems with quantifiable upper bounds on truncation errors. The authors provide numerical evidence and conjecture that LRFD RBMs may exactly represent the ground states of various quantum systems, even in cases where other state representations become less efficient. They also propose a paradigm for analyzing the complexity of generic long-range RBMs and characterize their nonlocal structures, which may apply to higher-dimensional systems and deep neural-network quantum states.

Authors

Ruizhi Pan Joint Quantum Institute, NIST/University of Maryland, College Park, MD, USA, Charles W. Clark Joint Quantum Institute, NIST/University of Maryland, College Park, MD, USA National Institute of Standards and Technology, Gaithersburg, Maryland, USA

Abstract

Neural-network state representations of quantum many-body systems are attracting great attention and more rigorous quantitative analysis about their expressibility and complexity is warranted. Our analysis of the restricted Boltzmann machine (RBM) state representation of one-dimensional (1D) quantum spin systems provides new insight into their computational complexity. We define a class of long-range-fast-decay (LRFD) RBM states with quantifiable upper bounds on truncation errors and provide numerical evidence for a large class of 1D quantum systems that may be approximated by LRFD RBMs of at most polynomial complexities. These results lead us to conjecture that the ground states of a wide range of quantum systems may be exactly represented by LRFD RBMs or a variant of them, even in cases where other state representations become less efficient. At last, we provide the relations between multiple typical state manifolds. Our work proposes a paradigm for doing complexity analysis for generic long-range RBMs which naturally yields a further classification of this manifold. This paradigm and our characterization of their nonlocal structures may pave the way for understanding the natural measure of complexity for quantum many-body states described by RBMs and are generalizable for higher-dimensional systems and deep neural-network quantum states.

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