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

• URL: http://arxiv.org/abs/2302.00173v1
• Date: Wed, 1 Feb 2023 01:44:03 GMT
• Title: Efficiency of neural-network state representations of one-dimensional quantum spin systems
• Authors: Ruizhi Pan (1), Charles W. Clark (1 and 2) ((1) Joint Quantum Institute, NIST/University of Maryland, College Park, MD, USA, (2) National Institute of Standards and Technology, Gaithersburg, Maryland, USA)
• Abstract summary: We study the Boltzmann machine (RBM) state representation of one-dimensional (1D) quantum spin systems. We define a class of long-rangefastdecay (LRFD) RBM states with quantifiable upper bounds on truncation errors. We conjecture that the ground states of a wide range of quantum systems may be exactly represented by LRFD RBMs or a variant of them.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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