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arXiv:2405.03797 · Physical Review Letters

Tensor Network Computations That Capture Strict Variationality, Volume Law Behavior, and the Efficient Representation of Neural Network States

Wen-Yuan Liu, Si-Jing Du, Ruojing Peng, Johnnie Gray, Garnet Kin‐Lic Chan

Published Dec 30, 2024• 9 citations
arXivHTMLPDF

Abstract

We introduce a change of perspective on tensor network states that is defined by the computational graph of the contraction of an amplitude. The resulting class of states, which we refer to as tensor network functions, inherit the conceptual advantages of tensor network states while removing computational restrictions arising from the need to converge approximate contractions. We use tensor network functions to compute strict variational estimates of the energy on loopy graphs, analyze their expressive power for ground states, show that we can capture aspects of volume law time evolution, and provide a mapping of general feed-forward neural nets onto efficient tensor network functions. Our work expands the realm of computable tensor networks to ones where accurate contraction methods are not available, and opens up new avenues to use tensor networks.

DOI
10.1103/physrevlett.133.260404

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We introduce a change of perspective on tensor network states that is defined by the computational graph of the contraction of an amplitude. The resulting class of states, which we refer to as tensor network functions, inherit the conceptual advantages of tensor network states while removing computational restrictions arising from the need to converge approximate contractions. We use tensor network functions to compute strict variational estimates of the energy on loopy graphs, analyze their expressive power for ground states, show that we can capture aspects of volume law time evolution, and provide a mapping of general feed-forward neural nets onto efficient tensor network functions. Our work expands the realm of computable tensor networks to ones where accurate contraction methods are not available, and opens up new avenues to use tensor networks.
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We study tensor network states defined on an underlying graph which is sparsely connected. Generic sparse graphs are expander graphs with a high probability, and one can represent volume law entangled states efficiently with only polynomial resources. We find that message-passing inference algorithms such as belief propagation can lead to efficient computation of local expectation values for a class of tensor network states defined on sparse graphs. As applications, we study local properties of square root states, graph states, and also employ this method to variationally prepare ground states of gapped Hamiltonians defined on generic graphs. Using the variational method we study the phase diagram of the transverse field quantum Ising model defined on sparse expander graphs.
Abstract Computing free energy is a fundamental problem in statistical physics. Recently, two distinct methods have been developed and have demonstrated remarkable success: the tensor-network-based contraction method and the neural-network-based variational method. Tensor networks are accurate, but their application is often limited to low-dimensional systems due to the high computational complexity in high-dimensional systems. The neural network method applies to systems with general topology. However, as a variational method, it is not as accurate as tensor networks. In this work, we propose an integrated approach, tensor-network-based variational autoregressive networks (TNVAN), that leverages the strengths of both tensor networks and neural networks: combining the variational autoregressive neural network’s ability to compute an upper bound on free energy and perform unbiased sampling from the variational distribution with the tensor network’s power to accurately compute the partition function for small sub-systems, resulting in a robust method for precisely estimating free energy. To evaluate the proposed approach, we conducted numerical experiments on spin glass systems with various topologies, including two-dimensional lattices, fully connected graphs, and random graphs. Our numerical results demonstrate the superior accuracy of our method compared to existing approaches. In particular, it effectively handles systems with long-range interactions and leverages GPU efficiency without requiring singular value decomposition, indicating great potential in tackling statistical mechanics problems and simulating high-dimensional complex systems through both tensor networks and neural networks.
We analyze the tensor network loop cluster expansion, introduced in [G. Park, J. Gray, and G. K.-L. Chan, Phys. Rev. B 112, 174310 (2025)] as a systematic correction to belief propagation, in the context of general quantum many-body problems. We provide numerical examples of the accuracy and practical applicability of the approach for the computation of ground-state observables for high bond dimension tensor networks, in two- and three-dimensions, with open and periodic boundary conditions, and for spin and fermion problems. We find that the contraction error converges approximately exponentially with cluster size, enabling accurate local observable and energy estimates for many systems where standard contraction methods are otherwise impractical.
Tensor network methods, most prominently matrix product states (MPS), have become fundamental tools in modern quantum many-body physics. While MPS and extensions like the multiscale entanglement renormalization ansatz (MERA) and tree tensor networks (TTN) efficiently capture area-law entanglement and its logarithmic violations, they inherently struggle to represent highly entangled wavefunctions. Specifically, reaching the volume-law regime typically demands exponential resources within these conventional frameworks. Motivated by this challenge, we propose holographic isometric tensor network states (holographic isoTNS) that simulate quantum lattice models in DDD spatial dimensions via (D+1)(D+1)(D+1)-dimensional networks of tensors. The additional dimension substantially enlarges the representational manifold, while isometric constraints on each tensor ensure efficient contractibility. Using one-dimensional systems as testbeds, we analyze the properties of holographic isoTNS. First, we show that randomly initialized holographic isoTNS typically display volume-law entanglement at fixed bond dimension. Second, through analytic constructions and variational optimization, we demonstrate that holographic isoTNS can faithfully describe a broad class of highly entangled yet low-complexity states. In particular, the ansatz can represent arbitrary fermionic Gaussian states, Clifford states, extensions of rainbow states, and certain short-time-evolved states under local evolution. Third, to exploit this expressivity in broader contexts, we implement a time-evolving block decimation (TEBD) algorithm on holographic isoTNS. While the method remains efficient and scalable, error accumulation over TEBD sweeps suggests the need for further algorithmic improvement. Overall, holographic isoTNS broaden the scope of tensor-network methods, opening new avenues to study physics in the volume-law regime.