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  • Open Access

Alternating and Gaussian Fermionic Isometric Tensor Network States

Yantao Wu1,*, Zhehao Dai2,3, Sajant Anand2, Sheng-Hsuan Lin4, Qi Yang1,5, Lei Wang1, Frank Pollmann4, and Michael P. Zaletel2,6

  • *Contact author: yantaow@iphy.ac.cn
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PRX Quantum 6, 040324 – Published 5 November, 2025

DOI: https://doi.org/10.1103/8ypw-c8t4

Abstract

Isometric tensor networks in two dimensions enable efficient and accurate study of quantum many-body states, yet the effect of the isometric restriction on the represented quantum states is not fully understood. We address this question in two main contributions. First, we introduce an improved variant of isometric tensor network states (isoTNS) in two dimensions, where the isometric arrows on the columns of the network alternate between pointing upward and downward; hence the name alternating isometric tensor network states. Second, we introduce a numerical tool—the isometric Gaussian fermionic TNS (isoGfTNS)—that incorporates isometric constraints into the framework of Gaussian fermionic tensor network states. We demonstrate in numerous ways that alternating isoTNSs represent many-body ground states of two-dimensional quantum systems significantly better than the original isoTNSs. First, we show that the entanglement in an isoTNS is mediated along the isometric arrows and that alternating isoTNSs mediate entanglement more efficiently than conventional isoTNSs. Second, alternating isoTNSs correspond to a deeper, and thus more representative, sequential-circuit construction of depth O(Lx⋅Ly) compared to the original isoTNSs of depth O(Lx+Ly). Third, using the Gaussian framework and gradient-based energy minimization, we provide numerical evidence of better bond-dimension scaling and variational energy of alternating isoGfTNSs for ground states of various free-fermionic models, including the Fermi surface, the band insulator, and the px+ipy mean-field superconductor. Finally, benchmarking on the transverse-field Ising model, we demonstrate that an alternating isoTNS provides substantially improved performance and stability relative to the original isoTNS for the ground-state search algorithm in interacting systems.

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Popular Summary

Quantum many-body systems—ensembles of many interacting particles—are at the heart of some of the most intriguing phenomena in physics, from superconductors to magnets. The difficulty of simulating these systems on classical computers is related to the amount of “quantum entanglement” in a system. Tensor network methods offer a promising way to tackle this by compressing quantum states into manageable chunks. However, an open question remains: How is entanglement mediated by tensor networks?

In this work, we explore a special class of tensor networks known as “isometric tensor network states,” which can be thought of as providing a blueprint for how a quantum state could be built or “prepared” as a quantum circuit. We introduce a new layout—called alternating isometric tensor networks—that we show can more effectively channel entanglement in two dimensions. To show this, we also develop a related framework called isometric Gaussian fermion tensor networks, which allows us to test how well these representations capture the ground states of various fermionic quantum systems. Across a variety of models, we find that our new layout more accurately represents the true quantum states, without any added computational cost.

Looking ahead, this approach could open the door to more efficient simulations of complex quantum materials and could even inform new ways to build quantum circuits on actual quantum computers. By optimizing how we structure these tensor networks, we hope to push the frontier of classical simulations of quantum systems—and better understand which exotic states of matter can be realistically modeled today.

Article Text

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PRX Quantum 6, 040324– Published 5 November, 2025
  • Received 28 February 2025
  • Revised 18 June 2025
  • Accepted 30 September 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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