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arXiv:2502.10695 · PRX Quantum

Alternating and Gaussian Fermionic Isometric Tensor Network States

Yantao Wu, Zhehao Dai, Sajant Anand, Sheng-Hsuan Lin, Qi Yang, Lei Wang, Frank Pollmann, Michael P. Zaletel

Published Sep 30, 2025• 2 citations

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 . Second, we introduce a numerical tool—the (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 ( L x ⋅ L y ) compared to the original isoTNSs of depth O ( L x + L y ) . 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 p x + i p y 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.

DOI: 10.1103/8ypw-c8t4

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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 . Second, we introduce a numerical tool—the (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 ( L x ⋅ L y ) compared to the original isoTNSs of depth O ( L x + L y ) . 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 p x + i p y 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.

Isometric tensor network states (isoTNS) generalize the isometric form of the one-dimensional matrix product states to tensor networks in two and higher dimensions. Here, we introduce an alternative isometric form for isoTNS by incorporating auxiliary tensors to represent the orthogonality hypersurface. We implement the time evolving block decimation algorithm on this new isometric form and benchmark the method by computing ground states and the real time evolution of the transverse field Ising model in two dimensions on large square lattices of up to 1250 sites. Our results demonstrate that isoTNS can efficiently capture the entanglement structure of two-dimensional area law states. The short-time dynamics is also accurately reproduced even at the critical point. Our isoTNS formulation further allows for a natural extension to different lattice geometries, such as the honeycomb or kagome lattice.

Simulating the dynamics of finite two-dimensional (2D) quantum many-body systems poses challenges to numerical methods due to the exponential growth of the many-body Hilbert space. In this work, the authors explore the variational power of recently introduced isometric tensor network states (isoTNS) --- a subset of general tensor-network states that allow an efficient numerical simulation. As a efficient tool to find ground states, the variational density matrix renormalization group (DMRG

{}_{2}

) is introduced. It is then demonstrated that dynamical correlation functions for the 2D transverse field Ising model and the Kitaev Honeycomb model can be obtained.

Tensor-network states (TNS) are a promising but numerically challenging tool for simulating two-dimensional (2D) quantum many-body problems. We introduce an isometric restriction of the TNS ansatz that allows for highly efficient contraction of the network. We consider two concrete applications using this ansatz. First, we show that a matrix-product state representation of a 2D quantum state can be iteratively transformed into an isometric 2D TNS. Second, we introduce a 2D version of the time-evolving block decimation algorithm for approximating of the ground state of a Hamiltonian as an isometric TNS-which we demonstrate for the 2D transverse field Ising model.

Recently, a class of tensor networks called isometric tensor network states (isoTNS) was proposed which generalizes the canonical form of matrix product states to tensor networks in higher dimensions. While this ansatz allows for efficient numerical computations, it remained unclear which phases admit an isoTNS representation. In this work, we show that two-dimensional string-net liquids, which represent a wide variety of topological phases including discrete gauge theories, admit an exact isoTNS representation. We further show that the isometric form can be preserved after applying a finite-depth local quantum circuit. Taken together, these results show that long-range entanglement by itself is not an obstruction to isoTNS representation and suggest that all two-dimensional gapped phases with gappable edges admit an isoTNS representation.

Sampling a quantum system's underlying probability distributions is an important computational task, e.g., for quantum advantage experiments and quantum Monte Carlo algorithms. Tensor networks are an invaluable tool for efficiently representing states of large quantum systems with limited entanglement. Algorithms for sampling one-dimensional (1D) tensor networks are well-established and utilized in several 1D tensor network methods. In this paper we introduce two novel sampling algorithms for two-dimensional (2D) isometric tensor network states (isoTNS) that generalize existing 1D tensor networks sampling algorithms. Our first proposed algorithm performs independent sampling and yields a single configuration together with its associated probability. The second algorithm employs a greedy search strategy to identify

K

high-probability configurations and their corresponding probabilities. Numerical results demonstrate the effectiveness of these algorithms across quantum states with varying entanglement and system size.

We determine the computational power of isometric tensor-network states (isoTNSs), a variational ansatz originally developed to numerically find and compute properties of gapped ground states and topological states in two dimensions. By mapping two-dimensional isoTNSs to ( 1 + 1 )D unitary quantum circuits, we find that the computation of local expectation values in isoTNSs (including those with translation-invariant bulk) is -complete. We then introduce isoTNSs, which are those isoTNSs that are the unique ground states of frustration-free Hamiltonians and which are characterized by an injectivity parameter δ ∈ ( 0 , 1 / D ] , where D is the bond dimension of the isoTNS. We show that injectivity necessarily adds depolarizing noise to the circuit at a rate η = δ 2 D 2 . We show that weakly injective isoTNS (small δ ) are still -complete but that there exists an efficient classical algorithm to compute local expectation values in strongly injective isoTNSs ( η ≥ 0.41 ). We also show that while weakly injective isoTNSs can support long-range correlations, strongly injective isoTNSs are approximate Markov states with exponentially decaying correlations. Sampling from isoTNSs corresponds to monitored quantum dynamics and we exhibit a family of isoTNSs that undergo a phase transition from a hard regime to an easy phase in which the monitored circuit can be sampled efficiently. Our results can be used to design provable algorithms to contract isoTNSs. Our mapping between ground states of certain frustration-free Hamiltonians to open circuit dynamics in one dimension fewer may be of independent interest.