We introduce a class of tensor-network states in which physical legs are shared among local tensors, termed leg-tied tensor ansätze (LETTA). Physical leg ties encode long-range correlations directly, while a virtual matrix product state (MPS) backbone retains short-range multipartite entanglement. The linear virtual backbone allows us to develop a deterministic density matrix renormalization group-like variational optimization algorithm using exact contractions over the active tie-boundary sets and local minimization. We demonstrate the advantages of LETTA for the two-dimensional frustrated $J_1$--$J_2$ Heisenberg model and the three-dimensional transverse-field Ising model. Our results show that LETTA is substantially more accurate than same-bond-dimension MPS calculations and can typically reach the accuracy of much larger MPS calculations using one order of magnitude fewer variational parameters. LETTA thus opens the door for explicitly correlated tensor-network states that can encode long-range correlation beyond virtual bonds.
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14 pages, 5 figures, including Supplemental Material
We introduce a class of tensor-network states in which physical legs are shared among local tensors, termed leg-tied tensor ansätze (LETTA). Physical leg ties encode long-range correlations directly, while a virtual matrix product state (MPS) backbone retains short-range multipartite entanglement. The linear virtual backbone allows us to develop a deterministic density matrix renormalization group-like variational optimization algorithm using exact contractions over the active tie-boundary sets and local minimization. We demonstrate the advantages of LETTA for the two-dimensional frustrated J1--J2 Heisenberg model and the three-dimensional transverse-field Ising model. Our results show that LETTA is substantially more accurate than same-bond-dimension MPS calculations and can typically reach the accuracy of much larger MPS calculations using one order of magnitude fewer variational parameters. LETTA thus opens the door for explicitly correlated tensor-network states that can encode long-range correlation beyond virtual bonds.
Abstract We present a new variational tree tensor network state (TTNS) ansatz, the three-legged tree tensor network state (T3NS). Physical tensors are interspersed with branching tensors. Physical tensors have one physical index and at most two virtual indices, as in the matrix product state (MPS) ansatz of the density matrix renormalization group (DMRG). Branching tensors have no physical index, but up to three virtual indices. In this way, advantages of DMRG, in particular a low computational cost and a simple implementation of symmetries, are combined with advantages of TTNS, namely incorporating more entanglement. Our code is capable of simulating quantum chemical Hamiltonians, and we present several proof-of-principle calculations on LiF, N2, and the bis(μ-oxo) and μ–η2:η2 peroxo isomers of [Cu2O2]2+.
We introduce Neural Tensor Network States (νTNS), a variational many-body wave-function ansatz that integrates deep neural networks with tensor-network architectures. In the νTNS framework, a neural network serves as a disentangler of the wave-function, transforming the physical degrees of freedom into renormalized variables with much less entanglement. The renormalized state is then efficiently encoded by a back-flow tensor network. This construction yields a compact yet highly expressive representation of strongly correlated quantum states. Using convolutional neural networks combined with matrix product states as a concrete implementation, we obtain state-of-the-art variational energies for the spin-1/2J1-J2 Heisenberg model on the square lattice at the highly frustrated point J2/J1=0.5, for systems up to 20×20 with periodic boundary conditions. Finite-size scaling of spin, dimer, and plaquette correlations exhibits power-law decay without magnetic or valence-bond long-range order, consistent with a gapless quantum spin-liquid ground state at that point.This νTNS framework is flexible and naturally extensible to other neural and tensor-network structures, offering a general platform for investigating strongly correlated quantum many-body systems.
These notes provide an introduction to tensor-network methods in quantum many-body physics, with an emphasis on matrix-product states (MPS). They develop the basic tensor-network language, including graphical notation, virtual indices, bond dimensions, gauge freedom, canonical forms, QR and singular-value decompositions, and the role of entanglement in controlling the efficiency of the representation. The main MPS algorithms are then introduced, including contractions, correlation functions, matrix-product operators, DMRG, and time-evolution methods. The notes also briefly discuss projected entangled-pair states (PEPS) as a higher-dimensional generalization of MPS, together with the basic ideas behind approximate PEPS contraction. Finally, tensor-network representations of mixed states, quantum channels, and Lindblad dynamics are presented, with applications to thermal states and open quantum systems. The presentation is accompanied by short Julia code examples based on ITensor, ITensorMPS, and TensorMixedStates. These notes were written for the 9th Les Houches Summer School on Computational Physics: Open Quantum Systems, held in June 2026.
Multipartite entanglement offers a powerful framework for understanding the complex collective phenomena in quantum many-body systems that are often beyond the description of conventional bipartite entanglement measures. Here, we propose a class of multipartite entanglement measures that incorporate the matrix product state (MPS) representation, enabling the characterization of the optimality of quantum circuits for state preparation. These measures are defined as the minimal distances from a target state to the manifolds of MPSs with specified virtual bond dimensions χ, and thus are dubbed as χ-specified matrix product entanglement (χ-MPE). We demonstrate superlinear, linear, and sublinear scaling behaviors of χ-MPE with respect to the negative logarithmic fidelity F in state preparation, which correspond to excessive, optimal, and insufficient circuit depth D for preparing χ-virtual-dimensional MPSs, respectively. Specifically, a linearly growing χ-MPE with F suggests H_{χ}≃H_{D}, where H_{χ} denotes the manifold of the χ-virtual-dimensional MPSs and H_{D} denotes that of the states accessible by the D-layer circuits. We provide an exact proof that H_{χ=2}≡H_{D=1}. Our results establish tensor networks as a powerful and general tool for developing parametrized measures of multipartite entanglement. The matrix product form adopted in χ-MPE can be readily extended to other tensor network Ansätze, whose scaling behaviors are expected to assess the optimality of quantum circuit in preparing the corresponding tensor network states.
By taking inspiration from the backflow transformation for correlated systems, we introduce a tensor network Ansatz which extends the well-established matrix product state representation of a quantum many-body wave function. This structure provides enough resources to ensure that states in dimensions larger than or equal to one obey an area law for entanglement. It can be efficiently manipulated to address the ground-state search problem by means of an optimization scheme which mixes tensor-network and variational Monte Carlo algorithms. We benchmark the Ansatz against spin models both in one and two dimensions, demonstrating high accuracy and precision. We finally employ our approach to study the challenging S=1/2 two-dimensional (2D) ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, demonstrating that it is competitive with the state-of-the-art methods in 2D.
Recent progress in combining Clifford circuits with tensor-network (TN) methods has shown that local Clifford disentanglers can reduce bipartite entanglement across TN bonds prior to tensor compression, thereby improving the efficiency of TN simulations. In this work, we embed local Clifford disentanglers in the Grassmann-tensor language to define a Clifford-augmented Grassmann matrix product state (CAGMPS) ansatz, and develop a density-matrix renormalization group (DMRG) framework based on this ansatz while preserving locality and fermion-parity structure. We benchmark the resulting CAGMPS--DMRG method on representative fermionic lattice systems, including the tight-binding, t-V, and t-V-V′ models. In all cases, Clifford augmentation systematically suppresses bipartite entanglement and improves the accuracy of the ground-state energy at a fixed bond dimension. We further show that the Grassmann-evenness condition, together with equivalence under entangling action, restricts the relevant two-site Clifford candidates to 12 inequivalent representatives, enabling a more economical disentangling search than approaches based on the standard two-qubit Clifford gate set. Our results suggest that the CAGMPS--DMRG method provides a scalable and efficient variational tool for strongly correlated fermionic systems.