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arXiv:2511.08407 · Physical review. E

Exploring the performance of superposition of product states: from 1D to 3D quantum spin systems

Apimuk Sornsaeng, Itai Arad, Dario Poletti

quant-phSubmitted Nov 11, 2025• Announced Nov 12, 2025• 0 citations
arXivHTMLPDF

Abstract

Tensor networks (TNs) are one of the best available tools to study many-body quantum systems. TNs are particularly suitable for one-dimensional local Hamiltonians, while their performance for generic geometries is mainly limited by two aspects: the limitation in expressive power and the approximate extraction of information. Here we investigate the performance of superposition-of-product-states (SPS) ansatz, a variational framework structurally related to canonical polyadic tensor decomposition. The ansatz does not compress information as effectively as tensor networks, but it has the advantages (i) of allowing accurate extraction of information, (ii) of being structurally independent of the geometry of the system, (iii) of being readily parallelizable, and (iv) of allowing analytical shortcuts. We first study the typical properties of the SPS ansatz for spin-$1/2$ systems, including its entanglement entropy, and its trainability. We then use this ansatz for ground state search in tilted Ising models -- including one-dimensional and three-dimensional with short- and long-range interaction, and a random network -- demonstrating that SPS can attain high accuracy.

DOI
10.1103/2p2k-3j2s
Comments
12 pages, 6 figures

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Tensor networks (TNs) are one of the best available tools to study many-body quantum systems. TNs are particularly suitable for one-dimensional local Hamiltonians, while their performance for generic geometries is mainly limited by two aspects: the limitation in expressive power and the approximate extraction of information. Here we investigate the performance of superposition-of-product-states (SPS) ansatz, a variational framework structurally related to canonical polyadic tensor decomposition. The ansatz does not compress information as effectively as tensor networks, but it has the advantages (i) of allowing accurate extraction of information, (ii) of being structurally independent of the geometry of the system, (iii) of being readily parallelizable, and (iv) of allowing analytical shortcuts. We first study the typical properties of the SPS ansatz for spin-1/21/21/2 systems, including its entanglement entropy, and its trainability. We then use this ansatz for ground state search in tilted Ising models -- including one-dimensional and three-dimensional with short- and long-range interaction, and a random network -- demonstrating that SPS can attain high accuracy.
We introduce Neural Tensor Network States (νννTNS), a variational many-body wave-function ansatz that integrates deep neural networks with tensor-network architectures. In the νννTNS 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-1/21/21/2 J1J_1J1​-J2J_2J2​ Heisenberg model on the square lattice at the highly frustrated point J2/J1=0.5J_2/J_1=0.5J2​/J1​=0.5, for systems up to 20×2020\times 2020×20 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 νννTNS 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.
This chapter introduces the tensor network ansatz for a quantum state whose entanglement entropy obeys the so-called area law with or without logarithmic corrections. This ansatz represents a quantum many-body wave function by a network product of local tensors defined on the lattice sites and treats all tensor elements as variational parameters. It includes, for example, one-dimensional matrix product states (MPS) and two-dimensional projected entangled pair states (PEPS) or projected entangled simplex states (PESS). A typical example is the spin-1 AKLT chain, whose ground state can be exactly represented as an MPS. If a logarithmic correction to the entanglement area law emerges, a tensor network state termed the multi-scale entanglement renormalization ansatz (MERA) describes the entanglement structure of the ground state more accurately in one dimension.
In this thesis, we use the new numerical method multiscale entanglement renormalization ansatz (MERA) to study the one dimensional spin system. This method, MERA, utilizes a network of tensors to represent quantum many-body systems on a lattice. Other examples of tensor network state are matrix product states (MPSs) for 1D systems, tree tensor networks (TTNs) for systems with a tree shape, and projected entangled-pair states (PEPSs) for 2D systems and be- yond. The three structures dier in the graph that denes how the tensors are interconnected into a network: The graphs for MPSs, TTNs, and 2D PEPSs are, respectively, a chain, a tree, and a 2D lattice. Importantly, from these tensor networks the expectation value of local observables can be computed eciently. Developing algorithms to simulate quantum many-body systems is important for understanding and studying the physical principles. And these may become powerful tools for the analysis of quantum computation and quantum information theory. But in general ecient simulation is dicult, due to the fundamental quantum physical laws. The main method we have studied and worked on is multiscale entangle- ment renormalization ansatz (MERA) which is a variational ansatz for many- body states. MERA is a tree tensor network for 1D, 2D systems and beyond. We have developed code of ternary-MERA to compute the ground state energy, expected values of local observables, and correlators for 1D lattice systems. In a nutshell, MERA is an algorithm for entanglement renormalization which is a numerical technique based on locally reorganizing the Hilbert space of a quantum many-body system with the aim to reduce the amount of entanglement in its wave function. We use this method to study 1D Ising model, XXZ model, and J1 - J2 model, and then comparing the numerical results with exact diagonalization (ED) and density matrix renormalization group (DMRG).
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=1/2S=1/2S=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.
It appears that the tensor network ansatz, while not quite complete, is an efficient coordinate system for the tiny subset of a many-body Hilbert space which can be realized as a low energy state of a local Hamiltonian.However, we don't fully understand precisely which phases are captured by the tensor network ansatz, how to compute their physical observables (even numerically), or how to compute a tensor network representation for a ground state given a microscopic Hamiltonian. These questions are algorithmic in nature, but their resolution is intimately related to understanding the nature of quantum entanglement in many-body systems.For this reason it is useful to compute the tensor network representation of various `model' wavefunctions representative of different phases of matter; this allows us to understand how the entanglement properties of each phase are expressed in the tensor network ansatz, and can serve as test cases for algorithm development.Condensed matter physics has many illuminating model wavefunctions, such as Laughlin's celebrated wave function for the fractional quantum Hall effect, the Bardeen-Cooper-Schrieffer wave function for superconductivity, and Anderson's resonating valence bond ansatz for spin liquids.This thesis presents some results on exact tensor network representations of these model wavefunctions. In addition, a tensor network representation is given for the time evolution operator of a long-range one-dimensional Hamiltonian, which allows one to numerically simulate the time evolution of power-law interacting spin chains as well as two-dimensional strips and cylinders.
We introduce the concept of concatenated tensor networks to efficiently describe quantum states. We show that the corresponding concatenated tensor network states can efficiently describe time evolution and possess arbitrary block-wise entanglement and long-ranged correlations. We illustrate the approach for the enhancement of matrix product states, i.e. one-dimensional (1D) tensor networks, where we replace each of the matrices of the original matrix product state with another 1D tensor network. This procedure yields a 2D tensor network, which includes—already for tensor dimension 2—all states that can be prepared by circuits of polynomially many (possibly non-unitary) two-qubit quantum operations, as well as states resulting from time evolution with respect to Hamiltonians with short-ranged interactions. We investigate the possibility of efficiently extracting information from these states, which serves as the basic step in a variational optimization procedure. To this aim, we utilize the known exact and approximate methods for 2D tensor networks and demonstrate some improvements thereof, which are also applicable e.g. in the context of 2D projected entangled pair states. We generalize the approach to higher dimensional and tree tensor networks.