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arXiv:2405.06534 · Physical Review Research

Automatic structural search of tensor network states including entanglement renormalization

Ryo Watanabe, Hiroshi Ueda

Published Sep 6, 2024• 3 citations
arXivHTMLPDF

Abstract

Tensor network (TN) states, including entanglement renormalization (ER), can encompass a wider variety of entangled states. When the entanglement structure of the quantum state of interest is nonuniform in real space, accurately representing the state with a limited number of degrees of freedom hinges on appropriately configuring the TN to align with the entanglement pattern. However, a proposal has yet to show a structural search of ER due to its high computational cost and the lack of flexibility in its algorithm. In this study, we conducted an optimal structural search of TN, including ER, based on the reconstruction of their local structures with respect to variational energy. First, we demonstrated that our algorithm for the spin- 1 / 2 tetramer singlets model could calculate exact ground energy using the multiscale entanglement renormalization ansatz (MERA) structure as an initial TN structure. Subsequently, we applied our algorithm to the random XY models with the two initial structures: MERA and the suitable structure underlying the strong disordered renormalization group. We found that, in both cases, our algorithm achieves improvements in variational energy, fidelity, and entanglement entropy. The degree of improvement in these quantities is superior in the latter case compared to the former, suggesting that utilizing an existing TN design method as a preprocessing step is important for maximizing our algorithm's performance. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.033259

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Tensor network (TN) states, including entanglement renormalization (ER), can encompass a wider variety of entangled states. When the entanglement structure of the quantum state of interest is nonuniform in real space, accurately representing the state with a limited number of degrees of freedom hinges on appropriately configuring the TN to align with the entanglement pattern. However, a proposal has yet to show a structural search of ER due to its high computational cost and the lack of flexibility in its algorithm. In this study, we conducted an optimal structural search of TN, including ER, based on the reconstruction of their local structures with respect to variational energy. First, we demonstrated that our algorithm for the spin- 1 / 2 tetramer singlets model could calculate exact ground energy using the multiscale entanglement renormalization ansatz (MERA) structure as an initial TN structure. Subsequently, we applied our algorithm to the random XY models with the two initial structures: MERA and the suitable structure underlying the strong disordered renormalization group. We found that, in both cases, our algorithm achieves improvements in variational energy, fidelity, and entanglement entropy. The degree of improvement in these quantities is superior in the latter case compared to the former, suggesting that utilizing an existing TN design method as a preprocessing step is important for maximizing our algorithm's performance. Published by the American Physical Society 2024
Tensor network states have proven to be powerful tools for investigating quantum many-body systems. Here, the authors propose a tensor network method, based on an adaptation of entanglement renormalization, for investigating strongly disordered systems. The method makes use of the strong disorder renormalization group to determine the order in which lattice sites are coarse grained, which sets the overall structure of the corresponding tensor network ansatz, before optimization using variational energy minimization. This approach leads to a new class of efficiently contractible tensor network ansatz for 1D systems, which may be understood as a generalization of the multiscale entanglement renormalization ansatz for disordered systems.
In this paper, we introduce a tensor network (TN) scheme into the entanglement augmentation process of the synergistic optimization framework by Rudolph [] to build its process systematically for inhomogeneous systems. Our synergistic approach first embeds the variational optimal solution of the TN state with the entropic area law, which can be perfectly optimized in conventional (classical) computers, in a quantum variational circuit ansatz inspired by the TN state with the entropic volume law. Next, the framework performs a variational quantum eigensolver (VQE) process with embedded states as the initial state. We applied the synergistic to the ground-state analysis of the all-to-all coupled random transverse-field Ising, XYZ, Heisenberg model, employing the binary multiscale entanglement renormalization ansatz (MERA) state and branching MERA states as TN states with entropic area law and volume law, respectively. We then show that the synergistic accelerates VQE calculations in the three models without an initial parameter guess of the branching-MERA-inspired ansatz and can avoid a local solution trapped by a standard VQE with the ansatz in the Ising model. The improvement of optimizers for MERA in all-to-all coupled inhomogeneous systems, enhancement, and potential synergistic applications are also discussed. Published by the American Physical Society 2024
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).
Strongly correlated quantum matter, such as the insufficiently understood high- temperature superconductors, are difficult to simulate numerically. The multiscale entanglement renormalization ansatz (MERA) is a hierarchical class of tensor networks that efficiently captures the typical ground-state entanglement structures. Motivated by recent advances to employ MERA in quantum-classical algorithms, the authors take a fresh look here, analyzing new variations of the approach including tensor Trotterization and importance sampling. Optimal contraction sequences and algorithmic phase diagrams show that these quantum-inspired methods can substantially reduce computational costs.
Tensor network states (TNSs) offer an efficient representation for the ground state of quantum many-body systems and play an important role in the simulations of them. Numerous TNSs have been proposed in the past few decades. However, due to the high cost of TNS for two-dimensional systems, a balance between the encoded entanglement and computational complexity of TNS is yet to be reached. In this work we introduce a new tree tensor network-based (TTN-based) TNS dubbed a ``fully augmented tree tensor network (FATTN) by releasing the constraint in augmented tree tensor network (ATTN). When disentanglers are augmented in the physical layer of TTN, FATTN can provide more entanglement than TTN and ATTN. At the same time, FATTN maintains the scaling of computational cost with bond dimension in TTN and ATTN. Benchmark results on the ground-state energy for the transverse Ising model are provided to demonstrate the improvement of accuracy of FATTN over TTN and ATTN. Moreover, FATTN is quite flexible, which can be constructed as an interpolation between TTN and the multiscale entanglement renormalization ansatz to reach a balance between the encoded entanglement and the computational cost.
Abstract In tensor-network analysis of quantum many-body systems, it is of crucial importance to employ a tensor network with a spatial structure suitable for representing the state of interest. In the previous work [Hikihara et al., Phys. Rev. Research 5, 013031 (2023)], we proposed a structural optimization algorithm for tree-tensor networks. In this paper, we apply the algorithm to the Rainbow-chain model, which has a product state of singlet pairs between spins separated by various distances as an approximate ground state. We then demonstrate that the algorithm can successfully visualize the spatial pattern of spin-singlet pairs in the ground state.