Tensor Network States Research Articles

Confinement in the Transverse Field Ising Model on the Heavy Hex Lattice.

Inspired by a recent quantum computing experiment [Y. Kim etal., Nature (London), 618, 500-5 (2023)NATUAS0028-083610.1038/s41586-023-06096-3], we study the emergence of confinement in the transverse field Ising model on a decorated hexagonal lattice. Using an infinite tensor network state optimized with belief propagation we show how a quench from a broken symmetry state leads to striking nonthermal behavior underpinned by persistent oscillations and saturation of the entanglement entropy. We explain this phenomenon by constructing a minimal model based on the confinement of elementary excitations. Our model is in excellent agreement with our numerical results. For quenches to larger values of the transverse field and/or from nonsymmetry broken states, our numerical results display the expected signatures of thermalization: a linear growth of entanglement entropy in time, propagation of correlations, and the saturation of observables to their thermal averages. These results provide a physical explanation for the unexpected classical simulability of the quantum dynamics.

Tensor Network State Algorithms on AI Accelerators.

We introduce novel algorithmic solutions for hybrid CPU-multiGPU tensor network state algorithms utilizing non-Abelian symmetries building on AI-motivated state-of-the-art hardware and software technologies. The presented numerical simulations on the FeMo cofactor, which plays a crucial role in converting atmospheric nitrogen to ammonia, are far beyond the scope of traditional approaches. Our large-scale SU(2) spin adapted density matrix renormalization group calculations up to bond dimension D = 216 on complete active space (CAS) size of 18 electrons in 18 orbitals [CAS(18, 18)] demonstrate that the current limit of exact solution, i.e. full-CI limit, can be achieved in fraction of time. Furthermore, benchmarks up to CAS(113, 76) demonstrate the utilization of NVIDIA's highly specialized AI accelerators via NVIDIA Tensor Cores, leading to performance around 115 TFLOPS on a single node supplied with eight NVIDIA A100 devices. As a consequence of reaching 71% of the full capacity of the hardware, the cubic scaling of computational time with bond dimension can be reduced to a linear form for a broad range of D values; thus, breaking the current computational limits of small CAS spaces in ab initio quantum chemistry and material science is becoming a reality. In comparison to strict U(1) implementations with matching accuracy, our solution has an estimated effective performance of 300-500 TFLOPS, which emphasizes the mutual need for both algorithmic and technological developments to push current frontiers on classical computation.

Unveiling correlated two-dimensional topological insulators through fermionic tensor network states—classification, edge theories and variational wavefunctions

The study of topological band insulators has revealed fascinating phases characterized by band topology indices and anomalous boundary modes protected by global symmetries. In strongly correlated systems, where the traditional notion of electronic bands becomes obsolete, it has been established that topological insulator phases persist as stable phases, separate from the trivial insulators. However, due to the inability to express the ground states of such systems as Slater determinants, the formulation of generic variational wave functions for numerical simulations is highly desirable. In this paper, we tackle this challenge for two-dimensional topological insulators by developing a comprehensive framework for fermionic tensor network states. Starting from simple assumptions, we obtain possible sets of tensor equations for any given symmetry group, capturing consistent relations governing symmetry transformation rules on tensor legs. We then examine the connection between these tensor equations andnon-chiraltopological insulators by constructing edge theories and extracting quantum anomaly data from each set of tensor equations. By exhaustively exploring all possible sets of equations, we achieve a systematic classification of non-chiral topological insulator phases. Imposing the solutions of a given set of equations onto local tensors, we obtain generic variational wavefunctions for the corresponding topological insulator phases. Our methodology provides an important step toward simulating topological insulators in strongly correlated systems. We discuss the limitations and potential generalizations of our results, paving the way for further advancements in this field.

Automatic structural search of tensor network states including entanglement renormalization

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

MPSDynamics.jl: Tensor network simulations for finite-temperature (non-Markovian) open quantum system dynamics.

The MPSDynamics.jl package provides an easy-to-use interface for performing open quantum systems simulations at zero and finite temperatures. The package has been developed with the aim of studying non-Markovian open system dynamics using the state-of-the-art numerically exact Thermalized-Time Evolving Density operator with Orthonormal Polynomials Algorithm based on environment chain mapping. The simulations rely on a tensor network representation of the quantum states as matrix product states (MPS) and tree tensor network states. Written in the Julia programming language, MPSDynamics.jl is a versatile open-source package providing a choice of several variants of the Time-Dependent Variational Principle method for time evolution (including novel bond-adaptive one-site algorithms). The package also provides strong support for the measurement of single and multi-site observables, as well as the storing and logging of data, which makes it a useful tool for the study of many-body physics. It currently handles long-range interactions, time-dependent Hamiltonians, multiple environments, bosonic and fermionic environments, and joint system-environment observables.

TNSP: A framework supporting symmetry and fermion tensors for tensor network state methods

Recent advancements have established tensor network states (TNS) as formidable tools for exploring the complex realm of strongly-correlated many-particle systems in both one and two dimensions. To tackle the challenges presented by strongly-correlated fermion systems, various fermion tensor network states (f-TNS) methodologies have been developed. However, implementing f-TNS methods poses substantial challenges due to their particularly complex nature, making development efforts significantly difficult. This complexity is further exacerbated by the lack of underlying software packages that facilitate the development of f-TNS. Previously, we developed TNSPackage, a software package designed for TNS methods [1]. Initially, this package was only capable of handling spin and boson models. To confront the challenges presented by f-TNS, TNSPackage has undergone significant enhancements in its latest version, incorporating support for both symmetry and fermion tensors. This updated version provides a uniform interface for the consistent management of tensors across boson, fermion, and various symmetry types, maintaining its user-friendly and versatile nature. This greatly facilitates the development of programs based on f-TNS. The new TNSP framework consists of two principal components: a low-level tensor package named TAT, which supports sophisticated tensor operations, and a high-level interface package called tetragono that is built upon TAT. The tetragono package is designed to significantly simplify the development of complex physical models on square lattices. The TNSPackage framework enables users to implement a wide range of physical models with greater ease, without the need to pay close attention to the underlying implementation details.

Optimal tree tensor network operators for tensor network simulations: Applications to open quantum systems.

Tree tensor network states (TTNS) decompose the system wavefunction to the product of low-rank tensors based on the tree topology, serving as the foundation of the multi-layer multi-configuration time-dependent Hartree method. In this work, we present an algorithm that automatically constructs the optimal and exact tree tensor network operators (TTNO) for any sum-of-product symbolic quantum operator. The construction is based on the minimum vertex cover of a bipartite graph. With the optimal TTNO, we simulate open quantum systems, such as spin relaxation dynamics in the spin-boson model and charge transport in molecular junctions. In these simulations, the environment is treated as discrete modes and its wavefunction is evolved on equal footing with the system. We employ the Cole-Davidson spectral density to model the glassy phonon environment and incorporate temperature effects via thermo-field dynamics. Our results show that the computational cost scales linearly with the number of discretized modes, demonstrating the efficiency of our approach.

Variational manifolds for ground states and scarred dynamics of blockade-constrained spin models on two- and three-dimensional lattices

We introduce a variational manifold of simple tensor network states for the study of a family of constrained models that describe spin-12 systems as realized by Rydberg atom arrays. Our manifold permits analytical calculation via perturbative expansion of one- and two-point functions in arbitrary spatial dimensions and allows for efficient computation of the matrix elements required for variational energy minimization and variational time evolution in up to three dimensions. We apply this framework to the PXP model on the hypercubic lattice in one, two, and three dimensions and show that, in each case, it exhibits quantum phase transitions breaking the sublattice symmetry in equilibrium, and hosts quantum many-body scars out of equilibrium. We demonstrate that our variational ansatz qualitatively captures all these phenomena and predicts key quantities with an accuracy that increases with the dimensionality of the lattice, and conclude that our method can be interpreted as a generalization of mean-field theory to constrained spin models. Published by the American Physical Society 2024

Variational quantum eigensolver with embedded entanglement using a tensor-network ansatz

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

Variational Tensor Wave Functions for the Interacting Quantum Spin Hall Phase.

The quantum spin hall (QSH) phase, also known as the 2D topological insulator, is characterized by protected helical edge modes arising from time reversal symmetry. While initially proposed as band insulators, this phase can also manifest in strongly correlated systems where conventional band theory fails. To overcome the challenge of simulating this phase in realistic correlated models, we propose a novel framework utilizing fermionic tensor network states. Our approach involves constructing a tensor representation of the fixed-point wave function based on an exact solvable model, enabling us to derive a set of tensor equations governing the transformation rules of local tensors under symmetry operations. These tensor equations lead to the anomalous edge theory, which provides a comprehensive description of the QSH phase. By solving these tensor equations, we obtain variational ansatz for the QSH phase, which we subsequently verify its topological properties through numerical calculations. This method serves as an initial step toward employing tensor algorithms to simulate the QSH phase in strongly correlated systems, opening new avenues for investigating and understanding topological phenomena in complex materials.

Transforming two-dimensional tensor networks into quantum circuits for supervised learning

There have been numerous quantum neural networks reported, but they struggle to match traditional neural networks in accuracy. Given the huge improvement of the neural network models’ accuracy by two-dimensional tensor network (TN) states in classical tensor network machine learning (TNML), it is promising to explore whether its application in quantum machine learning can extend the performance boundary of the models. Here, we transform two-dimensional TNs into quantum circuits for supervised learning. Specifically, we encode two-dimensional TNs into quantum circuits through rigorous mathematical proofs for constructing model ansätze, including string-bond states, entangled-plaquette states and isometric TN states. In addition, we propose adaptive data encoding methods and combine with TNs. We construct a tensor-network-inspired quantum circuit (TNQC) supervised learning framework for transferring TNML from classical to quantum, and build several novel two-dimensional TN-inspired quantum classifiers based on this framework. Finally, we propose a parallel quantum machine learning method for multi-class classification to construct 2D TNQC-based multi-class classifiers. Classical simulation results on the MNIST benchmark dataset show that our proposed models achieve the state-of-the-art accuracy performance, significantly outperforming other quantum classifiers on both binary and multi-class classification tasks, and beat simple convolutional classifiers on a fair track with identical inputs. The noise resilience of the models makes them successfully run and work in a real quantum computer.

A tensor network view of multilayer multiconfiguration time-dependent Hartree methods

The multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method and the density matrix renormalisation group (DMRG) are powerful workhorses applied mostly in different scientific fields. Although both methods are based on tensor network states, very different mathematical languages are used for describing them. This severely limits knowledge transfer and sometimes leads to re-inventions of ideas well known in the other field. Here, we review ML-MCTDH and DMRG theory using both MCTDH expressions and tensor network diagrams. We derive the ML-MCTDH equations of motions using diagrams and compare them with time-dependent and time-independent DMRG algorithms. We further review two selected recent advancements. The first advancement is related to optimising unoccupied single-particle functions in MCTDH, which corresponds to subspace enrichment in the DMRG. The second advancement is related to finding optimal tree structures and on highlighting similarities and differences of tensor networks used in MCTDH and DMRG theories. We hope that this contribution will foster more fruitful cross-fertilisation of ideas between ML-MCTDH and DMRG.

Solving the Electronic Schrödinger Equation by Pairing Tensor-Network State with Neural Network Quantum State

Neural network methods have shown promise for solving complex quantum many-body systems. In this study, we develop a novel approach through incorporating the density-matrix renormalization group (DMRG) method with the neural network quantum state method. The results demonstrate that, when tensor-network pre-training is introduced into the neural network, a high efficiency can be achieved for quantum many-body systems with strong correlations.

2500 vibronic eigenstates of the NO3 radical.

The nitrate radical NO3 plays an important role in atmospheric chemistry, yet many aspects of its coupled and anharmonic vibronic structure remain elusive. Here, using an accurate, coupled full-dimensional diabatic potential that includes five electronic states, we revisit the vibronic spectrum associated with the electronic state. Using recently developed tensor network state methods, we are able to compute more than 2500 vibronic states, thereby increasing the number of computed full-dimensional states by a factor of 50, compared to previous work. While we obtain good agreement with experiment for most of the assigned vibronic levels, for several others, we observe striking disagreement. Further, for the antisymmetric bending motion we find remarkably large symmetry-induced level splittings that are larger than the zero-order reference. We discuss non-negligible nonadiabatic effects and show that the Born-Oppenheimer approximation leads to significant errors in the spectrum.

Finite-Entanglement Scaling of 2D Metals.

We extend the study of finite-entanglement scaling from one-dimensional gapless models to two-dimensional systems with a Fermi surface. In particular, we show that the entanglement entropy of a contractible spatial region with linear size L scales as S∼Llog[ξf(L/ξ)] in the optimal tensor network, and hence area-law entangled, state approximation to a metallic state, where f(x) is a scaling function which depends on the shape of the Fermi surface and ξ is a finite correlation length induced by the restricted entanglement. Crucially, the scaling regime can be realized with numerically tractable bond dimensions. We also discuss the implications of the Lieb-Schultz-Mattis theorem at fractional filling for tensor network state approximations of metallic states.

Quantum phase transition between symmetry enriched topological phases in tensor-network states

Quantum phase transition between symmetry enriched topological phases in tensor-network states

Typical Correlation Length of Sequentially Generated Tensor Network States

Typical Correlation Length of Sequentially Generated Tensor Network States

Tensor Network Efficiently Representing Schmidt Decomposition of Quantum Many-Body States.

Efficient methods to access the entanglement of a quantum many-body state, where the complexity generally scales exponentially with the system size N, have long been a concern. Here we propose the Schmidt tensor network state (Schmidt TNS) that efficiently represents the Schmidt decomposition of finite- and even infinite-size quantum states with nontrivial bipartition boundary. The key idea is to represent the Schmidt coefficients (i.e., entanglement spectrum) and transformations in the decomposition to tensor networks (TNs) with linearly scaled complexity versus N. Specifically, the transformations are written as the TNs formed by local unitary tensors, and the Schmidt coefficients are encoded in a positive-definite matrix product state (MPS). Translational invariance can be imposed on the TNs and MPS for the infinite-size cases. The validity of Schmidt TNS is demonstrated by simulating the ground state of the quasi-one-dimensional spin model with geometrical frustration. Our results show that the MPS encoding the Schmidt coefficients is weakly entangled even when the entanglement entropy of the decomposed state is strong. This justifies the efficiency of using MPS to encode the Schmidt coefficients, and promises an exponential speedup on the full-state sampling tasks.

Evidence for the utility of quantum computing before fault tolerance

Quantum computing promises to offer substantial speed-ups over its classical counterpart for certain problems. However, the greatest impediment to realizing its full potential is noise that is inherent to these systems. The widely accepted solution to this challenge is the implementation of fault-tolerant quantum circuits, which is out of reach for current processors. Here we report experiments on a noisy 127-qubit processor and demonstrate the measurement of accurate expectation values for circuit volumes at a scale beyond brute-force classical computation. We argue that this represents evidence for the utility of quantum computing in a pre-fault-tolerant era. These experimental results are enabled by advances in the coherence and calibration of a superconducting processor at this scale and the ability to characterize1 and controllably manipulate noise across such a large device. We establish the accuracy of the measured expectation values by comparing them with the output of exactly verifiable circuits. In the regime of strong entanglement, the quantum computer provides correct results for which leading classical approximations such as pure-state-based 1D (matrix product states, MPS) and 2D (isometric tensor network states, isoTNS) tensor network methods2,3 break down. These experiments demonstrate a foundational tool for the realization of near-term quantum applications4,5.

Classification of measurement-based quantum wire in stabilizer PEPS

We consider a class of translation-invariant 2D tensor network states with a stabilizer symmetry, which we call stabilizer PEPS. The cluster state, GHZ state, and states in the toric code belong to this class. We investigate the transmission capacity of stabilizer PEPS for measurement-based quantum wire, and arrive at a complete classification of transmission behaviors. The transmission behaviors fall into 13 classes, one of which corresponds to Clifford quantum cellular automata. In addition, we identify 12 other classes.