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.
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.
Tensor network states are an indispensable tool for the simulation of strongly correlated quantum many-body systems. In recent years, tree tensor network states (TTNS) have been successfully used for two-dimensional systems and to benchmark quantum simulation approaches for condensed matter, nuclear, and particle physics. In comparison to the more traditional approach based on matrix product states (MPS), the graph distance of physical degrees of freedom can be drastically reduced in TTNS. Surprisingly, it turns out that, for large systems in D>1 spatial dimensions, MPS simulations of low-energy states are nevertheless more efficient than TTNS simulations. With a focus on D=2 and 3, the scaling of computational costs for different boundary conditions is determined under the assumption that the system obeys an entanglement (log-)area law, implying that bond dimensions scale exponentially in the surface area of the associated subsystems.
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).
Tensor networks are used to efficiently approximate states of strongly-correlated quantum many-body systems. More generally, tensor network approximations may allow to reduce the costs for operating on an order-N tensor from exponential to polynomial in N, and this has become a popular approach for machine learning. We introduce the idea of imposing low-rank constraints on the tensors that compose the tensor network. With this modification, the time and space complexities for the network optimization can be substantially reduced while maintaining high accuracy. We detail this idea for tree tensor network states (TTNS) and projected entangled-pair states. Simulations of spin models on Cayley trees with low-rank TTNS exemplify the effect of rank constraints on the expressive power. We find that choosing the tensor rank r to be on the order of the bond dimension m, is sufficient to obtain high-accuracy groundstate approximations and to substantially outperform standard TTNS computations. Thus low-rank tensor networks are a promising route for the simulation of quantum matter and machine learning on large data sets.
The numerical simulation of two-dimensional quantum many-body systems away from equilibrium constitutes a major challenge for all known computational methods. We investigate the utility of tree tensor network (TTN) states to solve the dynamics of the quantum Ising model in two dimensions. Within the perturbative regime of small transverse fields, TTNs faithfully reproduce analytically known, but nontrivial and physically interesting dynamics of magnetic domains, for lattices up to 16 × 16 sites. Limitations of the method related to the rapid growth of entanglement entropy are explored within more general, paradigmatic quench settings. We provide and discuss comprehensive benchmarks regarding the benefit of GPU acceleration and the impact of using local operator sums on the performance.
We introduce a novel tensor network structure augmenting the well-established tree tensor network representation of a quantum many-body wave function. The new structure satisfies the area law in high dimensions remaining efficiently manipulatable and scalable. We benchmark this novel approach against paradigmatic two-dimensional spin models demonstrating unprecedented precision and system sizes. Finally, we compute the ground state phase diagram of two-dimensional lattice Rydberg atoms in optical tweezers observing nontrivial phases and quantum phase transitions, providing realistic benchmarks for current and future two-dimensional quantum simulations.
We investigate tree tensor network states for quantum chemistry. Tree tensor network states represent one of the simplest generalizations of matrix product states and the density matrix renormalization group. While matrix product states encode a one-dimensional entanglement structure, tree tensor network states encode a tree entanglement structure, allowing for a more flexible description of general molecules. We describe an optimal tree tensor network state algorithm for quantum chemistry. We introduce the concept of half-renormalization which greatly improves the efficiency of the calculations. Using our efficient formulation we demonstrate the strengths and weaknesses of tree tensor network states versus matrix product states. We carry out benchmark calculations both on tree systems (hydrogen trees and π-conjugated dendrimers) as well as non-tree molecules (hydrogen chains, nitrogen dimer, and chromium dimer). In general, tree tensor network states require much fewer renormalized states to achieve the same accuracy as matrix product states. In non-tree molecules, whether this translates into a computational savings is system dependent, due to the higher prefactor and computational scaling associated with tree algorithms. In tree like molecules, tree network states are easily superior to matrix product states. As an illustration, our largest dendrimer calculation with tree tensor network states correlates 110 electrons in 110 active orbitals.