Tensor networks provide a powerful and elegant approach to quantum manybody simulation. The simplest example is the density matrix renormalisation group (DMRG), which is based on the variational update of a matrix product state (MPS). It has proved to be the most accurate approach for the numerical study of strongly correlated one dimensional systems. We use DMRG to study the one dimensional disordered Bose-Hubbard model at fillings N=L = 1=2, 1 and 2 and show that the whole phase diagram for each can be successfully obtained by analysing entanglement properties alone. We �nd that the average entanglement is insufficient to accurately locate all of the phases, however using the standard error on the mean we are able to construct a phase diagram that is consistent with previous studies. It has recently been shown that there is a connection between the geometry of tensor networks and the entanglement and correlation properties that it can encode, which is a generalisation of the so called area law for entanglement entropy. This suggests that whilst gapped quantum systems can be accurately modeled using an MPS, a tensor network with a holographic geometry is natural to capture the logarithmic entanglement scaling and power law decaying correlation functions of critical systems. We create an algorithm for the disordered Heisenberg Hamiltonian that self assembles a tensor network based on the disorder in the couplings. The geometry created is that of a disordered tree tensor network (TTN) that when averaged has the holographic properties characteristic of critical systems. We continue the analysis of holographic tensor network geometry by considering the average length of leaf-to-leaf paths in various tree graphs, which is related to two-point correlation functions in tensor networks. For regular, complete trees we analytically calculate the average path length and all statistical moments, and generalise it for any splitting number. We then turn to the Catalan trees, which is the set of unique binary trees with n vertices, as it has a similar geometry to the disordered TTNs. We calculate the average depth of a leaf and show that it is equal to the average path length. We compare these analytic results with the structures found in the TTN and randomly constructed trees to show that the renormalisation involved in the TTN algorithm is crucial in the selection of the tree structure.
Tensor networks provide a powerful and elegant approach to quantum manybody simulation. The simplest example is the density matrix renormalisation group (DMRG), which is based on the variational update of a matrix product state (MPS). It has proved to be the most accurate approach for the numerical study of strongly correlated one dimensional systems. We use DMRG to study the one dimensional disordered Bose-Hubbard model at fillings N=L = 1=2, 1 and 2 and show that the whole phase diagram for each can be successfully obtained by analysing entanglement properties alone. We �nd that the average entanglement is insufficient to accurately locate all of the phases, however using the standard error on the mean we are able to construct a phase diagram that is consistent with previous studies. It has recently been shown that there is a connection between the geometry of tensor networks and the entanglement and correlation properties that it can encode, which is a generalisation of the so called area law for entanglement entropy. This suggests that whilst gapped quantum systems can be accurately modeled using an MPS, a tensor network with a holographic geometry is natural to capture the logarithmic entanglement scaling and power law decaying correlation functions of critical systems. We create an algorithm for the disordered Heisenberg Hamiltonian that self assembles a tensor network based on the disorder in the couplings. The geometry created is that of a disordered tree tensor network (TTN) that when averaged has the holographic properties characteristic of critical systems. We continue the analysis of holographic tensor network geometry by considering the average length of leaf-to-leaf paths in various tree graphs, which is related to two-point correlation functions in tensor networks. For regular, complete trees we analytically calculate the average path length and all statistical moments, and generalise it for any splitting number. We then turn to the Catalan trees, which is the set of unique binary trees with n vertices, as it has a similar geometry to the disordered TTNs. We calculate the average depth of a leaf and show that it is equal to the average path length. We compare these analytic results with the structures found in the TTN and randomly constructed trees to show that the renormalisation involved in the TTN algorithm is crucial in the selection of the tree structure.
We show that the numerical strong disorder renormalization group algorithm of Hikihara et al. [Phys. Rev. B 60, 12116 (1999)] for the one-dimensional disordered Heisenberg model naturally describes a tree tensor network (TTN) with an irregular structure defined by the strength of the couplings. Employing the holographic interpretation of the TTN in Hilbert space, we compute expectation values, correlation functions, and the entanglement entropy using the geometrical properties of the TTN. We find that the disorder-averaged spin-spin correlation scales with the average path length through the tensor network while the entanglement entropy scales with the minimal surface connecting two regions. Furthermore, the entanglement entropy increases with both disorder and system size, resulting in an area-law violation. Our results demonstrate the usefulness of a self-assembling TTN approach to disordered systems and quantitatively validate the connection between holography and quantum many-body systems.
The research work in this thesis is based on strongly interacting quantum lattice systems. The biggest part of research was conducted using state-of-the-art tensor network simulations. Tensor networks provide efficient and highly accurate representations of quantum states when any simply connected patch of the quantum state is slightly entangled. Matrix Product States (MPS) is a tensor network representation for quantum states which is quasi-exact for one-dimensional systems when entanglement entropy of any bipartition of the state follows ``area-law". Projected Entangled Pair States (PEPS) is the extension of MPS for higher dimensional systems, where entanglement entropy area-law is non-trivial. These formalisms and their relevant ground state optimization techniques, Density Matrix Renormalization Group (DMRG) for MPS and Simple Update (SU) with Tensor Renormalization Group (TRG) for PEPS are thoroughly analysed. Entanglement entropy area-law is usually obeyed by the ground state of local Hamiltonians, while generic highly excited states follow a volume-law (they span a finite part of the Hilbert space). Recently, a class of interacting system was shown to undergo a dynamical phase transition (Many Body Localization) where the entropy of every eigenstate follows an area-law. This transition is achieved when the system is highly disordered and the quantum many body state becomes localized similarly to the free particle Anderson localization. The area-law property makes highly excited eigenstates of Many Body Localized (MBL) systems efficiently represented by the MPS ansatz. We develop a highly optimized algorithm (eDMRG) which goes beyond ground state optimization and successfully target eigenstates in any part of the spectrum. This algorithm is used together with analytical calculations, based on local integrability of MBL systems, to identify the universal behaviour of the entanglement spectrum of highly excited eigenstates in MBL systems. In the second part we study interacting bosons in 2D optical lattices. Strongly interacting bosons are simulated using the Bose-Hubbard (BH) model when the interactions are strictly local and the Extended Bose-Hubbard (EBH) model, when additional dipolar interactions are present. The ground states of BH/EBH Hamiltonians in a hexagonal lattice are studied. The phase spectrum includes various insulating and critical phases which are studied in detail using infinite-PEPS ansatz. Additional results on entanglement entropy scaling with respect to the filling of the lattice are presented. Finally, a strongly interacting Harper-Hofstadter Hamiltonian is realized by combining synthetic fields with the BH model. The homogeneity of the system is then broken by a parabolic trapping potential, similar to the ones used in cold atom experiments. Using time-dependent Gutzwiller ansatz (GA) the expansion dynamics of the cloud in large square lattices are studied. In contrast to the expansion dynamics of the BH model, it is found that the synthetic fields generate a self-trapping effect. Using phenomenology and simulations the dynamics are studied in the hard-core and soft-core limit.
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.
Quantum computing offers the potential for computational abilities that can go beyond classical machines. However, they are still limited by several challenges such as noise, decoherence, and gate errors. As a result, efficient classical simulation of quantum circuits is vital not only for validating and benchmarking quantum hardware but also for gaining deeper insights into the behavior of quantum algorithms. A promising framework for classical simulation is provided by tensor networks. Recently, the density-matrix renormalization group (DMRG) algorithm was developed for simulating quantum circuits using matrix product states (MPS). Although MPS are efficient for representing quantum states with one-dimensional correlation structures, the fixed linear geometry restricts the expressive power of the MPS. In this work, we extend the DMRG algorithm for simulating quantum circuits to tree tensor networks (TTNs). The framework employs a variational compression scheme that optimizes the TTN to approximate the evolved quantum state. To benchmark the method, we simulate random circuits and the quantum approximate optimization algorithm (QAOA) with various two-qubit gate connectivities. For the random circuits, we devise treelike gate layouts that are suitable for TTN and show that TTN requires less memory than MPS for the simulations. For the QAOA circuits, a naive TTN construction that exploits graph structure significantly improves the simulation fidelities. Our findings show that the DMRG algorithm with TTNs provides a promising framework for simulating quantum circuits, particularly when gate connectivities exhibit clustering or a hierarchical structure.
This thesis treats the classical simulation of strongly-interacting many-body quantum-mechanical systems in more than one dimension using matrix product states and the more general tensor product states. Contrary to classical systems, quantum many-body systems possess an exponentially larger number of degrees of freedom, thereby significantly complicating their numerical treatment on a classical computer. For this thesis two different representations of quantum many-body states were employed. The first, the so-called matrix product states (MPS) form the basis for the extremely successful density matrix renormalization group (DMRG) algorithm. While originally conceived for one-dimensional systems, MPS are in principle capable of describing arbitrary quantum many-body states. Using concepts from quantum information theory it is possible to show that MPS provide a representation of one-dimensional quantum systems that scales polynomially in the number of particles, therefore allowing an efficient simulation of one-dimensional systems on a classical computer. One of the key results of this thesis is that MPS representations are indeed efficient enough to describe even large systems in two dimensions, thereby enabling the simulation of such systems using DMRG. As a demonstration of the power of the DMRG algorithm, it is applied to the Heisenberg antiferromagnet with spin S=1/2 on the kagome lattice. This model's ground state has long been under debate, with proposals ranging from static spin configurations to so-called quantum spin liquids, states where quantum fluctuations destroy conventional order and give rise to exotic quantum orders. Using a fully SU(2)-symmetric implementation allowed us to handle the exponential growth of entanglement and to perform a large-scale study of this system, finding the ground state for cylinders of up to 700 sites. Despite employing a one-dimensional algorithm for a two-dimensional system, we were able to compute the spin gap (i.e. the energy gap to the first spinful excitation) and study the ground state properties, such as the decay of correlation functions, the static spin structure factors, and the structure and distribution of the nearest-neighbor spin-spin correlations. Additionally, by applying a new tool from quantum information theory, the topological entanglement entropy, we could also with high confidence demonstrate the ground state of this model to be the elusive gapped Z2 quantum spin liquid with topological order. To complement this study, we also considered the extension of MPS to higher dimensions, known as tensor product states (TPS). We implemented an optimization algorithm exploiting symmetries for this class of states and applied it to the bilinear-biquadratic-bicubic Heisenberg model with spin S=3/2 on the z=3 Bethe lattice. By carefully analyzing the simulation data we were able to determine the presence of both conventional and symmetry-protected topological order in this model, thereby demonstrating the analytically predicted existence of the Haldane phase in higher dimensions within an extended region of the phase diagram. Key properties of this symmetry-protected topological order include a doubling of the levels in the entanglement spectrum and the presence of edge spins, both of which were confirmed in our simulations. This finding simultaneously validated the applicability of the novel TPS algorithms to the search for exotic order.
Tensor network methods, most prominently matrix product states (MPS), have become fundamental tools in modern quantum many-body physics. While MPS and extensions like the multiscale entanglement renormalization ansatz (MERA) and tree tensor networks (TTN) efficiently capture area-law entanglement and its logarithmic violations, they inherently struggle to represent highly entangled wavefunctions. Specifically, reaching the volume-law regime typically demands exponential resources within these conventional frameworks. Motivated by this challenge, we propose holographic isometric tensor network states (holographic isoTNS) that simulate quantum lattice models in D spatial dimensions via (D+1)-dimensional networks of tensors. The additional dimension substantially enlarges the representational manifold, while isometric constraints on each tensor ensure efficient contractibility. Using one-dimensional systems as testbeds, we analyze the properties of holographic isoTNS. First, we show that randomly initialized holographic isoTNS typically display volume-law entanglement at fixed bond dimension. Second, through analytic constructions and variational optimization, we demonstrate that holographic isoTNS can faithfully describe a broad class of highly entangled yet low-complexity states. In particular, the ansatz can represent arbitrary fermionic Gaussian states, Clifford states, extensions of rainbow states, and certain short-time-evolved states under local evolution. Third, to exploit this expressivity in broader contexts, we implement a time-evolving block decimation (TEBD) algorithm on holographic isoTNS. While the method remains efficient and scalable, error accumulation over TEBD sweeps suggests the need for further algorithmic improvement. Overall, holographic isoTNS broaden the scope of tensor-network methods, opening new avenues to study physics in the volume-law regime.