Tensor-network (TN) states supply one of the most powerful variational tools to simulate quantum many-body systems. Though classical simulation of TN states (with approximation) is efficient, the required computational (classical) resources are still beyond our current capability when the size of the system and bond dimension becomes large (which is necessary for studying complicated quantum many-body systems). The TN contraction, which is the dominant cost in TN algorithms, can be replaced by measuring the corresponding physical observables directly in the experimentally prepared TN states. The computational cost is thus dramatically reduced. Here, we propose a scheme to generate TN states by combining multiple phonon modes and qubits in the ion trap platform. With the full connectivity and parallelism of the phonon modes operation, we can generate TN states with rough but complex entanglement structures [such as the multiscale entanglement renormalization Ansatz on the one-dimensional (1D) lattice and the projected entangled pair states on the kagome lattice] by shallow generation circuits. With the abundant local free parameters of the qubits operation, we can refine the local details of the generated TN states. We further benchmark the expressive power of the generated TN states by optimizing the ground energy, which is very close to the exact results of the 1D transverse-field Ising model near the critical point and the 1D Heisenberg model. Our method supplies a quantum-classical hybrid way to simulate the ground states of many-body systems and paves a promising way to demonstrate the advantage of quantum simulation.
Tensor-network (TN) states supply one of the most powerful variational tools to simulate quantum many-body systems. Though classical simulation of TN states (with approximation) is efficient, the required computational (classical) resources are still beyond our current capability when the size of the system and bond dimension becomes large (which is necessary for studying complicated quantum many-body systems). The TN contraction, which is the dominant cost in TN algorithms, can be replaced by measuring the corresponding physical observables directly in the experimentally prepared TN states. The computational cost is thus dramatically reduced. Here, we propose a scheme to generate TN states by combining multiple phonon modes and qubits in the ion trap platform. With the full connectivity and parallelism of the phonon modes operation, we can generate TN states with rough but complex entanglement structures [such as the multiscale entanglement renormalization Ansatz on the one-dimensional (1D) lattice and the projected entangled pair states on the kagome lattice] by shallow generation circuits. With the abundant local free parameters of the qubits operation, we can refine the local details of the generated TN states. We further benchmark the expressive power of the generated TN states by optimizing the ground energy, which is very close to the exact results of the 1D transverse-field Ising model near the critical point and the 1D Heisenberg model. Our method supplies a quantum-classical hybrid way to simulate the ground states of many-body systems and paves a promising way to demonstrate the advantage of quantum simulation.
Tensor-network (TN) states are efficient parametric representations of ground states of local quantum Hamiltonians extensively used in numerical simulations.Employing TN Ansatz states directly on a quantum simulator can potentially offer an exponential computational advantage over purely numerical simulation.We implement a quantum-encoded TN Ansatz state using a variational quantum eigensolver on an ion-trap quantum computer that approximates the ground states of the extended Su-Schrieffer-Heeger model.The generated states are characterized by estimating the topological invariants, verifying their topological order.Our TN encoding as a trapped-ion circuit employs only single-site optical pulses-the native operations naturally available on the platform.We reduce nearest-neighbor crosstalk by selecting different magnetic sublevels with well-separated transition frequencies to encode the qubits in neighboring ions.
We adopt a two-dimensional tensor-network (TN) ansatz to simulate variational quantum algorithms on two-dimensional qubit architectures, demonstrating its capability to accurately simulate deep circuits through the Quantum Approximate Optimization Algorithm (QAOA) applied to Ising spin-glass problems on heavy-hexagonal and square lattices. For heavy-hexagonal problems with up to three-body interactions, parameters trained on small instances and transferred to systems an order of magnitude larger improve the sampled energy distribution only up to intermediate depths, indicating a fundamental limit of parameter concentration as a transfer strategy. By extending the training itself with TN simulations on larger system sizes, we avoid local minima and obtain lower-energy samples. Analyses of entanglement growth and importance sampling show that the simulation remains classically feasible with moderate bond dimension. We find that parameter concentration also persists on square lattices, albeit at substantially higher computational cost to perform reliable sampling. Overall, our TN framework not only provides an efficient and controlled framework for benchmarking variational quantum algorithms on two-dimensional lattices, but also serves as an effective surrogate model for training variational algorithms.
The study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.
Hybrid tensor networks (hTNs) offer a promising solution for encoding variational quantum states beyond the capabilities of efficient classical methods or noisy quantum computers alone. However, their practical usefulness and many operational aspects of hTN-based algorithms, like the optimization of hTNs, the generalization of standard contraction rules to an hybrid setting, and the design of application-oriented architectures have not been thoroughly investigated yet. In this work, we introduce a novel algorithm to perform ground-state optimizations with hybrid tree tensor networks (hTTNs), discussing its advantages and roadblocks, and identifying a set of promising applications. We benchmark our approach on two paradigmatic models, namely the Ising model at the critical point and the Toric-code Hamiltonian. In both cases, we successfully demonstrate that hTTNs can improve upon classical equivalents with equal bond dimension in the classical part.
Simulating many-body quantum systems on a classical computer is difficult due to the large number of degrees of freedom, causing the computational complexity to grow exponentially with system size. Tensor Networks (TN) is a framework that breaks down large tensors into a network of smaller tensors, enabling efficient simulation of certain many-body quantum systems. To calculate expectation values of local observables or simulate nearest-neighbor interactions, a contraction of the entire network is needed. This is a known hard problem, which cannot be done exactly for systems with spatial dimension D>1 and is the major bottleneck in all tensor-network based algorithms. Various approximate-contraction algorithms have been suggested, all with their strengths and weaknesses. Nevertheless, contracting a 2D TN remains a major numerical challenge, limiting the use of TN techniques for many interesting systems.
Recently, a close connection between TN and Probabilistic Graphical Models (PGM) has been shown. In the PGM framework, marginals of complicated probability distributions can be approximated using iterative message passing algorithms such as Belief Propagation (BP). The BP algorithm can be adapted to the TN framework as an efficient contraction algorithm. While BP is extremely efficient and easy to parallelize, it often yields inaccurate results for highly correlated quantum states or frustrated systems. To overcome this, we suggest the BlockBP algorithm, which coarse-grains the system into blocks and performs BP between them.
This thesis focuses on: (i) development and implementation of the BlockBP algorithm for infinite lattices; (ii) using this algorithm to study the anti-ferromagnetic Heisenberg model on the Kagome lattice in the thermodynamic limit - a frustrated 2D model that is difficult to simulate using existing numerical methods.
Strongly correlated quantum systems are one of the most interesting and challenging phenomena in modern condensed matter physics. Unfortunately, the theoretical access even to their low energy properties is limited by the exponential growth of the Hilbert space in a many-body setting. To mitigate this computational barrier, tensor network methods are one of the most competitive approaches to classically simulate complex quantum systems. In this thesis we will discuss recent developments in the field of tensor networks to efficiently and accurately simulate two-dimensional quantum many-body states in the thermodynamic limit. Hereby, we introduce the variational update globally optimizing the wave-function, encoded as tensor network in the form of an infinite projected entangled pair state (iPEPS), in order to obtain ground states of many-body Hamiltonian models. We focus in the discussion on the well-established method to approximately contract this two-dimensional structure, the corner transfer matrix renormalization group (CTMRG), and discuss its native application on square lattice and triangular lattice based models. Furthermore, we report the development of a refined numerical algorithm, the split-CTMRG, to significantly reduce its computational complexity, pushing the limits of feasible simulations within modern hardware capabilities. Finally, the method advancements are applied to a currently widely studied model of frustrated magnetism, the Heisenberg maple-leaf antiferromagnet. We are able to find a large phase of vanishing magnetization in contradiction to previous works and, to the best of our knowledge, the lowest reported variational ground state energy at the isotropic point of the model. As an outlook, we describe a potential adaption of the previously developed methods to variationally simulate two-dimensional many-body thermal states.