Properties of quantum many-body systems at finite energy densities, such as microcanonical averages, are in general difficult to compute with classical algorithms. Quantum algorithms that combine quantum dynamics with filtering and sampling techniques can however access some of these quantities. Here, the authors show that simulating the time evolution part with tensor networks provides quantum-inspired classical algorithms that reach much larger systems than other numerical methods.
Properties of quantum many-body systems at finite energy densities, such as microcanonical averages, are in general difficult to compute with classical algorithms. Quantum algorithms that combine quantum dynamics with filtering and sampling techniques can however access some of these quantities. Here, the authors show that simulating the time evolution part with tensor networks provides quantum-inspired classical algorithms that reach much larger systems than other numerical methods.
Energy filter methods in combination with quantum simulation can efficiently access the properties of quantum many-body systems at finite energy densities [S. Lu , ]. Classically simulating this algorithm with tensor networks can be used to investigate the microcanonical properties of large spin chains, as recently shown by Y. Yang []. Here we extend this strategy to explore the properties of off-diagonal matrix elements of observables in the energy eigenbasis, fundamentally connected to the thermalization behavior and the eigenstate thermalization hypothesis. We test the method on integrable and nonintegrable spin chains of up to 60 sites, much larger than accessible with exact diagonalization. Our results allow us to explore the scaling of the off-diagonal functions with the size and energy difference, and to establish quantitative differences between integrable and nonintegrable cases. Published by the American Physical Society 2024
In this thesis, quantum many-body systems are studied by means of tensor network methods. After introducing the quantum many-body problem and discussing paradigmatic models as well as established methods for their solution, the focus is set on a practical approach to tensor network methods, always keeping in mind the realistic numerical applicability of the tools devised. In order to achieve this objective, we focus on loop-free network geometries, which allow to boost the computational performance to the greatest possible extent. The developed algorithms are described and benchmarked in detail, starting from their elementary constituents, namely the multidimensional arrays called tensors which are at the heart of any tensor network algorithm. We then apply the developed simulation techniques to a diverse collection of quantum many-body models of current research interest. We are interested, in particular, in a characterization of the ground-state properties of these models, but we also investigate problems beyond the zero-temperature equilibrium setting, such as time evolution or finite-temperature behavior. Specifically, novel findings for the phase diagrams of the disordered Bose-Hubbard chain and of the Heisenberg chain with additional biquadratic interactions are presented, as well as compelling new evidence for the existence of fractional quantum Hall states in the interacting Harper-Hofstadter model.
Computing dynamical distributions in quantum many-body systems represents one of the paradigmatic open problems in theoretical condensed matter physics. Despite the existence of different techniques both in real-time and frequency space, computational limitations often dramatically constrain the physical regimes in which quantum many-body dynamics can be efficiently solved. Here we show that the combination of machine-learning methods and complementary many-body tensor network techniques substantially decreases the computational cost of quantum many-body dynamics. We demonstrate that combining kernel polynomial techniques and real-time evolution, together with deep neural networks, allows to compute dynamical quantities faithfully. Focusing on many-body dynamical distributions, we show that this hybrid neural-network many-body algorithm, trained with single-particle data only, can efficiently extrapolate dynamics for many-body systems without prior knowledge. Importantly, this algorithm is shown to be substantially resilient to numerical noise, a feature of major importance when using this algorithm together with noisy many-body methods. Ultimately, our results provide a starting point towards neural-network powered algorithms to support a variety of quantum many-body dynamical methods, that could potentially solve computationally expensive many-body systems in a more efficient manner.
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.
The accurate simulation of complex systems, encompassing fields like materials science, quantum chemistry, and finance, often presents a formidable computational challenge. Traditional methods frequently struggle to scale effectively, particularly when dealing with high-dimensionality or intricate correlations. This paper introduces a novel simulation approach leveraging tensor networks (TN) in conjunction with quantum-inspired sampling techniques. The core idea is to approximate the dynamics of these complex systems by exploiting the inherent structure of tensor networks and employing sampling strategies inspired by quantum measurement processes. We demonstrate that this hybrid approach offers a pathway towards scalable simulations, achieving high accuracy while mitigating the computational bottlenecks associated with conventional methods. The key innovations lie in the adaptive construction of TNs based on the system's correlation structure and the utilization of quantum-inspired sampling methods to efficiently capture the relevant dynamics. This work presents a theoretical framework and outlines a methodology for implementing such simulations, paving the way for advancements in the modeling and understanding of complex systems.