Ongoing Research

Tensor Network Approach for Stochastic dynamics (PI: Prof. Hong-Yan Shih and Yi-Ping Huang, Academia Sinica)


This research focuses on utilizing the tensor network approach to find the steady state of evolutionary dynamic system. The one-dimensional SIS model is a simple but classic and useful model that can describe the non-equilibrium phase transitions. Tensor network approach is highly accurate and efficient method for solving many-body quantum system. Recently, physicists found that this approach can also be applied to solving differential equations and analyzing stochastic dynamics. It has been known that the TN approach may be a powerful method that can solve the one-dimensional model effectively. We want to explore the principles behind applying the TN approach to the one-dimensional SIS model and extend it to other stochastic dynamical systems.




Pattern Formation and Dynamics in Quantum System (PI: Prof. Yi-Ping Huang, NTHU)

This research focuses on bridging the gap between quantum dynamics and non-linear phenomena to reveal pattern formation mechanisms in quantum systems. Employing the quantum trajectories method and phase representation, I investigate pattern formation in both two-site and many-site quantum systems. Anticipated outcomes involve characterizing patterns in phase space, comprehending non-linear effects, and assessing the influence of wave function patterns on quantum statistics. In essence, this study can contribute to a more profound understanding of the interplay between non-linear dynamics, pattern formation, and statistical mechanics in quantum systems.

Topics of Interest

The Density Matrix Renormalization Group Approach for the SIS Model in the Rennormalization Group Language


The density matrix renormalization group (DMRG) was originally introduced by Steven White using the language of renormalization group theory, which provides a physical description of the method. However, contemporary applications of DMRG typically utilize matrix product states and matrix product operators. This approach is also used for simulating the SIS model. Consequently, the application of DMRG in its original renormalization group framework for the SIS model may not be entirely clear. In contrast, the renormalization group description of the STS model offers valuable insights that can aid in understanding the behavior of the SIS model. By drawing on this description, we can enhance our comprehension of the SIS model's dynamics and properties.

References:
[1] Density matrix formulation for quantum renormalization groups
[2] Entanglement and tensor network states
[3] Efficient simulations of epidemic models with tensor networks: application to the one-dimensional SIS model



Analyzing Critical Behaviors in Genetic Algorithms Through Tensor Networks

It is well-established that genetic algorithms exhibit critical behaviors, which are pivotal for understanding their performance and efficiency. Tensor network approaches offer a powerful and sophisticated theoretical framework for analyzing such critical behaviors within stochastic dynamics. These approaches enable us to capture and interpret the complex interactions and phase transitions that occur in these systems. Given the conceptual parallels between genetic algorithms and biological evolution, it is plausible to apply tensor network methodologies to genetic algorithms. By leveraging tensor networks, we can gain deeper insights into the underlying mechanisms of genetic algorithms, elucidate their critical behaviors, and enhance our understanding of their operational dynamics. This approach could potentially lead to more effective optimization strategies and improved algorithmic performance.

References:
[1] Efficient simulations of epidemic models with tensor networks: application to the one-dimensional SIS model
[2] Phase transitions and symmetry breaking in genetic algorithms with crossover


Stochastic Analysis of Quantum Trajectories Using Tensor Networks

The quantum trajectory method is a numerical technique designed to simulate the behavior of open quantum systems, capturing the evolution of these systems over time. Often, this evolutionary process can be modeled as a Markovian process, where the future state of the system depends only on its current state and not on its past history. Given that tensor network approaches have demonstrated considerable efficacy in the analysis of stochastic processes, they may offer a powerful tool for exploring quantum trajectories. By applying tensor network methods, we could gain deeper insights into the dynamics of quantum systems, particularly in understanding how individual quantum systems evolve and how non-equilibrium phase transitions occur. The ability of tensor networks to handle high-dimensional data and complex correlations makes them a promising candidate for enhancing the simulation and analysis of quantum trajectories and related phenomena.

References:
[1] Tensor Network Simulation Methods for Open Quantum Lattice Models
[2] Phase transitions and symmetry breaking in genetic algorithms with crossover


Computing Feynman Path Integral through Tensor Networks

The Feynman path integral is a powerful tool for describing quantum dynamical systems. In this approach, the functional integral is often computed using the Monte Carlo method. However, given that tensor network techniques have also proven effective for evaluating high-dimensional integrals, they may offer a valuable numerical alternative for simulating quantum dynamical systems. By leveraging tensor networks, we could potentially enhance the accuracy and efficiency of such simulations.

References:
[1] A Monte Carlo method for high dimensional integration
[2] Mean Field Simulation for Monte Carlo Integration
[3] TT-cross approximation for multidimensional arrays


(Possible) High Energy Physics and Tensor Networks

Tensor networks probaly can offer a robust framework for tackling various complex problems in high-energy physics, providing new insights and computational techniques that complement traditional methods. Since I am not currently a student specializing in high-energy physics, I have not yet provided a detailed proposal. I will develop and present a concrete framework as I gain further knowledge in the field. Here are some possible topics:
• Quantum Field Theory (QFT)

Tensor networks are used to study quantum field theories, including lattice gauge theories. They provide a powerful tool for analyzing complex quantum systems and can help in understanding the dynamics of fields and particles in a non-perturbative regime.


• Quantum Gravity and Black Holes

Tensor networks have been applied to the study of quantum gravity and black holes. For example, the AdS/CFT correspondence, a significant area in theoretical high-energy physics, has connections to tensor network approaches. Tensor networks are used to model the structure of spacetime and understand the information paradox and other aspects of black hole physics.


• Many-Body Quantum Systems

In high-energy physics, tensor networks are employed to analyze many-body quantum systems and their phases. They are used to study the entanglement properties and phase transitions of such systems, which can be relevant for understanding quantum chromodynamics (QCD) and other high-energy phenomena.


• Conformal Field Theory (CFT)

Tensor networks have applications in conformal field theory, where they are used to study scaling behaviors and critical phenomena. This helps in understanding the symmetry properties and critical exponents of different high-energy systems.