多體動力學在張量網路與量子電路中的計算

Abstract

本論文探討了利用張量網路與量子電路來模擬量子多體系統以及求解偏微分方程（PDE）的方法。第一部分著重於在無限與有限長度格點的張量網路架構中使用時間依賴變分原理（Time-Dependent Variational Principle, TDVP）演算法，以研究自旋模型的動力學行為。我們透過將 Thirring 模型轉換為spin-lattice Hamiltonian，並進行複時間演化，探討其動態量子相變（Dynamical Quantum Phase Transitions, DQPTs）。本研究建立了 DQPT 與 Fisher 零點（Fisher zeros）之間的關聯。此外，我們利用有限系統的 TDVP 方法，在 XXZ 模型中進行 quantum quench 後的模擬，觀察到隨著實時間演化而出現的 coarsening 現象。 接著，本論文探討以極座標下的Quantic Tensor Train (QTT) 表示法求解二維偏微分方程。為處理 Laplacian 算符在原點處的 singularity，我們提出一種方法，使得所有相關運算皆可轉換為矩陣乘積算符（Matrix Product Operators, MPOs）的形式。此架構使得我們能夠利用 DMRG 及 TDVP 等演算法，模擬二維簡諧振子與 Gross–Pitaevskii（GP）方程的基態與動態行為。 最後一部分則研究量子電路方法在多體量子動力學中的應用，特別是 Ising 模型。我們利用受 TDVP 啟發的參數化量子電路進行無限長度 Ising 模型的虛時間演化，將矩陣乘積態（Matrix Product States, MPS）轉換為量子電路的表示，並於實際量子硬體上執行模擬。同時，我們 cost function 與期望值的分佈進行統計分析，並結合 quantum Lanczos 演算法以提升變分最佳化的效果。此外，我們模擬具縱向與橫向磁場的 Ising 模型之實時間演化，以探討其所呈現的 E_8 對稱性。為提升量子電路實作效率，我們亦應用基於張量網路的壓縮技術，優化電路深度，並於真實量子裝置上進行驗證。 綜合而言，本論文展示了張量網路與量子電路方法在模擬複雜量子系統與求解偏微分方程中的高效性與靈活性，為量子模擬的理論研究與實際應用開啟了新的方向。

This dissertation explores methods for simulating quantum many-body systems and solving partial differential equations (PDEs) using tensor networks and quantum circuits. The first part focuses on the use of the Time-Dependent Variational Principle (TDVP) algorithm in both infinite- and finite-size lattice systems in tensor networks to study dynamical behavior in spin models. We investigate dynamical quantum phase transitions (DQPTs) in the Thirring and XXZ models by mapping the Thirring model to a spin-lattice Hamiltonian and performing complex-time evolution. A connection between Fisher zeros and DQPTs is established. In addition, we study the coarsening phenomenon in the XXZ model by applying finite-size TDVP following a quantum quench, revealing the emergence of coarsening behavior through real-time evolution. Then, addresses the solution of two-dimensional partial differential equations using the Quantic Tensor Train (QTT) representation under polar coordinates. To manage the singularity at the origin of the Laplacian operator, we propose a regularization method that enables all relevant operations to be expressed as Matrix Product Operators (MPOs). This framework allows for ground state and dynamical simulations of the two-dimensional simple harmonic oscillator and the Gross–Pitaevskii (GP) equation using DMRG and TDVP algorithms. The final part investigates quantum circuit approaches to many-body dynamics, with a particular focus on the Ising model. We perform imaginary-time evolution of the infinite-size Ising model using parameterized quantum circuits inspired by TDVP, converting Matrix Product States (MPS) into circuit representations and executing simulations on real quantum hardware. Statistical analysis of cost function distributions and expectation values is conducted, and a quantum Lanczos algorithm is used to improve variational optimization. Additionally, we simulate the real-time evolution of the transverse-field Ising model with a longitudinal field to explore signatures of emergent E_8 symmetry. Tensor network-based compression techniques are applied to optimize the depth of the quantum circuit, further enhancing the efficiency of the implementation on quantum devices. Together, these studies demonstrate the effectiveness of tensor network and quantum circuit frameworks in simulating complex quantum systems and solving PDEs, paving the way for both theoretical advances and practical applications in quantum simulation.