标题： LREI：量子兰道-利夫希茨方程的快速数值求解器 显示英文标题

标题： LREI: A fast numerical solver for quantum Landau-Lifshitz equations

摘要： 我们开发了LREI（低秩本征模积分），这是一种用于求解量子兰道-利夫希茨（q-LL）和量子兰道-利夫希茨-吉尔伯特（q-LLG）方程的内存和时间高效的方案，这些方程描述开放量子系统中的自旋动力学。虽然系统尺寸随着自旋数量呈指数增长，但我们的方法利用密度矩阵的低秩结构和哈密顿量的稀疏性来避免完整的矩阵计算。通过使用低秩因子表示密度矩阵，并应用Krylov子空间方法进行部分特征分解，我们将龙格-库塔和亚当斯-巴什福思方案的每步复杂度从$\mathcal{O}(N^3)$降低到$\mathcal{O}(r^2N)$，其中$N = 2^n$是$n$个自旋的希尔伯特空间维度，$r \ll N$是有效秩。 类似地，内存成本从$\mathcal{O}(N^2)$减少到$\mathcal{O}(rN)$，因为没有形成完整的$N\times N$矩阵。一个关键的进展是处理零特征值的不变子空间。通过使用为主要特征空间构建的豪斯霍尔德反射，我们完全避免了大矩阵的使用。例如，一个二十自旋系统的时步，其密度矩阵大小超过一百万，现在在标准笔记本电脑上仅需几秒钟。龙格-库塔方法和阿达姆斯-巴什福思方法都被重新表述，以在整个演化过程中保持密度矩阵的物理特性。这种低秩算法使得对更大自旋系统的模拟成为可能，这些系统之前是不可行的，为比较 q-LL 和 q-LLG 动力学提供了强大的工具，测试每个模型的有效性，并探究量子特性如关联和纠缠如何在不同系统尺寸和阻尼区域中演化。 显示英文摘要

摘要： We develop LREI (Low-Rank Eigenmode Integration), a memory- and time-efficient scheme for solving quantum Landau-Lifshitz (q-LL) and quantum Landau-Lifshitz-Gilbert (q-LLG) equations, which govern spin dynamics in open quantum systems. Although system size grows exponentially with the number of spins, our approach exploits the low-rank structure of the density matrix and the sparsity of Hamiltonians to avoid full matrix computations. By representing density matrices via low-rank factors and applying Krylov subspace methods for partial eigendecompositions, we reduce the per-step complexity of Runge-Kutta and Adams-Bashforth schemes from $\mathcal{O}(N^3)$ to $\mathcal{O}(r^2N)$, where $N = 2^n$ is the Hilbert space dimension for $n$ spins and $r \ll N$ the effective rank. Similarly, memory costs shrink from $\mathcal{O}(N^2)$ to $\mathcal{O}(rN)$, since no full $N\times N$ matrices are formed. A key advance is handling the invariant subspace of zero eigenvalues. By using Householder reflectors built for the dominant eigenspace, we perform the solution entirely without large matrices. For example, a time step of a twenty-spin system, with density matrix size over one million, now takes only seconds on a standard laptop. Both Runge-Kutta and Adams-Bashforth methods are reformulated to preserve physical properties of the density matrix throughout evolution. This low-rank algorithm enables simulations of much larger spin systems, which were previously infeasible, providing a powerful tool for comparing q-LL and q-LLG dynamics, testing each model validity, and probing how quantum features such as correlations and entanglement evolve across different regimes of system size and damping.