发表机构
Laboratoire de Physique de l’École Normale Supérieure, Mines Paris, Inria, CNRS, ENS-PSL, Sorbonne Université, PSL Research University（巴黎高等师范学院物理实验室，巴黎矿业学院，法国国家信息与自动化研究所，法国国家科学研究中心，巴黎高等师范学院-巴黎文理研究大学，索邦大学，巴黎文理研究大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对大型开放量子系统的林德布拉德主方程，提出基于无跳演化逆映射的预条件迭代方法，可高效求解稳态、低能谱及刚性系统时间演化，每次迭代规模为$O(n^3)$，在CPU和GPU上表现最优。
AI 中文摘要
马尔可夫开放量子系统可由林德布拉德主方程（Master Equation, ME）$\frac{\text{d}}{\text{d} t} \rho_t = \boldsymbol{\rho}_t$ 精准建模，其中 $\boldsymbol{\rho}_t$ 为线性（超）算子，$\rho_t$ 为系统态，即正矩阵。在设计或表征量子系统时，人们通常关注稳态 $\rho_\boldsymbol{\rho}$（满足 $\boldsymbol{\rho} \rho_\boldsymbol{\rho} = 0$）、前几个激发态以及轨迹 $t \to \rho_t$。在有限维度下，$\rho_t$ 是 $n \times n$ 矩阵，因此将 $\boldsymbol{\rho}$ 显式存储为稠密矩阵通常需要 $O(n^4)$ 的存储量，而精确对角化或求逆则需要 $O(n^6)$ 的运算量，这使得标准线性代数技术对于大型系统而言成本高昂。不过，应用 $\boldsymbol{\rho}$ 通常仅需 $O(n^3)$ 的运算量，这使得迭代方法颇具吸引力，但迭代方法若无良好的预条件子则无法生效。本文的主要发现是，林德布拉德方程中对应于所谓无跳演化 $\boldsymbol{\rho}$ 的部分可高效求逆。利用该逆映射，我们引入了一个辅助的完全正保迹（CPTP）映射 $\boldsymbol{\rho}$，其不动点与 $\rho_\boldsymbol{\rho}$ 直接相关，且所有其他特征值的模更小。因此，映射 $\boldsymbol{\rho}$ 非常适合迭代方法，且可通过数次 Arnoldi 迭代求得 $\rho_\boldsymbol{\rho}$。使用相同的逆映射 $\boldsymbol{\rho}^{-1}$ 作为预条件子，我们通过位移-逆 Arnoldi 高效计算低能谱，并作为概念验证，构建了一个隐式时间积分器，该积分器在低精度区域的刚性系统上具有竞争力。对于稳态和低激发态问题，我们的方法每次迭代的规模为 $O(n^3)$，并在 CPU 和 GPU 上提供了最先进的性能。
英文摘要
Markovian open quantum systems are well modeled by the Lindblad Master Equation (ME) $\frac{\mathrm{d}}{\mathrm{d} t} ρ_t = \mathcal{L} ρ_t$, where $\mathcal{L}$ is a linear (super-)operator and $ρ_t$ is the system state, a positive matrix. When designing or characterizing a quantum system, one is usually interested in the steady state $ρ_\infty$ (such that $\mathcal{L} ρ_\infty = 0$), the first few excited states, and trajectories $t\mapsto ρ_t$. In finite dimension, $ρ_t$ is an $n\times n$ matrix, $\mathcal{L}$ thus typically costs $n^4$ to store explicitly as a dense matrix, and $O(n^6)$ to diagonalize or invert exactly, making standard linear algebraic techniques expensive for large systems. However, $\mathcal{L}$ usually costs only $O(n^3)$ to apply. This makes iterative methods appealing, but they do not work without a good preconditioner. In this article, our main observation is that a part of the Lindblad equation, corresponding to the so-called no-jump evolution $\mathcal{S}$, can be inverted efficiently. Using this inverse map, we introduce an auxiliary completely positive trace-preserving (CPTP) map $Φ$ whose fixed point is directly related to $ρ_\infty$, all the other eigenvalues having smaller magnitude. The map $Φ$ is thus well suited to iterative methods, and $ρ_\infty$ can be found in a few Arnoldi iterations. Using the same inverse map $\mathcal{S}^{-1}$ as preconditioner, we compute the low-lying spectrum efficiently via shift-invert Arnoldi, and, as a proof of concept, build an implicit time integrator that is competitive on stiff systems in the low-precision regime. For the steady-state and low excited states problems, our methods scale like $O(n^3)$ per iteration and offer state-of-the-art performance on CPU and GPU.
