一种用于非马尔可夫量子动力学的李代数方法
A Lie-algebraic approach to non-Markovian quantum dynamics
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中文总结 AI 辅助
研究量子计算中非马尔可夫量子动力学，基于李代数方法，通过矢量化密度矩阵用线性时变方程表示演化，研究马格努斯展开及截断误差，还考虑测量噪声下的随机微分方程情况，数值模拟展示方法效率及李代数对截断误差的影响。
中文摘要 AI 辅助
本文从基于数值分析的李代数方法视角研究量子计算中的非马尔可夫量子动力学。通过矢量化量子态密度矩阵，非马尔可夫演化可用高维线性时变方程表示，时变参数源于量子系统与环境的非马尔可夫相互作用。研究了此类线性时变量子动力学的马格努斯展开，阐明一阶和二阶马格努斯展开的截断误差如何受非马尔可夫性质影响。此外，测量量子态进行滤波时，因测量噪声存在，动力学可建模为时变随机微分方程。基于伊藤或斯特拉托诺维奇方法对量子测量噪声建模时，基于量子随机滤波的马格努斯展开不同，截断误差也不同。数值模拟进一步证明马格努斯展开在模拟有或无随机性的非马尔可夫量子动力学方面的效率，以及截断误差如何受刘维尔空间中李代数影响。
英文摘要
In this paper, we study the non-Markovian quantum dynamics in quantum computations from the perspective of a Lie algebraic approach based on numerical analysis. By vectorizing the density matrix of quantum states, the non-Markovian evolutions can be represented with high-dimensional linear time-varying equations, where the time-varying parameters arise from the non-Markovian interactions between the quantum system and environment. We study the Magnus expansion of such linear time-varying quantum dynamics and clarify how the truncation errors for the first- and second-order Magnus expansions are influenced by the non-Markovian properties. Besides, when the quantum states are measured for filtering, the dynamics can be modeled as time-varying stochastic differential equations due to the existence of measurement noise. The Magnus expansions based on quantum stochastic filtering are different when the quantum measurement noises are modeled in an {Itô} or Stratonovich approach, rendering different truncation errors. Based on this, numerical simulations further demonstrate the efficiency of Magnus expansions in simulating non-Markovian quantum dynamics without or with stochasticity, and how the truncation errors are influenced by the Lie algebras in the Liouville space.
