A Lie-algebraic approach to non-Markovian quantum dynamics

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 {It\^{o}} 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.