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arXiv:2607.13865

A Lie-algebraic approach to non-Markovian quantum dynamics

Haijin Ding, Stephen S. -T. Yau, Zhiwen Zhang

quant-phSubmitted Jul 15, 2026• Announced Jul 16, 2026• 0 citations
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Abstract

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.

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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.
The evolution of a quantum system under time-dependent driving exhibits phenomena that are absent in its stationary counterpart. However, the high dimensionality and non-commutative nature of quantum dynamics make this a challenging problem. The Magnus expansion provides an analytic framework to approximate the effective dynamics on short time-scales, but computing high-order terms with existing methods is computationally expensive. We introduce a scalable approach that reduces the computational effort to depend only on the degrees of freedom defining the time-dependent control function. We focus specifically on Hamiltonians consisting of a constant drift term and a controllable term. Our method provides a polynomial expression for the Magnus expansion which can be evaluated several orders of magnitude faster than previous techniques, enabling broad applications in the realms of quantum simulation and quantum optimal control. We showcase an application of the method by designing control pulses for the 5-qubit phase gate on a neutral-atom platform utilizing Rydberg atoms.
The numerical simulation of non-Markovian open quantum systems based on the non-Markovian quantum state diffusion (NMQSD) equation is complicated by functional derivatives with respect to the stochastic process. A general numerical framework that directly treats these functional derivatives without relying on prescribed decompositions of the bath correlation function is still lacking. In this work, we derive an analytical solution of the linear NMQSD equation that reveals three elementary structures of the non-Markovian stochastic dynamics: stochastic propagation, functional-derivative insertion, and memory pairing. Based on this structure, we construct a general auxiliary-state framework for arbitrary bath correlation functions. The framework separates the numerical construction into time discretization, memory quadrature, and hierarchy truncation. We then construct first- and second-order schemes and provide diagrammatic transition rules for their explicit implementation. Numerical results verify the expected temporal accuracy and demonstrate the applicability of the proposed methods to different bath correlation functions and multi-level quantum systems.
In this thesis, I focus on the quantum Fisher information (QFI) - a central quantity in parameter estimation theory for quantum systems, which provides a lower bound on the precision one can achieve. In the first part of the thesis, I explore a Lie algebraic approach to evaluate the quantum Fisher information for unitary dynamics. I show how the Lie algebra associated with the unitary evolution, together with the initial state, allows one to compute various quantities of interest, circumventing the computation of the time-evolved state. In particular, I apply this shortcut approach to calculate the QFI and use it for the QFI evaluation in various prototype models, both finiteand infinite-dimensional. In the second part of the thesis, I shift focus to non-unitary dynamics, such as those described by a quantum master equation. In this setting, I theoretically study the description of open system dynamics in the Liouville space of vectorized density matrices and investigate the so-called dissipative quantum Fisher information (DQFI) within that framework. Specifically, I demonstrate how the metric defined in Liouville space differs from its counterpart in Hilbert space. I establish a map between the Hilbert space QFI and the Liouville dissipative quantum Fisher information for qubit and qudit systems, expressed via the quantum state’s purity. Furthermore, I discuss the physical meaning of the DQFI by studying its behavior under completely positive trace-preserving (CPTP) maps. The present results demonstrate that the Lie algebraic approach provides a computationally efficient framework for the QFI evaluation. This approach allows to avoid the explicit solution of quantum dynamics, and remains well-defined in cases when conventional approaches like WeiNorman expansion exhibit singularities.
This report is concerned with the efficiency of numerical methods for simulating quantum spin systems, with the aim to implement an improved method for simulation of a time-dependent Hamiltonian that displays chirped pulses at a high frequency. Working in the density matrix formulation of quantum systems, we study evolution under the Liouville-von Neumann equation, presenting analysis of and benchmarking current numerical methods. The accuracy of existing techniques is assessed in the presence of chirped pulses. We also discuss the Magnus expansion and detail how a truncation of it is used to solve differential equations. The results of this work are implemented in the Python package MagPy to provide a better error-to-cost ratio than current approaches allow for time-dependent Hamiltonians.
Abstract We present a general numerical simulation method to solve non-Markovian dynamics of an open quantum system influenced by quantum Brownian motion. Based on the determined memory kernel function, this method enables the resolution of non-Markovian dynamics for a wide range of system Hamiltonians and spectral densities. The system dynamics are described by exact integro-differential operator equations without any common approximations and they are simulated in this work by definite-number equations with stochastic initial conditions. This approach ensures the applicability of mature numerical methods and maintains computational complexity that remains largely invariant, even when dealing with more complex models. The high accuracy of our simulation is evident from a comparison with the results obtained from corresponding exact master equations, underscoring the reliability and precision of our method.
Based on a Liouville-space formulation of open systems, we present two methods to solve the quantum dynamics of coupled harmonic oscillators experiencing Markovian loss. The starting point is the quantum master equation in Liouville space which is generated by a Liouvillian that induces a Lie algebra. We show how this Lie algebra allows to define ladder operators that construct Fock-like eigenstates of the Liouvillian. These eigenstates are used to decompose the time-evolved density matrix and, together with the accompanying eigenvalues, provide insight into the transport properties of the lossy system. Additionally, a Wei-Norman expansion of the generated time evolution can be found by a structure analysis of the algebra. This structure analysis yields a construction principle to implement effective non-Hermitian Hamiltonians in lossy systems.