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arXiv:2609.16352

qdmag: A Python package for simulating nonequilibrium magnetization of molecules using quantum master equations

Shuanglong Liu, Xiao Chen, Andrew Cupo, James N. Fry, Hai-Ping Cheng

quant-phSubmitted Sep 14, 2026• Announced Sep 16, 2026
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Abstract

We present qdmag, a computational utility for calculating the magnetization of magnetic molecules in a time-varying external magnetic field by solving a generalized Lindblad quantum master equation with spin-phonon coupling treated as a dissipation term. The package relies on the spin Hamiltonian formalism, which includes both the magnetic exchange interactions and the zero-field splitting (ZFS) interaction. Different types of magnetic exchange interaction and ZFS terms up to 12th order, which are written in terms of the extended Stevens operators, are supported. qdmag enables long-time evolution of magnetization for up to a few milliseconds, which is achieved through a staircase approximation of the magnetic field as a function of time. For high-dimensional spin Hamiltonians, a lower-dimensional effective Hamiltonian is constructed to make the calculations computationally feasible. Three case studies of one-spin, two-spin, and three-spin systems are presented to demonstrate the usage of qdmag. Our work offers a computational tool to bridge the gap between theoretical development and numerical calculations of spin dynamics in the presence of spin-phonon coupling.

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We present qdmag, a computational utility for calculating the magnetization of magnetic molecules in a time-varying external magnetic field by solving a generalized Lindblad quantum master equation with spin-phonon coupling treated as a dissipation term. The package relies on the spin Hamiltonian formalism, which includes both the magnetic exchange interactions and the zero-field splitting (ZFS) interaction. Different types of magnetic exchange interaction and ZFS terms up to 12th order, which are written in terms of the extended Stevens operators, are supported. qdmag enables long-time evolution of magnetization for up to a few milliseconds, which is achieved through a staircase approximation of the magnetic field as a function of time. For high-dimensional spin Hamiltonians, a lower-dimensional effective Hamiltonian is constructed to make the calculations computationally feasible. Three case studies of one-spin, two-spin, and three-spin systems are presented to demonstrate the usage of qdmag. Our work offers a computational tool to bridge the gap between theoretical development and numerical calculations of spin dynamics in the presence of spin-phonon coupling.
We consider the dynamical properties of a gapped quantum spin system coupled to the electric field of a laser, which drives the resonant excitation of specific phonon modes that modulate the magnetic interactions. We deduce the quantum master equations governing the time-evolution of both the lattice and spin sectors, by developing a Lindblad formalism with bath operators providing an explicit description of their respective phonon-mediated damping terms. We investigate the nonequilibrium steady states (NESS) of the spin system established by a continuous driving, delineating parameter regimes in driving frequency, damping, and spin-phonon coupling for the establishment of physically meaningful NESS and their related nontrivial properties. Focusing on the regime of generic weak spin-phonon coupling, we characterize the NESS by their frequency and wave-vector content, explore their transient and relaxation behavior, and discuss the energy flow, the system temperature, and the critical role of the type of bath adopted. Our study lays a foundation for the quantitative modeling of experiments currently being designed to control coherent many-body spin states in quantum magnetic materials.
The dynamics of the one-dimensional spin model (s = 1/2) XXZ in the quantum “quenching” mode is investigated on a classical computer using the Qiskit and LindbladMPO software. The quantitative description of the time behavior of local magnetizations and spin correlation functions are proposed to be considered as a standard for comparative analysis with similar results previously obtained on existing noisy intermedium-scale quantum (NISQ) computers. Quantum quenching of the spin system is created at the initial time, t=0t = 0t=0 , by an abrupt (nonadiabatic) change of the spin Hamiltonian parameter. The subsequent ( t>0t > 0t>0 ) time evolution of the reduced density matrix ρ(t)\rho (t)ρ(t) of the system is traced from the solutions of the quantum Liouville equation in the case of an isolated system and its generalization in the form of the Lindblad equation for an open system interacting with the environment. In the case of an open quantum system, dissipation processes are caused by the influence of fluctuating environmental fields on the spin system.
The dynamics of open quantum systems is governed by the Lindblad master equation, which provides a consistent framework for incorporating environmental effects into the evolution of the system. Since exact solutions are rarely available, numerical methods become essential tools for analyzing such systems. This article presents a step-by-step implementation of the fourth-order Runge-Kutta method in Python to solve the Lindblad equation for a single quantized field mode subject to decay. A coherent state is used as the initial condition, and the time evolution of the average photon number is investigated. The proposed methodology enables transparent and customizable simulations of dissipative quantum dynamics, emphasizing a pedagogical approach that helps readers understand the numerical structure without relying on external libraries such as QuTiP. This standalone implementation offers full control over each integration step, making it particularly suitable for educational contexts and for exploring non-standard dynamics or introducing custom modifications to the Liouvillian.
Simulating quantum dynamics at finite temperature in complex chemical systems remains a central challenge in quantum chemistry and materials science. In many cases, it is advantageous to focus on the dynamics of a subsystem of interest, represented by its reduced density matrix, whose interaction with the surrounding environment gives rise to open-system behavior. The Nakajima-Zwanzig generalized quantum master equation (GQME) provides a formally exact framework for capturing such non-Markovian dynamics, where memory effects play a defining role. Here, in Part VI of the QFlux tutorial series, we explore methods for simulating non-Markovian open quantum systems described by the GQME on quantum computers. The approach leverages the Sz.-Nagy dilation theorem to embed memory-dependent, non-unitary evolution into an enlarged unitary space suitable for quantum simulation. Using the spin-boson model as a prototypical example, we demonstrate how to formulate, implement, and analyze GQME dynamics within the QFlux open-source platform. This tutorial provides readers with both the theoretical foundation and practical tools required to study non-Markovian quantum dynamics, bridging fundamental concepts with executable workflows for quantum simulation.
We study the effect of antiferromagnetic interactions on the single spin-flip Glauber dynamics of two different one-dimensional Ising models with spin $\ifmmode\pm\else\textpm\fi{}1$. The first model is an Ising chain with antiferromagnetic exchange interaction limited to nearest neighbors and subject to an oscillating magnetic field. The system of master equations describing the time evolution of sublattice magnetizations can easily be solved within a linear field approximation and a long time limit. Resonant behavior of the magnetization as a function of temperature (stochastic resonance) is found, at low frequency, only when spins on opposite sublattices are uncompensated owing to different gyromagnetic factors (i.e., in the presence of a ferrimagnetic short range order). The second model is the axial next-nearest-neighbor Ising (ANNNI) chain, where an antiferromagnetic exchange between next-nearest neighbors is assumed to compete with a nearest-neighbor exchange interaction of either sign. The long time response of the model to a weak, oscillating magnetic field is investigated in the framework of a decoupling approximation for three-spin correlation functions, which is required to close the system of master equations. The calculation, within such an approximate theoretical scheme, of the dynamic critical exponent zzz, defined as $1∕\ensuremath{\tau}\ensuremath{\approx}{(1∕\ensuremath{\xi})}^{z}$ (where $\ensuremath{\tau}$ is the longest relaxation time and $\ensuremath{\xi}$ is the correlation length of the chain), suggests that the T=0T=0T=0 single spin-flip Glauber dynamics of the ANNNI chain is in a different universality class than that of the unfrustrated Ising chain.
The work develops an approach to researching the dynamics of qubits that makes it possible to monitor the behavior of the system in the process of quantum computing using computer simulation. The study deals with individual and interacting qubits based on spin ½ in a magnetic field. The goal of the work is to obtain evolutionary equations (equations of motion) for observables that adequately describe the state of the system and provide opportunities for computer modeling of processes. The dynamics of qubits are described using the formalism of the density matrix of pure states. The paper shows derivation of the equations of motion for observables being the average values of operators of physical quantities that describe the state of the system. For spin ½ in magnetic fields, the derivation of the dynamic equation for the spin polarization vector, which is a unit vector in real three-dimensional space, is shown. The shape and parameters of radiofrequency pulses that implement single-qubit operations are presented. In the case of two interacting spins ½, an expression is given for the density matrix and the Hamiltonian of the system for scalar indirect interaction of the magnetic moments of nuclei through the electron shells of atoms in the molecule. Evolution equations are derived for the average dipole moments and quadrupole moment of the system. It is shown that in the case of weak interaction the dynamics of the system can be described using a closed system of equations for the average dipole moments, which in this case coincide with the spin polarization vectors of individual spins. The presentation is accompanied by examples of numerical solutions of the resulting equations describing one-qubit and two-qubit operations. A conclusion is made about the adequacy of the created mathematical model and the possibility of its use for computer modeling of quantum computing processes, the influence of the environment on them and the stabilization of states to prevent or reduce errors in quantum computing.