---
abstract: 'We propose a general framework of quantum kinetic Monte Carlo algorithm, based on a stochastic representation of a series expansion of the quantum evolution. Two approaches have been developed in the context of quantum many-body spin dynamics, using different decomposition of the Hamiltonian. The effectiveness of the methods is tested for many-body spin systems up to $40$ spins.'
author:
- 'Zhenning Cai[^1]'
- 'Jianfeng Lu[^2]'
bibliography:
- 'SpinDynamics.bib'
title: 'A quantum kinetic Monte Carlo method for quantum many-body spin dynamics[^3]'
---

Spin dynamics, quantum kinetic Monte Carlo, Dyson series

74S60

Introduction
============

We consider the system with $N$ spins in a magnetic field, which is given by the Hamiltonian $$\label{eq:hamiltonian}
H(t) = \sum_{k=1}^N m^{(k)}(t) \cdot \sigma^{(k)} + \sum_{1 \leq j < k \leq N}
  \gamma_{jk}(t) \sigma^{(j)} \cdot \sigma^{(k)}$$ with Hilbert space of the quantum system given by $\mathcal{H} = (\mathbb{C}^2)^{\otimes N}$. For every $k = 1,\cdots,N$, the operator $\sigma^{(k)}$ denotes the Pauli matrices acting on the $k$-th spin: $$\sigma^{(k)} = (\sigma_1^{(k)}, \sigma_2^{(k)}, \sigma_3^{(k)})^{\top}, \qquad
\sigma_i^{(k)} = {\mathrm{Id}}^{\otimes (k-1)} \otimes \sigma_i \otimes {\mathrm{Id}}^{\otimes (N-k)};$$ where ${\mathrm{Id}}$ stands for the identity operator acting on a single spin ($\mathbb{C}^2$). The first term on the right hand side of gives the single-body 