  Based oh the properties of Lie algebras, in this work we develop a general
framework to linearize the von-Neumann equation rendering it in a suitable form
for quantum simulations. We show that one of these linearizations of the
von-Neumann equation corresponds to the standard case in which the state vector
becomes the column stacked elements of the density matrix and the Hamiltonian
superoperator takes the form $I\otimes H-H^\top \otimes I$ where $I$ is the
identity matrix and $H$ is the standard Hamiltonian. It is proven that this
particular form belongs to a wider class of ways of linearizing the von Neumann
equation that can be categorized by the algebra from which they originated.
Particular attention is payed to Hermitian algebras that yield real density
matrix coefficients substantially simplifying the quantum tomography of the
state vector. Based on this ideas, a quantum algorithm to simulate the dynamics
of the density matrix is proposed. It is shown that this method, along with the
unique properties of the algebra formed by Pauli strings allows to avoid the
use of Trotterization hence considerably reducing the circuit depth. Even
though we have used the special case of the algebra formed by the Pauli
strings, the algorithm can be readily adapted to other algebras. The algorithm
is demonstrated for two toy Hamiltonians using the IBM noisy quantum circuit
simulator.
