# Unfolding quantum master equation into a system of real-valued equations: computationally effective expansion over the basis of $SU(N)$ generators. (arXiv:1812.11626v2 [quant-ph] UPDATED)

Dynamics of an open $N$-state quantum system is typically modeled with a
Markovian master equation describing the evolution of the system's density
operator. By using generators of $SU(N)$ group as a basis, the density operator
can be transformed into a real-valued 'Bloch vector'. The Lindbladian, a
super-operator which serves a generator of the evolution, can be expanded over
the same basis and recast in the form of a real matrix. Together, these
expansions result is a non-homogeneous system of $N^2-1$ real-valued linear
differential equations for the Bloch vector. Now one can, e.g., implement a
high-performance parallel simplex algorithm to find a solution of this system
which guarantees preservation of the norm (exactly), Hermiticity (exactly), and
positivity (with a high accuracy) of the density matrix. However, when
performed on in a straightforward way, the expansion turns to be an operation
of the time complexity $\mathcal{O}(N^{10})$. The complexity can be reduced
substantially when the number of dissipative operators is independent of $N$,
which is often the case for physically meaningful models. Here we present an
algorithm to transform quantum master equation into a system of real-valued
differential equations and propagate it forward in time. By using a scalable
model, we evaluate computational efficiency of the algorithm and demonstrate
that it is possible to handle the model system with $N = 10^3$ states on a
regular-size computer cluster.