Unfolding quantum master equation into a system of real-valued equations: computationally effective expansion over the basis of SU(N) generators

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, %in the master equation, 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 N²-1 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 exact preservation of the norm and Hermiticity of the density matrix. However, when performed in a straightforward way, the expansion turns to be an operation of the time complexity mathcal{O}\(N¹⁰\). The complexity can be reduced 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 = 10³ states on a single node of a computer cluster.