Asymptotic state of an open quantum system can undergo qualitative changes upon small variation of system parameters. We demonstrate it that such 'quantum bifurcations' can be appropriately defined and made visible as changes in the structure of the asymptotic density matrix. By using an N -boson open quantum dimer, we present quantum diagrams for the pitchfork and saddlenode bifurcations in the stationary case and visualize a period-doubling transition to chaos for the periodically modulated dimer. In the latter case, we also identify a specific bifurcation of purely quantum nature.

Quantum systems out of equilibrium are presently a subject of active research, both in theoretical and experimental domains. In this work we consider time-periodically modulated quantum systems which are in contact with a stationary environment. Within the framework of a quantum master equation, the asymptotic states of such systems are described by time-periodic density operators. Resolution of these operators constitutes a non-trivial computational task. Approaches based on spectral and iterative methods are restricted to systems with the dimension of the hosting Hilbert space dimH = N 300, while the direct long-time numerical integration of the master equation becomes increasingly problematic for N 400, especially when the coupling to the environment is weak. To go beyond this limit, we use the quantum trajectory method which unravels master equation for the density operator into a set of stochastic processes for wave functions. The asymptotic density matrix is calculated by performing a statistical sampling over the ensemble of quantum trajectories, preceded by a long transient propagation. We follow the ideology of eventdriven programming and construct a new algorithmic realization of the method. The algorithm is computationally efficient, allowing for long 'leaps' forward in time, and is numerically exact in the sense that, being given the list of uniformly distributed (on the unit interval) random numbers, {η1, η2, ..., ηn}, one could propagate a quantum trajectory (with ηi's as norm thresholds) in a numerically exact way. By using a scalable N -particle quantum model, we demonstrate that the algorithm allows us to resolve the asymptotic density operator of the model system with N = 2000 states on a regular-size computer cluster, thus reaching the scale on which numerical studies of modulated Hamiltonian systems are currently performed.

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 nonhomogeneous 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 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 O(N 10 ). 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 = 10 3 states on a single node of a computer cluster.

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Due to complexity of the systems and processes it addresses, the development of computational quantum physics is influenced by the progress in computing technology. Here we overview the evolution, from the late 1980s to the current year 2020, of the algorithms used to simulate dynamics of quantum systems. We put the emphasis on implementation aspects and computational resource scaling with the model size and propagation time. Our minireview is based on a literature survey and our experience in implementing different types of algorithms.