Open many-body quantum systems play an important role in quantum optics and condensed-matter physics, and capture phenomena like transport, interplay between Hamiltonian and incoherent dynamics, and topological order generated by dissipation. We introduce a versatile and practical method to numerically simulate one-dimensional open quantum many-body dynamics using tensor networks. It is based on representing mixed quantum states in a locally purified form, which guarantees that positivity is preserved at all times. Moreover, the approximation error is controlled with respect to the trace norm. Hence, this scheme overcomes various obstacles of the known numerical open-system evolution schemes. To exemplify the functioning of the approach, we study both stationary states and transient dissipative behaviour, for various open quantum systems ranging from few to many bodies.Open quantum systems are ubiquitous in physics. To some extent any quantum system is coupled to an environment, and in many instances this interaction significantly alters the system's dynamics. Traditionally such decoherence processes are seen as an enemy to coherent state manipulation. However, suitably engineered dissipation can also have beneficial effects and can be exploited for state preparation [1][2][3][4][5][6], even of states containing strong entanglement or featuring topological order [7,8]. In condensed matter physics, many concepts such as transport are often studied within the closed systems paradigm, but, it is becoming increasingly clear that some familiar concepts may have to be revisited in the open system setting [9], where the interplay between coherent quantum many-body and open systems dynamics, i.e. the competition between Hamiltonian interactions and dissipation leads to interesting physical effects. Since few analytical methods are available for such systems, the design of novel numerical tools for the simulation of dissipative quantum many-body systems is of the utmost importance. In this work, we present a new algorithm which captures the open many-body dynamics in one spatial dimension -for both transient and steady regimes -based on a locally purified tensor network ansatzclass. It comprises a new approach in that the positivity of the operators is maintained during the whole simulation. Importantly, the approximation errors can be controlled in a way that yields a trace-norm certificate. Hence, the algorithm provides not only a conceptually new approach to the problem, but also combines several desired features of the existing schemes and overcomes previous limitations.Tensor-network ansatz-classes have proven to be widely successful to capture the physics of many-body states [10][11][12][13][14][15]. They rely on the idea that relevant quantum states lie in the very small sub-manifold with local correlations, which in turn can be efficiently captured in terms of tensor networks. The density-matrix renormalisation method [16] can indeed be viewed as a variational principle over matrix-product states [11,13,[17][18][...

We present a compendium of numerical simulation techniques, based on tensor network methods, aiming to address problems of many-body quantum mechanics on a classical computer. The core setting of this anthology are lattice problems in low spatial dimension at finite size, a physical scenario where tensor network methods, both Density Matrix Renormalization Group and beyond, have long proven to be winning strategies. Here we explore in detail the numerical frameworks and methods employed to deal with low-dimension physical setups, from a computational physics perspective. We focus on symmetries and closed-system simulations in arbitrary boundary conditions, while discussing the numerical data structures and linear algebra manipulation routines involved, which form the core libraries of any tensor network code. At a higher level, we put the spotlight on loop-free network geometries, discussing their advantages, and presenting in detail algorithms to simulate low-energy equilibrium states. Accompanied by discussions of data structures, numerical techniques and performance, this anthology serves as a programmer's companion, as well as a self-contained introduction and review of the basic and selected advanced concepts in tensor networks, including examples of their applications.