This paper is a set of lecture notes that introduces tensor‑network methods in quantum many‑body physics, with a strong focus on matrix‑product states (MPS). It was written by Grégoire Misguich for the 9th Les Houches Summer School on Computational Physics: Open Quantum Systems in June 2026. The notes aim to teach the basic language and algorithms that practitioners use to represent and simulate quantum states on a computer.

The author develops the core ideas step by step. He explains graphical notation and the building blocks of tensor networks, such as virtual indices and bond dimensions (the numbers that control how much correlation the network can represent). The notes cover practical matrix factorizations used in these methods, including QR and singular‑value decompositions, and discuss gauge freedom and canonical forms that help stabilize numerical algorithms. The main MPS algorithms are presented in detail: how to contract networks, compute correlation functions, represent operators as matrix‑product operators (MPO), run the density‑matrix renormalization group (DMRG) for ground states, and perform time evolution using methods such as time‑evolving block decimation (TEBD) and the time‑dependent variational principle (TDVP). The notes also touch on projected entangled‑pair states (PEPS) for higher dimensions and on tensor‑network descriptions of mixed states, quantum channels, and Lindblad dynamics for open systems.

At a high level, tensor networks work by compressing a very large quantum state into a set of smaller tensors connected by virtual indices. The compression is guided by entanglement: states with limited entanglement need only small bond dimensions and can be represented compactly. This makes many low‑energy or equilibrium states of local quantum models tractable on a classical computer. The notes explain the key tradeoffs: increasing the bond dimension lets you represent more entanglement but costs more time and memory.