Beijing Institute of Mathematical Sciences and Applications

This course provide an comprehensible introduction to two pivotal tensor network states—the Matrix Product States (MPS)​​ and the ​Projected Entangled Pair States (PEPS)​—for simulating complex quantum systems in condensed matter physics. First originated from the density matrix renormalization group (DMRG) method, the MPS excels at modeling one-dimensional (1D) systems by efficiently representing ground states, excited states, and dynamics through low-rank tensor decomposition. Its computational tractability makes it ideal for studying spin chains, quantum phase transitions, and critical phenomena in 1D lattices.

PEPS extends these principles to ​two-dimensional (2D) and higher-dimensional systems, capturing intricate entanglement structures that evade traditional methods. PEPS utilizes tensor networks to simulate statistical models, topological order, quantum spin liquids, and anyonic excitations in high-dimensional lattices. However, PEPS also brings higher computational complexity to the algorithm and provides greater challenges to algorithm design.