Quantum many-body systems are hard to study because the associated Hilbert space, containing all possible many-body states, is huge: its dimension grows exponentially in the system size. In recent years, however, progress in our understanding of quantum entanglement has revealed that a large class of many-body states of interest are highly atypical and such that we can actually efficiently represent them with a mathematical structure called tensor network. As a result, it is now possible to accurately simulate, say, a quantum spin chain made of thousands of interacting spins. In this Colloquium I will (i) review the computational challenge posed by quantum many-body systems; (ii) summarize our current understanding of many-body entanglement; and (iii) give a gentle introduction to tensor networks as an efficient description of many-body states.
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