The past few years have witnessed the concrete and fast spreading of quantum technologies for practical computation and simulation. In particular, quantum computing platforms based on either trapped ions or superconducting qubits have become available for simulations and benchmarking, with up to few tens of qubits that can be reliably initialized, controlled, and measured. The present review aims at giving a comprehensive outlook on the state of art capabilities offered from these near-term noisy devices as universal quantum simulators, i.e. programmable quantum computers potentially able to calculate the time evolution of many physical models. First, we give a pedagogic overview on the basic theoretical background pertaining digital quantum simulations, with a focus on hardware-dependent mapping of spin-type Hamiltonians into the corresponding quantum circuit model as a key initial step towards simulating more complex models. Then, we review the main experimental achievements obtained in the last decade regarding the digital quantum simulation of such spin models, mostly employing the two leading quantum architectures. We compare their performances and outline future challenges, also in view of prospective hybrid technologies, towards the ultimate goal of reaching the long sought quantum advantage for the simulation of complex many body models in the physical sciences.
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