  We introduce novel schemes for quantum computing based on local measurements
on entangled resource states. This work elaborates on the framework established
in [Phys. Rev. Lett. 98, 220503 (2007), quant-ph/0609149]. Our method makes use
of tools from many-body physics - matrix product states, finitely correlated
states or projected entangled pairs states - to show how measurements on
entangled states can be viewed as processing quantum information. This work
hence constitutes an instance where a quantum information problem - how to
realize quantum computation - was approached using tools from many-body theory
and not vice versa. We give a more detailed description of the setting, and
present a large number of new examples. We find novel computational schemes,
which differ from the original one-way computer for example in the way the
randomness of measurement outcomes is handled. Also, schemes are presented
where the logical qubits are no longer strictly localized on the resource
state. Notably, we find a great flexibility in the properties of the universal
resource states: They may for example exhibit non-vanishing long-range
correlation functions or be locally arbitrarily close to a pure state. We
discuss variants of Kitaev's toric code states as universal resources, and
contrast this with situations where they can be efficiently classically
simulated. This framework opens up a way of thinking of tailoring resource
states to specific physical systems, such as cold atoms in optical lattices or
linear optical systems.
