One-way quantum computer
The one-way or measurement based quantum computer (MBQC) is a method of quantum computing that first prepares an entangled resource state, usually a cluster state or graph state, then performs single qubit measurements on it. It is "one-way" because the resource state is destroyed by the measurements.
The outcome of each individual measurement is random, but they are related in such a way that the computation always succeeds. In general the choices of basis for later measurements need to depend on the results of earlier measurements, and hence the measurements cannot all be performed at the same time.
Equivalence to quantum circuit model
Any one-way computation can be made into a quantum circuit by using quantum gates to prepare the resource state. For cluster and graph resource states, this requires only one two-qubit gate per bond, so is efficient.
Conversely, any quantum circuit can be simulated by a one-way computer using a two-dimensional cluster state as the resource state, by laying out the circuit diagram on the cluster; Z measurements ( basis) remove physical qubits from the cluster, while measurements in the X-Y plane ( basis) teleport the logical qubits along the "wires" and perform the required quantum gates. This is also polynomially efficient, as the required size of cluster scales as the size of the circuit (qubits x timesteps), while the number of measurement timesteps scales as the number of circuit timesteps.
Topological cluster state quantum computer
Measurement-based computation on a periodic 3D lattice cluster state can be used to implement topological quantum error correction. Topological cluster state computation is closely related to Kitaev's toric code, as the 3D topological cluster state can be constructed and measured over time by a repeated sequence of gates on a 2D array.
One-way quantum computation has been demonstrated by running the 2 qubit Grover's algorithm on a 2x2 cluster state of photons. A linear optics quantum computer based on one-way computation has been proposed.
AKLT state as a resource
It has been shown that the (spin ) AKLT state on a 2D Honeycomb lattice can be used as a resource for MBQC.   More recently it has been shown that a spin-mixture AKLT state can be used as a resource. 
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