What is the difference between "cluster states" and "graph states"? I wonder about the difference between the cluster state and the graph state.
I guess the only difference is the graph of the cluster state is limited to a two-dimensional square lattice
The concept of the cluster state and the graph state can be known here:
https://en.wikipedia.org/wiki/Graph_state
https://en.wikipedia.org/wiki/Cluster_state
 A: I don't think there is really a difference, though in general, for "cluster state" people would think of a regular lattice (not only 2D - e.g. 3D cluster states are used for fault-tolerant quantum computation), while for graph state, any graph would work (and there are systems which map to graph states with non-local underlying graphs).
A: There's no difference really. For example Michael Nielsen's review paper discusses this terminology a little bit (https://arxiv.org/abs/quant-ph/0504097):

Note that the states we have called cluster states are sometimes also known as
graph states. Originally, the term “cluster state” was introduced by Raussendorf and
Briegel [28] to refer to the case where the graph G is a two-dimensional square lattice.
This was the class of states which they showed in [27] could be used as a substrate for
quantum computation. The term “graph state” originally referred to the family of states
associated with more general graphs G. This distinction was blurred by the introduction
of schemes for quantum computing based on Raussendorf and Briegel’s ideas, but using
different graphs.
I believe it makes most sense to have a single terminology for the entire class of states,
and then to specify in any instance what graph is being used (e.g. a two-dimensional
square lattice with boundary). I suggest using the term “cluster state” for this purpose,
and will follow this terminology throughout this paper.

