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=== Definition ===
The general form of graph states was introduced as a
generalization of cluster states, which have been shown
to be a resource for one-way quantum computation. The importance of
graph states stems from the fact that universality of quantum
computer based on these states is one of the fundamental
applications of entanglement in quantum computation theory.
Definition: A graph state is a pure
m-qubit state |G⟩ corresponding to a
graph G(V, E). The
graph is described by the set V of vertices with cardinality
|V| = m,
representing the qubits of |G⟩, and the set E of edges, i.e. pairs of vertices,
representing pairs of qubits of |G⟩.
Construction
In order to construct |G⟩
one takes |+⟩⊗m,
with $\; |+\rangle=(|0\rangle + |1\rangle)/
\sqrt{2}$, as the initial state. Then, according to a given graph
G(V, E),
one applies a controlled phase gate UC − phase = |0⟩⟨0| ⊗ 1 + |1⟩⟨1| ⊗ σ3
to any pair of qubits corresponding to vertices connected by an edge
from E.
Note that, since all such controlled phase operations commute even if
performed according to the edges with a common vertex, the order in
which the operations are applied is arbitrary.
Properties
- Any connected graph state |G⟩ is a fully entangled
m-particle state and
violates some Bell inequality.
- From the construction it follows that the set of graph states is
described by a polynomial number m(m − 1)/2 of discrete
parameters (while in general the set of all states in the m-qubit Hilbert space is described
by an exponential 2m number of continuous
parameters).
- Two graph states are locally unitarily interconvertible under the
transformation ⊗i = 1mUi,
and this is equivalent to convertibility under stochastic local
operations and classical communication (SLOCC).
- R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum
entanglement, e-print .
- R. Raussendorf, D. Browne, H.-J. Briegel, Phys. Rev. A
68, 022312 (2003).
- H.-J. Briegel, R. Raussendorf, Phys. Rev. Lett
86, 910 (2001).
- R. Raussendorf, H.-J. Briegel, Phys. Rev. Lett
86, 5188 (2001).
- Hein et al., Proceedings of the International of
Physics School Enrico Fermi on Quantum Computers, Algorithms and
Chaos (2005) e-print .
Category:Quantum
States
Category:Mathematical
Structure
Category:Models
of Quantum Computation
Last modified:
Monday, October 26, 2015 - 17:56
