Multi-party entanglement in graph states2003 · M. Hein, J. Eisert, H. J. Briegel
Derives the stabilizer formalism for graph states on arbitrary graphs, including cycles.
arxiv.org/abs/quant-ph/0307130 ↗Exact & formalBasic circuitsStabilizer states
Graph state on a four-node ring
A graph state built on a 4-cycle rather than an open chain, illustrating how stabilizer generators follow directly from graph adjacency.
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Graph states generalize cluster states to arbitrary graphs: any simple graph G defines a stabilizer state, and the ring is the smallest topology where every qubit has exactly two neighbors, making the generator pattern fully translationally symmetric.
Circuit & simulation
q[0]
HCZCZ
q[1]
HCZCZ
q[2]
HCZCZ
q[3]
HCZCZ
Stabilized by K₀…K₃100%
Uniform basis-string support6%
What this takes and returns
TakesNothingWhat joins here
Nothing goes in, and that is what this entry is: a pipeline starts here. There is no upstream to choose — what you pick is what comes after.
Nothing in the Atlas meets this end.
Returns4 qubitsWhat joins here
Returns a 4-qubit register a next stage can take. This is the end that joins.
1 entry lines up on shape, composition unverified. Named below.
Prepares a state. Nothing goes in, and what comes out is a register another stage can take. See all 12 →
Shapes fit after this, composition unverified 1
The widths and types line up. What is not established is everything a width does not carry — the basis convention, the normalisation, the state each was written to start from — so this is not a claim that the two compose.
Where the map uses this
This record is an instance of an object the map names, so these are the processes that consume or produce one. None of them is about this record in particular.
State you can prepare 5 of 33 processes
- Quantum linear solve hands one back
- Prepare an input state hands one back
- Amplify a success branch hands one back
- Estimate an observable takes one
- Minimise the objective over the parameters hands one back
How it works
For a graph , the graph state is
For the 4-cycle with edges , every vertex has exactly two neighbors, and the stabilizer generators are
Contrast with the open chain. The path-graph cluster state's end qubits have only one neighbor (generators and have weight 2), while every ring generator has weight 3 — the extra wrap-around edge removes the boundary effect and makes the stabilizer group translationally invariant under cyclic qubit relabeling.
Verification structure. Because is bipartite and CZ is diagonal, 's commute pairwise: two generators either act on disjoint qubits, or share exactly the pair of sites where one contributes and the other on the same site, and on that one shared site is compensated by the same relation on the other shared site, giving an overall commutator of . This is the general argument for why any graph state's generator set is a valid stabilizer group — it does not depend on the specific graph topology, only on CZ being diagonal and self-inverse.
Significance. Ring graph states are the standard test case for verifying entanglement-witness and stabilizer-measurement protocols on ring-connectivity hardware (many superconducting and trapped-ion layouts are natively rings or short chains), and they are the resource state for small quantum repeater and secret-sharing demonstrations.
Implementation
Native
graph_state_ring.py
from qiskit import QuantumCircuit
from qiskit.quantum_info import StabilizerState
def graph_state_ring(n: int) -> QuantumCircuit:
qc = QuantumCircuit(n)
qc.h(range(n))
for i in range(n):
qc.cz(i, (i + 1) % n) # wrap-around edge closes the ring
return qc
qc = graph_state_ring(4)
stab = StabilizerState(qc)
print(stab.clifford.to_labels(mode="S"))
# ['+ZIZX', '+IZXZ', '+ZXZI', '+XZIZ'] — the weight-3 ring generators, in
# Qiskit's little-endian order.
FINAL_CIRCUIT = qcQuantum vs classical
Classical baseline
Use a classical state-vector or matrix simulation at the same width, precision, and measurement objective.
Quantum claim
The quantum record demonstrates a state or operator behavior; it does not make classical simulation or communication costs disappear.
How to compare
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