Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Wolfram`QuantumFramework`
GraphState  |
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A stores a stabilizer state in the graph-state representation introduced by Anders & Briegel (arXiv:quant-ph/0504117). For a graph the associated state is , i.e. the application of a controlled- on every edge to the all-plus state.
G=(V,E)
|G〉=|〉
∏
e∈E
CZ
e
⊗n
+
Z
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The internal representation is the Association . The field stores a list of integer indices into the 24-element single-qubit Clifford group; currently only the identity (index ) is exercised.
"Graph"g,"VOPs"{0,…}
"VOPs"
0
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The stabilizer generators of the graph state are read off the graph: for each vertex ,=⊗, where is the neighbour set of .
i
K
i
X
i
∏
j∈N(i)
Z
j
N(i)
i
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The graph-state representation has memory cost where is the average vertex degree, which is asymptotically smaller than the tableau for sparsely-connected codes.
O(n·)
d
d
O()
2
n
▪
The conversion succeeds only on graph-form stabilizers (one on the diagonal, / elsewhere); other inputs emit and return . Use a local-Clifford transformation (AndBri05 Lemma 1) to bring an arbitrary stabilizer state into graph form before calling.
X
Z
I
GraphState::nongraph
$Failed
▪
Two graph states (with identity VOPs) are local-Clifford-equivalent if and only if their graphs are related by a sequence of local complementations (); this is the basis of the graph-state classification of stabilizer states (AndBri05 Theorem 1, Hein-Eisert-Briegel arXiv:quant-ph/0602096).
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Property access uses the dispatch syntax ; the full list of recognised properties is given by . Conversion to a is done via .
gs["property"]
gs["Properties"]
PauliStabilizer
gs["PauliStabilizer"]
Examples
(24)
Basic Examples
(1)
The graph state of a 4-cycle:
In[1]:=
 | GraphState |
Out[2]=
GraphState
 | |
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The graph state of a 3-path:
In[1]:=
| GraphState |
Out[2]=
GraphState
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The graph state of a 4-vertex star (GHZ-equivalent under local Cliffords):
In[1]:=
| GraphState |
Out[2]=
GraphState
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A graph-form PauliStabilizer is converted directly:
In[1]:=
| GraphState |
 | PauliStabilizer |
Out[2]=
GraphState
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Inspecting stabilizers and the underlying graph:
In[1]:=
Module{gs},gs=
[CycleGraph[4]];{gs["VertexCount"],gs["EdgeCount"],gs["Stabilizers"]}
 | GraphState |
Out[2]=
{4,4,{XZIZ,ZXZI,IZXZ,ZIZX}}
Scope
(4)
Generalizations & Extensions
(1)
Options
(8)
Applications
(2)
Properties & Relations
(4)
Possible Issues
(3)
Neat Examples
(1)
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