Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Wolfram`QuantumFramework`
LocalComplement  |
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implements the local complementation operation of Anders & Briegel (arXiv:quant-ph/0504117, Definition 1): for the chosen vertex , every pair of distinct neighbours has the edge toggled (added if absent, removed if present).
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a∼b
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Local complementation generates the equivalence of graph states under local Clifford operations. The Anders-Briegel theorem (AndBri05 Theorem 1) states that two graph states and are local-Clifford-equivalent if and only if their graphs lie in the same orbit under successive local complementations.
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On a input, modifies the underlying graph but currently keeps the same vertex-operator-Pauli (VOP) assignments. Tracked-VOP local complementation per AndBri05 Eq (8) is on the roadmap; use the resulting 's graph for combinatorial reasoning and the original for unitary evolution.
GraphState
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Local complementation is involutive: applied twice at the same vertex restores the original graph.
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The local-complementation orbit of an -vertex graph is finite (bounded by the -vertex graph count) and is the equivalence class used to classify graph states up to local Cliffords; small representatives are tabulated in Hein et al. (arXiv:quant-ph/0602096).
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Examples
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Basic Examples
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Local complementation at an isolated vertex is the identity:
In[1]:=
 | LocalComplement |
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Apply local complementation to the centre of a 4-vertex star, producing a complete graph on the leaves:
In[1]:=
| LocalComplement |
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At the middle of a path graph, local complementation closes the path's outer arms into a triangle:
In[1]:=
Module{p3},p3=PathGraph[{1,2,3}];
[p3,2]
 | LocalComplement |
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Local complementation on a GraphState returns a new GraphState:
In[1]:=
| LocalComplement |
 | GraphState |
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GraphState
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Involutive at a single vertex:
In[1]:=
Module{g,twice},g=StarGraph[5];twice=
[g,1],1;IsomorphicGraphQ[g,twice]
 | LocalComplement |
 | LocalComplement |
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True
Scope
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Generalizations & Extensions
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Options
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Applications
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Properties & Relations
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Possible Issues
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Neat Examples
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