Graph operations corresponding to Clifford conjugation for F_{2^h} stabiliser codes

Determine the graph-theoretic operations that correspond to conjugation by Clifford operators for stabiliser codes over finite fields F_{2^h} when represented as graphs whose vertices are partitioned into n groups of h vertices. Specifically, characterize the transformations on these grouped-vertex graphs that capture equivalence under local Clifford conjugation, extending the known correspondence between local complementation and equivalence for one-dimensional binary stabiliser (graph) states.

Background

For binary stabiliser codes (qubits), one-dimensional codes can be represented as graphs (graph states), and equivalence under local Clifford operations corresponds to graph transformations known as local complementation. This provides a powerful combinatorial framework to paper equivalence classes of binary stabiliser states.

In this work, stabiliser codes over fields of even order F_{2^h} are shown to correspond to binary codes via a trace-orthogonal basis and admit a geometric description as quantum sets of symplectic polar spaces. The authors observe that such codes can also be viewed as graphs where vertices are grouped into n sets of h vertices, reflecting the decomposition induced by the field isomorphism. However, the precise translation of Clifford conjugation into operations on these grouped graphs remains unresolved.