Weight isometries of graph states: the weight function up to GL(n,2) is a complete invariant for n ≤ 7, and a correction to an example of Pllaha

Abstract

A self-dual additive code over 𝔽₄ of length n is an n-qubit stabilizer state, and every such code is equivalent to the graph code C_G={(x,Γ_G x):x ∈ 𝔽₂ⁿ} of a graph G on n vertices; two graph codes are monomially equivalent — equivalently, the two states are related by local Clifford operations and a permutation of the qubits — exactly when the graphs are related by local complementation and vertex permutation. Pllaha's Problem 6.3 asks how far the symplectic isometries between two self-dual stabilizer codes can exceed the monomial ones, and Gluesing-Luerssen and Pllaha call their single-code form of the question "wide open". We settle the emptiness half of Problem 6.3 at lengths six and seven. If G and H are graphs on six vertices and some 𝔽₂-linear g:𝔽₂⁶ → 𝔽₂⁶ satisfies w_H(gx)=w_G(x) for all x — no invertibility assumed — then G and H are related by local complementations and vertex permutations; so the number of classes surviving the coarser weight equivalence is W(6)=t(6)=26. At length seven the same holds for the 59 class representatives, which is W(7)=59 once Danielsen and Parker's t(7)=59 is granted. The weight enumerator is strictly weaker at both lengths: only E(6)=23 enumerators occur among the 26 classes, and we exhibit an inequivalent pair sharing one. We also correct Example 5.10 of Pllaha's e-print arXiv:1807.09107v1: the two length-four codes it displays are monomially equivalent — exactly 32 of the 6⁴ · 4!=31 104 monomial maps carry one onto the other — although the displayed isometry between them is realised by none. Outside the formal development, the same search run in C against Danielsen's orbit database extends the conclusion to n ≤ 11, where 4 501 923 inequivalent pairs share a weight enumerator and none is weight-isometric. Everything not explicitly labelled as computed outside the formal development is machine-checked in Lean 4.

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