Minimal Permutation-Invariant Qudit Codes from Edge-Colorings of Complete Graphs

• URL: http://arxiv.org/abs/2605.22439v1
• Date: Thu, 21 May 2026 13:08:38 GMT
• Title: Minimal Permutation-Invariant Qudit Codes from Edge-Colorings of Complete Graphs
• Abstract summary: We study permutation-invariant quantum codes in the symmetric subspace $mathrmSymn(mathbbCq) $ of $n$ qudits of local dimension $q$.<n>Four physical qudits suffice to encode one logical qudit with distance two in the symmetric sector for every local dimension.
• Score: 4.297070083645049
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study permutation-invariant quantum codes in the symmetric subspace $\mathrm{Sym}^n(\mathbb{C}^q) $ of $n$ qudits of local dimension $q$. For every integer $q\geq 2$, we construct a permutation-invariant code with parameters $((4,q,2))_q$. Thus four physical qudits suffice to encode one logical qudit with distance two in the symmetric sector for every local dimension. We also show, using linear-programming constraints for permutation-invariant quantum codes, that no permutation-invariant code of dimension $q$ and distance at least $2$ exists in $\mathrm{Sym}^n(\mathbb{C}^q)$ for $n\leq 3$. Hence four qudits are necessary and sufficient. The construction has a simple representation-theoretic and combinatorial description. In the irreducible $\mathrm{SU}(q)$-module $\mathrm{Sym}^4(\mathbb{C}^q)$, the distance-two Knill-Laflamme conditions split into root and Cartan parts. By restricting supports to the even-entry occupation layer, all root-error conditions vanish automatically. The remaining Cartan conditions reduce to linear balancing constraints on packets of occupation vectors. These packets admit a natural graph-theoretic interpretation in terms of the vertices and edges of the complete graph $K_q$: for odd $q$, they are organized by the midpoint rule, while for even $q$, they are organized by a decomposition of $K_q$ into perfect matchings. In this way, the existence of minimal $((4,q,2))_q$ permutation-invariant codes is reduced to a parity-dependent edge-coloring problem on $K_q$.

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