Eric Kubischta

Eric Kubischta, Ian Teixeira

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 $K_q$.

Eric Kubischta, Ian Teixeira

We develop an intrinsic enumerator framework for quantum error correction in unitary representations of symmetry groups. An intrinsic quantum code is a subspace of a representation $V$ of a group $G$, and errors are organized by the decomposition of the conjugation representation on $\mathcal{L}(V)$ into isotypic subspaces. Associated with any orthogonal decomposition of $\mathcal{L}(V)$ we introduce two families of quadratic enumerators, called projector and twirl enumerators, which satisfy positivity, normalization, and Knill--Laflamme type inequalities. When the conjugation representation is multiplicity--free, these enumerators are related by a linear transform that we interpret as an intrinsic MacWilliams identity. For $G=\mathrm{SU}(2)$, we compute this transform explicitly in terms of Wigner $6j$-symbols. Applied to symmetric-power representations, this gives linear programming bounds for permutation-invariant qubit and qudit codes, including extremality results for the four-qubit, seven-qubit, and three-qutrit examples treated here. We also develop the general equivariant theory in the presence of multiplicities, where the enumerators become matrix-valued, the MacWilliams transform becomes block unitary, and the resulting feasibility problem becomes semidefinite; we illustrate this theory in a first non-multiplicity-free $\mathrm{SU}(3)$ example.