Ian Teixeira

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 $K_q$.

Ian Teixeira

We derive an explicit formula for the intrinsic MacWilliams transform for permutation-invariant qudit codes. Such codes naturally live in symmetric power representations, where the relevant error sectors are determined by the irreducible decomposition of the conjugation action on the associated operator space. Using the multiplicity-free structure of this decomposition and the corresponding intertwiner algebra, we identify the intrinsic MacWilliams matrix with a finite Racah transform. The entries are given by a terminating hypergeometric series, and the rows of the matrix are Racah orthogonal polynomials with parameters determined explicitly by the block length and local dimension. Computing the spectrum of the degree-one twirl reveals that this spectrum lies on an affine quadratic lattice. Then we derive a tridiagonal multiplication rule from the representation theory of the adjoint sector. As consequences, we obtain closed-form orthogonality, detailed-balance, and involutivity identities for the transform. The resulting formula supplies an explicit MacWilliams matrix for computing linear programming bounds on permutation-invariant qudit codes.

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.