MacWilliams Identities for Intrinsic Quantum Codes
This work provides a new algebraic framework for analyzing quantum error-correcting codes with symmetry, enabling linear programming bounds for permutation-invariant codes, which is significant for quantum coding theory.
The authors develop an intrinsic enumerator framework for quantum error correction in unitary representations of symmetry groups, introducing projector and twirl enumerators that satisfy positivity and Knill-Laflamme type inequalities. They derive an intrinsic MacWilliams identity for multiplicity-free cases, compute it explicitly for SU(2) using Wigner 6j-symbols, and demonstrate linear programming bounds for permutation-invariant qubit and qudit codes, including extremality results for four-qubit, seven-qubit, and three-qutrit codes.
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.