Invariant Theory and Magic State Distillation

We show that the performance of a linear self-orthogonal $GF(4)$ code for magic state distillation of Bravyi and Kitaev's $|T\rangle$-state is characterized by its simple weight enumerator. We compute weight enumerators of all such codes with fewer than 20 qubits and find none whose threshold exceeds that of the 5-qubit code. Using constraints on weight enumerators from invariant theory and linear programming, we establish bounds on the exponent characterizing noise suppression of a $|T\rangle$-state distillation protocol. We also obtain new non-negativity constraints on such weight enumerators by demanding consistency of the associated magic state distillation routine. These constraints yield new bounds on the distances of classical Hermitian self-dual and maximal self-orthogonal linear $GF(4)$ codes, notably proving the nonexistence of such codes with parameters $[12m, 6m, 4m+2]_{GF(4)}$.