Subsystem codes are a generalization of decoherence free subspaces, noiseless subsystems, and quantum error-correcting codes. The known constructions of subsystem codes from classical codes are limited to quantum systems that all have the same dimension, and this dimension must be a power of a prime. It is shown that one can remove these restrictions and obtain subsystem codes in quantum systems of arbitrary finite dimension from classical codes that are subgroups of an abelian group. The constructions are derived from Clifford codes over abstract error groups with abelian index groups.
We study cyclic triorthogonal codes in prime dimension
p, on which a transversal third-level diagonal gate acts as a logical non-Clifford gate. For odd
p, we use a definition of triorthogonality that does not require self-orthogonality. For cyclic codes of length coprime to
p, triorthogonality reduces to a sumset condition on the spectral support, which makes exhaustive search feasible: for two natural constructions, we enumerate all maximal codes of this kind for
p=3 up to length
121 and
p=5,7 up to length
96. For qubits, we prove that a cyclic code of odd length is triply even if and only if it is classically triorthogonal, giving a complete description of triply-even cyclic codes, and we determine all such codes up to length
763. For qudits, we find codes such as
[[19,1,3]]3 and
[[7,1,3]]7, which improve on the shortest Reed-Muller and Reed-Solomon codes in both overhead exponent and distillation threshold. The latter is the first member of an infinite family of
[[p,1,2⌊p/6⌋+1]]p codes.
Recent work has shown that quantum double models can be used to circumvent the Bravyi-Koenig bound. In particular, this is achieved through code-switching protocols in which non-Pauli stabilizer phases appear as intermediate codes. We introduce the
XCd(k) stabilizer formalism, which generalizes the Pauli stabilizer formalism by combining Pauli-
X operators with diagonal operators drawn from a fixed level
k of the Clifford hierarchy on prime-dimensional qudits. This yields a natural class of beyond-Pauli stabilizer models, including many twisted quantum double phases. We classify a large class of these
XCd(k) stabilizer states, showing that they are uniform superpositions over affine-linear subspaces with diagonal phases. In particular, when the stabilizers of a state require operators from level
k of the Clifford hierarchy, we show that the diagonal gates required to produce the phases in the wavefunction sit one level higher. We further show that every 2D twisted quantum double
Dω(Zpn) with
ω a product of type-III cocycles and
p an odd prime admits a realization using only Clifford (
k=2) stabilizers, and we conjecture that phases where
ω contains a type-I or type-II cocycle require level-
p Clifford-hierarchy stabilizers for
p-dimensional qudits. Finally, we demonstrate that a pair of qudit toric codes may be deterministically prepared in a logical magic state by measuring
XCd(k) stabilizers. Together, these results provide a concrete stabilizer framework for topological codes beyond Pauli stabilizer models and their logical gates.
This paper is motivated by the computer-generated nonadditive code described in Rains et al\cite{rains97nonadditive}. We describe a theory of non-stabilizer codes of which the nonadditive code of Rains et al is an example. Furthermore, we give a general strategy of constructing good nonstabilizer codes from good stabilizer codes and give some explicit constructions and asymptotically good nonstabilizer codes. Like in the case of stabilizer codes, we can design fairly efficient encoding and decoding procedures.