We investigate the quantum phase diagram of the XY toric code (XYTC) on the kagome lattice consisting of
XY hexagonal and triangular plaquette operators and conserved star operators. We demonstrate analytically by exploiting an exact local
Z2-symmetry on dodecagons that the kagome XYTC realizes a
Z2×Z2×Z2 topologically ordered phase in the limit of large hexagonal \mbox{plaquette} operators. This unconventional topological phase involves 64 quasi-particles - Abelian anyons - with restricted mobility on three colored sublattices establishing a quantum dimension of eight. The large number of topological superselection sectors originates from six independent Wilson loop operators, acting as emergent one-form symmetries. The resulting anyon structure is richer than that of established topological codes like the toric and the color code. A corresponding CSS topological stabilizer code is formulated, opening novel possibilities for the encoding and manipulation of topological quantum information.
The two-dimensional color code is an alternative to the toric code that encodes more logical qubits while maintaining crucial features of the
Z2×Z2 toric code in the long wavelength limit. However its short range physics include single qubit Pauli operations that violate either three or six stabilisers as opposed to the toric code where single qubit Pauli operations violate two or four stabilisers. Exploiting this fact we construct several non-CSS versions of the two-dimensional color code falling into two families - those where either three, four or five stabilisers are violated and those which violate exactly four stabilisers for all the three types of single qubit Pauli operations. These models are not equivalent to the original color code by a local unitary transformation. Nevertheless the code spaces of the CSS and non-CSS versions are related by local unitaries which reflects the fact that their long range physics coincide. This also implies that the non-CSS versions host transversal Clifford gates and hence support fault-tolerant computations. As a consequence of the non-CSS structure, the logical operators are of a mixed type which in some cases include all the three Pauli operators making them potentially useful for protection against biased Pauli noise.
Topological Calderbank-Shor-Steane (CSS) codes in more than two spatial dimensions offer features useful for fault-tolerant quantum computation, such as transversal non-Clifford gates and single-shot decoding, but a systematic understanding of them is still lacking. One approach is to decouple a code: map it to a stack of higher-form toric codes by a finite-depth local CSS-preserving Clifford circuit, possibly after adding ancillas. For translation-invariant
p-ary CSS codes, where
p is prime (
p=2 for qubits), we require the circuit to commute with the translations of a superlattice. In two dimensions, such a decoupling exists once these translations act trivially on every anyon type. In higher dimensions, excitations are sorted by their dimension into superselection layers, which can remain glued together even when translations act trivially on each layer. We show that this gluing is the only remaining obstruction: a code can be decoupled on a superlattice exactly when, after coarse-graining to it, translations act trivially on every layer and the gluing vanishes. Moreover, for a code with finitely many excitation types in each layer and
m nonzero layers, once translations act trivially on every layer, further coarse-graining by a factor of
pm−1 in every direction removes the aforementioned gluing. Physically, the same criterion determines when the code encodes the maximal number of logical qudits on a finite torus. Finally, we give an explicit algorithm that constructs the decoupling as a circuit of SUM (qudit CNOT) and SWAP gates.
Can excitation structure classify topological quantum codes up to finite-depth local circuits? For translation-invariant topological Calderbank--Shor--Steane (CSS) codes in arbitrary dimensions, we introduce excitation superselection profiles
Eσ (
σ=X,Z). Their cohomology layers
Eσℓ:=Hℓ(Eσ) encode excitation types, fusion rules, and translation action, while
Eσ retains cross-degree gluing. Pointlike, looplike, and higher-dimensional excitations correspond to degrees
ℓ=1,2,…, respectively. Our matrix-level Schanuel-lemma method proves single-degree superselection rigidity: if
Eσ is supported in a single degree
ℓ, then
Eσℓ and
ℓ fix the stable translation-invariant Clifford class under a finite-resolution hypothesis automatic for prime qudits. If
Eσℓ is also finite, prime-qudit codes decouple into toric-code stacks by Clifford circuits after coarse graining, with the exact minimal threshold set by the translation kernel. We develop a unified topological theory of Koszul-complex models (encompassing toric codes, bivariate-bicycle codes, and Haah's cubic code): for regular models, we compute
Eσℓ, prove
Eσ is single-degree, express finite-torus logical modules as
Tor groups, and obtain coarse-grained toric-stack normal forms at full length. Two constructions extend this framework to realize any admissible single-degree profile and build multi-degree extensions. The latter yields eight 3D qubit toric-code extensions with identical layers in both Pauli sectors and trivial translation action, yet distinct size-dependent ground-state degeneracies. The seven nonsplit models require blocking to decouple. Thus, cross-degree gluing carries translation-symmetry-enriched topological information invisible to individual layers.
The manipulation of topologically ordered phases of matter to encode and process quantum information forms the cornerstone of many approaches to fault-tolerant quantum computing. Here we demonstrate that fault-tolerant logical operations in these approaches can be interpreted as instances of anyon condensation. We present a constructive theory for anyon condensation and, in tandem, illustrate our theory explicitly using the color-code model. We show that different condensation processes are associated with a general class of domain walls, which can exist in both spacelike and timelike directions. This class includes semitransparent domain walls that condense certain subsets of anyons. We use our theory to classify topological objects and design novel fault-tolerant logic gates for the color code. As a final example, we also argue that dynamical “Floquet codes” can be viewed as a series of condensation operations. We propose a general construction for realizing planar dynamically driven codes based on condensation operations on the color code. We use our construction to introduce a new Calderbank-Shor-Steane–type Floquet code that we call the Floquet color code. Published by the American Physical Society 2024
In this paper, we introduce an algorithm for extracting topological data from translation invariant generalized Pauli stabilizer codes in two-dimensional systems, focusing on the analysis of anyon excitations and string operators. The algorithm applies to Z d qudits, including instances where d is a nonprime number. This capability allows the identification of topological orders that differ from the Z d toric codes. It extends our understanding beyond the established theorem that Pauli stabilizer codes for Z p qudits (with p being a prime) are equivalent to finite copies of Z p toric codes and trivial stabilizers. The algorithm is designed to determine all anyons and their string operators, enabling the computation of their fusion rules, topological spins, and braiding statistics. The method converts the identification of topological orders into computational tasks, including Gaussian elimination, the Hermite normal form, and the Smith normal form of truncated Laurent polynomials. Furthermore, the algorithm provides a systematic approach for studying quantum error-correcting codes. We apply it to various codes, such as self-dual CSS quantum codes modified from the two-dimensional honeycomb color code and non-CSS quantum codes that contain the double semion topological order or the six-semion topological order. Published by the American Physical Society 2024