3D Chiral Color Codes

• Chiral color codes are three-dimensional stabilizer codes using colored gauge-fixings to realize fermionic and chiral topological phases.
• They incorporate a chirality parameter that tunes anyon braiding statistics and support efficient single-shot error correction via measurement redundancy.
• Their structure enables seamless code switching with conventional color codes, providing a robust platform for quantum computation and anyon manipulation.

Chiral color codes are a class of three-dimensional stabilizer codes constructed within the gauge color code framework. These codes realize fermionic and chiral topological orders, notably supporting ℤ_d^α anyon theories with anomalous chiral surface topological order. On closed manifolds, the codes exhibit a unique ground state after condensation of bulk transparent excitations, and for qubit systems (d = 2), the model reduces to a single copy of the fermionic toric code. Chiral color codes permit single-shot error correction and code switching to other stabilizer color codes, making them platforms for realizing and manipulating fermions and chiral anyons (Lee et al., 22 Sep 2025).

1. Definition, Construction, and Stabilizer Structure

Chiral color codes are defined as specific gauge-fixings of the 3D gauge color code, itself a subsystem code built on four-colorable 3-manifolds. In conventional bosonic color codes, qubits are placed at vertices of the lattice, and stabilizers are uniformly Pauli X and Z operators acting on 2-cells (faces) and 3-cells (volumes). In contrast, chiral color codes employ "colored" stabilizer assignments that depend fundamentally on the color composition of the faces.

For the qubit case, the stabilizer generators on faces are:

• S(f_AB) = X(f_AB) for faces labeled AB or CD
• S(f_AC) = iY(f_AC) for faces labeled AC or BD
• S(f_AD) = Z(f_AD) for faces labeled AD or BC

Here, X(f), Y(f), Z(f) denote products of the corresponding Pauli operators on the face f. The volume stabilizers (composed of products of the face stabilizers on each 3-cell) are automatically generated.

Generalization to qudit systems (dimension d) requires use of bipartite lattices, assigning local signs λ(v) = ±1 to each vertex, and deploying nonuniform qudit Pauli operators. Crucially, a chirality parameter α ∈ ℤ_d (coprime to d for nondegeneracy) twists the assignments:

• AB/CD faces: X
• AC/BD faces: X^–1 Z^–α
• AD/BC faces: Z^α

The stabilizer group thus becomes: $$S_{\text{chiral-CC}} = \langle X(f_{\text{AB}}), X(f_{\text{CD}}), X^{-1} Z^{-\alpha}(f_{\text{AC}}), X^{-1} Z^{-\alpha}(f_{\text{BD}}), Z^{\alpha}(f_{\text{AD}}), Z^{\alpha}(f_{\text{BC}}) \rangle$$

The model admits reforms to a Majorana fermion picture, giving rise to fermionic excitations and chiral topological order.

2. Topological Phases and Anyon Theory

Chiral color codes realize topological phases described by ℤ_d^α anyon theories. The fundamental quasiparticle "a" satisfies a^d = 1. The modular data is controlled by α:

• Self statistics: θ(a^i) = ω^{α i^2}, with ω = e^{2πi/d}
• Mutual braiding: B(a^i, a^j) = ω^{2α i j}

In the d = 2, α = 1 case, the nontrivial excitation becomes a fermion (θ = –1), i.e., the code realizes the fermionic toric code. For higher d, α tunes the topological order from bosonic to chiral phases distinguished by their braiding and self-statistics. On open manifolds, boundaries host anomalous chiral topological orders (e.g., conjugate ℤ_d^α and ℤ_d^–α theories on opposite boundaries).

3. Ground State Structure and Bulk Entanglement

On closed manifolds (without boundaries), the ground state is unique after condensing transparent bulk excitations. For odd d and coprime α, the bulk is short-range entangled. This follows from an explicit construction of a local quantum channel that prepares the ground state. The procedure is:

1. Begin with the trivial product state |0⟩^{⊗n}.
2. Measure all face stabilizers S(f) using ancillary qudits; record syndrome s⃗.
3. Apply a local feedback correction V(s⃗) determined solely by nearby syndrome data.

$$|0\rangle^{\otimes n} \xrightarrow{\text{measure } S(f)} \sum_{s⃗} |\psi(s⃗)\rangle \otimes |s⃗\rangle \xrightarrow{\; V(s⃗) \;} |\psi\rangle$$

Each step relies only on local operations/metacheck conditions, ensuring short-range entanglement in the bulk even though the boundary supports chiral topological order.

4. Error Correction, Fault Tolerance, and Code Switching

Chiral color codes inherit subsystem redundancy from the gauge color code: the product of gauge face operators on a single volume is the identity, providing syndrome measurement redundancy crucial for single-shot error correction. This redundancy allows correction of measurement errors without repeated rounds—syndromes can be decoded after a single round even with noisy measurements.

Additionally, gauge color code framework supports code switching—transforming between the 3D chiral color code and conventional 2D (non-chiral) stabilizer color codes. This property is essential for implementing universal logical gates and converting between bosonic/fermionic encodings in quantum computation.

5. Applications: Quantum Computing, Fermion/Chiral Anyon Manipulation, MBQC

Chiral color codes serve as fault-tolerant quantum memories and computational substrates due to robust single-shot error correction against both physical and measurement errors. The model supports manipulation and paper of fermions (qubit case) and chiral anyons (qudit case with α ≠ 0). The tunable parameter α allows exploration of exotic braiding and topological phenomena.

In measurement-based quantum computation (MBQC), the anomalous order at boundaries can be used for teleporting logical information between spatially separated domains, taking advantage of conjugated chiral orders (ℤ_d^α vs. ℤ_d^–α) on thickened tori. This aligns with perspectives on universal resource states from SPT phases and Walker–Wang models.

Compared to bosonic color codes, which realize simpler (non-chiral) anyon orders, chiral color codes yield more complex topological orders due to twisted stabilizer assignments dependent on both face color and bipartition. This facilitates studies in interfaces between bosonic and fermionic codes and enhances logical gate set implementations through code switching.

6. Comparison with Other Topological Codes

Conventional stabilizer color codes realize bosonic ℤ₂ or ℤ_d topological orders with transversal Clifford gates and high error thresholds but lack chirality and fermionic excitations. Chiral color codes extend this family by incorporating non-uniform (color-dependent) assignments and the chirality parameter α, thereby supporting fermionic modes and ℤ_d^α topological phases.

The stabilizer group modification (assignment of X, X^–1 Z^–α, and Z^α based on face color and bipartition) is the key technical distinction. Fault-tolerance and decoding remains as robust as the subsystem gauge color code but with added capacity for exotic topological phenomena and code switching.

7. Relevant Mathematical Formulations

Stabilizer assignments for the qudit chiral color code: $$S_{\text{chiral-CC}} = \langle X(f_{\text{AB}}), X(f_{\text{CD}}), X^{-1} Z^{-\alpha}(f_{\text{AC}}), X^{-1} Z^{-\alpha}(f_{\text{BD}}), Z^{\alpha}(f_{\text{AD}}), Z^{\alpha}(f_{\text{BC}}) \rangle$$

Anyon modular data for ℤ_d^α theory: $$\theta(a^i) = \omega^{\alpha i^2}, \quad B(a^i, a^j) = \omega^{2\alpha i j}, \quad \omega = e^{2\pi i / d}$$

Local quantum channel preparation: $$|0\rangle^{\otimes n} \xrightarrow{\text{measure } S(f)} \sum_{s⃗} |\psi(s⃗)\rangle \otimes |s⃗\rangle \xrightarrow{V(s⃗)} |\psi\rangle$$

String operators for excitation creation: $O_\ell = \prod_{v \in \ell} P(f[\ell])_v$ where P(f[\ell])_v depends on color labels along the path ℓ.

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Chiral color codes combine subsystem code fault tolerance with explicit realization of chiral and fermionic topological phases, direct anyon manipulation, tunable boundary phenomena, and efficient error correction, thus offering a foundational framework for exotic and robust quantum error-correcting code architectures (Lee et al., 22 Sep 2025).