Color-Code Logical Qubit

Figure

Description

Color-code logical qubits are encoded in 2D or 3D topological stabilizer codes defined on trivalent, three-colorable lattices. Relative to surface codes, color codes can offer transversal implementation of a larger Clifford gate set (including the Hadamard and phase gates), reducing some lattice-surgery overheads for Clifford-heavy workloads.

The code is defined on a lattice where each face is assigned one of three colors (red, green, blue) such that no two adjacent faces share the same color. Physical qubits reside on the vertices. Both X-type and Z-type stabilizers are defined on the same faces, a key structural difference from the surface code.

Hamiltonian

For each face $f$ in a colorable lattice:

$S_f^X={\prod }_{i\in f}{X}_i,S_f^Z={\prod }_{i\in f}{Z}_i$

Code space is the +1 eigenspace of all face stabilizers. Logical operators correspond to colored string operators connecting boundaries of matching color.

Motivation

Color codes are a leading alternative to the surface code when gate-transversality and decoding tradeoffs favor reduced compilation overhead. The native transversal Clifford gate set eliminates the need for magic state distillation for H and S gates, which is especially advantageous for Clifford-heavy workloads common in many quantum algorithms.

Experimental Status

First fault-tolerant QEC with color code — Ryan-Anderson et al. (2021):

• Realized real-time fault-tolerant quantum error correction on a Quantinuum trapped-ion processor
• Demonstrated fault-tolerant parity readout and logical qubit persistence through repeated QEC rounds
• Color code on a distance-3 lattice with flag qubits

Transversal Clifford gates — Ryan-Anderson et al. (2024):

• Demonstrated native transversal Hadamard and phase gates on encoded color-code qubits
• Confirmed the architectural advantage of color codes for Clifford-heavy circuits
• arXiv:2404.02280

Key Metrics

Metric	Value	Notes	Fidelity reference
Logical lifetime	~10 QEC rounds (d=3)	Demonstrated logical qubit persistence through repeated error correction	Ryan-Anderson et al. 2021
1Q gate fidelity (transversal H/S)	99%+ (d=3)	Native Clifford advantage — transversal gates without magic state distillation	Ryan-Anderson et al. 2024
2Q gate fidelity (logical CNOT)	~97–99% (d=3, small scale)	Via code deformation; limited by physical gate fidelities	Postler et al. 2022
Threshold	~0.1–1%	Decoder/noise model dependent	—
Transversal Clifford support	Yes	Major architectural advantage	—
Qubit overhead	Comparable order to surface code	Constants depend on layout	—

Note: For QEC code entries, gate fidelities are logical-level operations on encoded information.

References

Original proposal

• H. Bombin and M. A. Martin-Delgado, “Topological Quantum Distillation,” Phys. Rev. Lett. 97, 180501 (2006) — arXiv:quant-ph/0605138

Experimental demonstrations

• C. Ryan-Anderson et al., “Realization of Real-Time Fault-Tolerant Quantum Error Correction,” Phys. Rev. X 11, 041058 (2021) — arXiv:2107.07505
• C. Ryan-Anderson et al., “High-fidelity and fault-tolerant teleportation of a logical qubit using transversal gates and lattice surgery on a trapped-ion quantum computer,” arXiv:2404.02280 (2024)

Linked Papers

• bombin-2006-color-codes

Related Entries

• surface-code-logical-qubit — primary alternative topological QEC code
• transmon — physical qubit platform for superconducting color code implementations
• trapped-ion-qubit — physical qubit platform for Quantinuum demonstrations