Decoding 3D color codes with boundaries

Practical large-scale quantum computation requires both efficient error correction and robust implementation of logical operations. Three-dimensional (3D) color codes are a promising candidate for fault-tolerant quantum computation due to their transversal non-Clifford gates, but efficient decoding remains challenging. In this work, we extend previous decoders for two-dimensional color codes [1], which are based on the restriction of the decoding problem to a subset of the qubit lattice, to three dimensions. Including boundaries of 3D color codes, we demonstrate that the 3D restriction decoder achieves optimal scaling of the logical error rate and a threshold value of 1.55(6)% for code-capacity bit- and phase-flip noise, which is almost a factor of two higher than previously reported for this family of codes [2, 3]. We furthermore present qCodePlot3D, a Python package for visualizing 2D and 3D color codes, error configurations, and decoding paths, which supports the development and analysis of such decoders. These advancements contribute to making 3D color codes a more practical option for exploring fault-tolerant quantum computation.

💡 Research Summary

The paper addresses a critical bottleneck in the deployment of three‑dimensional (3D) color codes for fault‑tolerant quantum computation: the lack of an efficient, high‑threshold decoder that works with realistic boundary conditions. While 2D color codes have benefited from restriction‑based decoders—where the syndrome is limited to a subset of stabilizers of two colors, enabling Minimum‑Weight Perfect Matching (MWPM)—extending this approach to 3D has been non‑trivial because of the richer topology and the presence of boundaries that break the regular 4‑valent structure of the lattice.

The authors first review the construction of 3D color codes in both primal and dual representations. In the primal picture qubits reside on vertices, X‑type stabilizers on cells, and Z‑type stabilizers on faces. The dual lattice swaps these roles, placing qubits on cells and stabilizers on vertices and edges. Both representations require that cells be 4‑colorable and that interior vertices be 4‑valent. When boundaries are introduced, corner vertices become 3‑valent, and new objects—borders and border‑orders—appear. The paper details how to modify the primal and dual lattices to accommodate these features while preserving the color‑consistency rules. Two concrete families are examined: the tetrahedral code (four boundaries, encoding a single logical qubit) and the cubic code (six boundaries, encoding three logical qubits). For each family the authors tabulate the number of physical qubits, independent faces, and cells as functions of the code distance d.

The core contribution is the “3D concatenated MWPM decoder.” The decoder proceeds in three hierarchical stages, each employing MWPM on a graph that captures a different subset of the syndrome information:

1. Restricted graph R_cd – All vertices of two chosen colors (e.g., red and green) are retained, while the other two colors are removed. MWPM on this graph yields a set of matched edges that corrects the syndrome restricted to those two colors.
2. First monochrome graph M_brg – Nodes are placed on all blue‑colored faces and cells (the “blue” subsystem). Edges correspond to the matched edges from the previous step together with any initially violated blue stabilizers. A second MWPM run produces a correction for the blue subsystem.
3. Second monochrome graph M_yrg – Analogously, nodes are placed on yellow‑colored faces and cells, and MWPM is applied using the outcome of the previous stage.

The final set of matched edges across the three stages is lifted back to the original qubit lattice, yielding a concrete Pauli correction. By iteratively matching on increasingly restricted color subsets, the decoder guarantees that each single‑qubit error produces an even number of violated stabilizers on the restricted graph, satisfying the perfect‑matching condition. Moreover, the hierarchical lifting procedure resolves the “odd‑parity” problem that plagued earlier higher‑dimensional restriction decoders, restoring the optimal sub‑threshold scaling ∝ p d³.

Numerical simulations under the code‑capacity model (independent bit‑ and phase‑flip errors) demonstrate that the decoder achieves a logical error rate that scales as O(p d³) and exhibits a threshold of 1.55 % ± 0.06 %, nearly double the best previously reported thresholds for 3D color codes with boundaries. This performance is validated on both tetrahedral and cubic families across a range of distances (d = 3–7). The results show that incorporating boundaries does not degrade the decoder’s efficacy; instead, the boundary‑aware construction contributes to the higher threshold.

In addition to the algorithmic advances, the authors release qCodePlot3D, a Python package that visualizes 2D and 3D color codes, error configurations, and the intermediate graphs (restricted and monochrome) used during decoding. The tool supports step‑by‑step inspection of the matching process, making it valuable for researchers developing new decoders, for educators teaching topological codes, and for experimentalists designing lattice geometries on ion‑trap or neutral‑atom platforms.

Overall, the paper delivers a comprehensive solution: a boundary‑compatible, restriction‑based decoder that attains near‑optimal thresholds for 3D color codes, together with open‑source software that lowers the barrier to further exploration. These contributions significantly advance the practicality of 3D color codes as a platform for fault‑tolerant quantum computation, especially given their native support for transversal non‑Clifford gates.