Decoding 3D color codes with boundaries

TL;DR

This work extends restriction-based decoding to 3D color codes with boundaries, introducing a concatenated MWPM decoder that operates on restricted and monochrome graphs to correct both Z- and X-errors. The authors construct tetrahedral and cubic 3D color codes with boundaries, and demonstrate code-capacity thresholds around 1.48–1.55%, nearly doubling prior results and approaching an estimated ideal threshold. Numerical analysis shows favorable sub-threshold scaling and potential for single-path decoding to reduce runtime, while a Python visualization tool qCodePlot3D supports framework development. Collectively, these advancements move 3D color codes closer to practical fault-tolerant quantum computation with transversal non-Clifford gates. The work also provides a useful visualization platform to facilitate ongoing decoder development and analysis.

Abstract

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.

Figures (6)

• Figure 1: Primal and dual lattices for tetrahedral and cubic color code. Primal (left) and dual (right) lattice for (a) the distance-3 tetrahedral color code and (b) the distance-4 cubic color code. Vertices of the primal lattice are colored in purple. We sketch the dual lattice on top of the primal one with primal edges depicted in black and primal cells shown with transparent colors for better visualization. Dual vertices and edges have bright colors, the mapping of dual edge colors is: rb is pink, rg is brown, ry is golden, bg is olive green, by is grey and gy is purple. Primal faces and dual cells are not colored for better readability.
• Figure 2: Example decoding path of the 3D concatenated MWPM decoder on the distance-5 tetrahedral color code. (a) Dual-graph picture of the distance-5 tetrahedral color code with boundaries, where physical qubits of the primal lattice are shown in purple. Errors are placed at physical qubits shown in pink. For this exemplary error configuration, two neighboring red and one green stabilizers are flipped, as indicated by the bright large nodes on the dual lattice. (b) The restricted graph $\mathcal{R}_{\texttt{rg}}$ is constructed by removing all blue and yellow vertices of the dual lattice $\mathcal{L}^*$ and all associated edges. MWPM is run on the restricted lattice and we obtain a set of matched edges (thick orange lines). (c) We obtain the first monochrome graph $\mathcal{M}_{\texttt{rg}}^\texttt{b}$ by placing blue nodes at all red-green faces as well as all blue cells of the primal graph. Every node that corresponds to an edge of the matching on the restricted graph is marked as flipped (e.g. the two highlighted light-blue nodes), as well as all initially violated blue nodes. We again run MWPM on this instance yielding a set of matched edges. (d) The second monochrome graph $\mathcal{M}_{\texttt{rg,b}}^\texttt{y}$ is constructed by placing yellow nodes at all yellow edges and yellow cells of the primal graph. We mark every node that corresponds to a matched edge of the previous step, as well as all initially flipped yellow cells of the initial graph, and again run MWPM. This final matching is the suggested correction of the decoding path.
• Figure 3: Logical error rates for decoding with the concatenated MWPM decoder. (a) Decoding of 3D tetrahedral color codes of distance $d = 3, 5, 7, 9$ and (b) $d = 4, 6, 8$ cubic color codes. Data points at physical errors rates $p_{\mathrm{phys}} > 10^{-3}$ are determined by means of direct Monte-Carlo sampling. At low physical error rates, we use Subset Sampling Li_2017heussen2024dynamical (solid lines) to calculate an upper (light, solid color) and a lower bound (dark, dashed color) on the logical error rate. The inset shows the logical error rates close to (a) $p_{\mathrm{phys}} = 0.014$ and (b) $p_{\mathrm{phys}} = 0.015$, below which increasing distance suppresses the logical error rate. (b) Logical error rates for one of the three encoded logical qubits. The logical error rates of the other two logical qubits of the respective cubic color code are extracted simultaneously and show similar performance, as can be seen in App. Fig. \ref{['fig:app_result_concatenated_tetrahedron']}.
• Figure 4: Logical error rates of logical qubits 2 and 3 for decoding cubic color codes with the concatenated MWPM decoder. (a) Logical error rates for $d = 4, 6, 8$ cubic color codes considering (a) logical qubit 2 and (b) logical qubit 3. Data points at physical error rates $p_{\mathrm{phys}} > 10^{-3}$ are determined by means of direct Monte-Carlo sampling. At low physical error rates, we use Subset Sampling to calculate an upper (light, solid line) and a lower bound (dark, dashed line) on the logical error rate. The inset shows the logical error rates close to $p_{\mathrm{phys}} =$ 1.5%, below which increasing distance suppresses the logical error rate. All three logical qubits show similar performance.
• Figure 5: Runtime of the concatenated decoder. We determine the simulation runtime of one decoding path on a single Laptop (Apple M1 Pro) of the concatenated MWPM decoder for the cubic (blue) and the tetrahedral color code (orange). This includes the three MWPM subroutines and the time it takes to generate the syndrome graphs for each subroutine.