Fault-tolerant quantum computing with color codes

TL;DR

This work demonstrates that color codes, particularly the 4.8.8 triangular family, enable fault-tolerant quantum computation with local 2D architectures. It introduces an integer-program-based MLE decoder, analyzes syndrome extraction schedules, and provides thresholds across code-capacity, phenomenological, and circuit-level noise models, revealing a hierarchy in which circuit-level thresholds are notably lower than surface-code counterparts. The study also connects decoding performance to classical spin-model phase transitions, proving a self-avoiding-walk bound and exposing a Nishimori-conjecture violation in the associated 3-body RBIM. It further develops two architecture paradigms—pancake and defect-based—for universal encoded computation, including transversal gates, code deformation, and magic-state management. Together, these results establish both quantitative thresholds and a practical, architecture-aware blueprint for implementing color-code–based FTQC in 2D layouts.

Abstract

We present and analyze protocols for fault-tolerant quantum computing using color codes. We present circuit-level schemes for extracting the error syndrome of these codes fault-tolerantly. We further present an integer-program-based decoding algorithm for identifying the most likely error given the syndrome. We simulated our syndrome extraction and decoding algorithms against three physically-motivated noise models using Monte Carlo methods, and used the simulations to estimate the corresponding accuracy thresholds for fault-tolerant quantum error correction. We also used a self-avoiding walk analysis to lower-bound the accuracy threshold for two of these noise models. We present and analyze two architectures for fault-tolerantly computing with these codes: one with 2D arrays of qubits are stacked atop each other and one in a single 2D substrate. Our analysis demonstrates that color codes perform slightly better than Kitaev's surface codes when circuit details are ignored. When these details are considered, we estimate that color codes achieve a threshold of 0.082(3)%, which is higher than the threshold of $1.3 \times 10^{-5}$ achieved by concatenated coding schemes restricted to nearest-neighbor gates in two dimensions but lower than the threshold of 0.75% to 1.1% reported for the Kitaev codes subject to the same restrictions. Finally, because the behavior of our decoder's performance for two of the noise models we consider maps onto an order-disorder phase transition in the three-body random-bond Ising model in 2D and the corresponding random-plaquette gauge model in 3D, our results also answer the Nishimori conjecture for these models in the negative: the statistical-mechanical classical spin systems associated to the 4.8.8 color codes are counterintuitively more ordered at positive temperature than at zero temperature.

Figures (32)

• Figure 1: The three possible face-three-colorable trivalent uniform tilings of the plane.
• Figure 2: Three distance $d=11$ triangular codes encoding one qubit, drawn from the $4.8.8$, $6.6.6$, and $4.6.12$ lattices respectively. For general $d$, these codes have length $n$ equal to $\frac{1}{2}d^2 + d - \frac{1}{2}$, $\frac{3}{4}d^2 + \frac{1}{4}$ and $\frac{3}{2}d^2 - 3d + \frac{5}{2}$ respectively. The asymptotic ratio of $d^2$ to $n$ is highest for the 4.8.8 codes.
• Figure 3: 4.8.8 color codes of sizes 3, 5, and 7.
• Figure 4: A green-colored chain in a triangular code. The chain connects a green-colored side of the 4.8.8 triangular code to a green octagonal face. If qubits are flipped (are in error) along this chain, it will only be detected by this terminal octagonal check operator.
• Figure 5: Six-step circuits for measuring $X^{\otimes 4}$ and $Z^{\otimes 4}$.