Fault-tolerant quantum computing with color codes

TL;DR

This paper develops fault-tolerant quantum computing protocols using color codes, including syndrome extraction and decoding algorithms, and evaluates their performance and thresholds through simulations and theoretical analysis.

Contribution

It introduces new fault-tolerant schemes for color codes, including decoding algorithms and architectures, with detailed threshold estimations and phase transition analysis.

Findings

Color codes outperform Kitaev's surface codes when circuit details are ignored.

Estimated fault-tolerance threshold of 0.082% for color codes under realistic noise models.

Decoder performance maps onto phase transitions in related statistical-mechanical models.

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.