Progressive-Proximity Bit-Flipping for Decoding Surface Codes

TL;DR

The paper tackles the challenge of efficiently decoding surface and toric codes with hardware-friendly decoders. It introduces Progressive-Proximity Bit-Flipping (PPBF), which combines a proximity-vector heuristic with an iterative matching stage to handle error degeneracy, all implemented with static memory and simple integer operations. PPBF achieves an $O(n^2)$ decoding complexity and demonstrates decoding thresholds around $7.5\%$ for toric codes and $7\%$ for rotated planar codes on the binary symmetric channel, offering near-MWPM/Uf performance at much lower hardware cost. The approach relies on offline precomputation of proximity influences and a shift-based mapping to apply these influences efficiently, enabling potential FPGA/cryogenic deployment. The work also outlines avenues for enhancement via decoding diversity and integration of soft information, with implications for scalable, hardware-viable quantum error correction.

Abstract

Topological quantum codes, such as toric and surface codes, are excellent candidates for hardware implementation due to their robustness against errors and their local interactions between qubits. However, decoding these codes efficiently remains a challenge: existing decoders often fall short of meeting requirements such as having low computational complexity (ideally linear in the code's blocklength), low decoding latency, and low power consumption. In this paper we propose a novel bit-flipping (BF) decoder tailored for toric and surface codes. We introduce the proximity vector as a heuristic metric for flipping bits, and we develop a new subroutine for correcting degenerate multiple errors on adjacent qubits. Our algorithm has quadratic complexity growth and it can be efficiently implemented as it does not require operations on dynamic memories, as do state-of-art decoding algorithms such as minimum weight perfect matching or union find. The proposed decoder shows a decoding threshold of 7.5% for the 2D toric code and 7% for the rotated planar code over the binary symmetric channel.

Figures (8)

• Figure 1: Tanner graph for the $\llbracket 9,1,3 \rrbracket$ rotated planar code. Circles (with black labels) correspond to variable nodes, while squares (with red labels) correspond to check nodes.
• Figure 2: Example of shifting of the proximity influence. The labeling on the variable (black) and check (red) nodes inside the parentheses is the labeling for the $L=3$ rotated code, while the labeling outside the parentheses is the one for the $4\times 5$ surface code. In this example, we compute the influence of $c_{15(2)}$ by shifting the influence of $c_{8(0)}$. The nodes colored in gray are the ones with a non-zero proximity metric; notice how some of the values are discarded after shifting, i.e. the values of $c_9$ and $c_{16}$.
• Figure 3: Example of the usage of the auxiliary proximity influence of checks in Algorithm \ref{['alg:iter-matching']}. In the example, $c_3$ and $c_7$ are unsatisfied checks, and the decoder has to find the shortest path between them, assuming it starts with $c_7$ as pivot.
• Figure 4: Addition of the extra check as described in Section \ref{['sec:matching']}. The blue variable nodes are the depth-1 influence of the extra check, and the darker ones are those which will be flipped. In the example, $\sigma_x=\sigma_y=1$.
• Figure 5: Performance of our decoder on planar rotated codes of different sizes, assuming a BSC. The threshold, which is around $7\%$, is highlighted.

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Definition 3