Progressive-Proximity Bit-Flipping for Decoding Surface Codes

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 a particular class of harmful degenerate errors. Our algorithm achieves linear complexity growth and it can be efficiently implemented as it only involves simple operations such as bit-wise additions, quasi-cyclic permutations and vector-matrix multiplications. 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.