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.

翻译：拓扑量子码（如环面码和表面码）因其鲁棒性及量子比特间的局部相互作用，成为硬件实现的理想候选方案。然而，高效解码这些码仍面临挑战：现有解码器往往难以满足低计算复杂度（理想情况下与码块长度呈线性关系）、低解码延迟和低功耗等要求。本文提出一种专为环面码和表面码设计的新型比特翻转（BF）解码器。我们引入邻近向量作为翻转比特的启发式度量，并开发出一个新子程序用于纠正特定类型的有害退化错误。该算法具有线性复杂度增长特性，且仅涉及按位加法、准循环置换和向量矩阵乘法等简单运算，因此可高效实现。所提解码器在二进制对称信道上对二维环面码和旋转平面码分别展现出7.5%和7%的解码阈值。