A stabilizer code over GF($q=2^l$) corresponds to a classical additive code over GF($q^2 = 2^{2l}$) that is self-orthogonal with respect to a binary symplectic inner product. We study the decoding of quantum low-density parity-check (LDPC) codes over finite fields GF($q=2^l$) for $l\ge 1$ by the sum-product algorithm (SPA), also known as belief propagation (BP). Previously, all the BP decoding of quantum codes are studied in linear domain. In this paper, we propose a BP decoding algorithm for quantum codes over GF($2^l$) in log domain by passing scalar messages derived from log-likelihood ratios (LLRs) of the channel statistics. Techniques such as message normalization or offset for improving BP performance can be naturally applied in this algorithm. Moreover, the implementation cost of this LLR-BP is relatively small compared to the linear-domain BP, which is of the same case as in the classical coding research. Several computer simulations are provided to demonstrate these advantages.
翻译：GF( q= 2 ⁇ l$) 上的一个稳定器代码( $q= 2 ⁇ l$), 相当于GF( q ⁇ 2 = 2 ⁇ 2 ⁇ l$) 上的一个古典添加码( q ⁇ 2= 2 ⁇ l$ $), 相对于二进制共振内产物而言, 这是一种自正反调的二进制内产物。 我们研究对限定域的量子低密度对等检查( LDPC) 代码进行解码( $q= 2 ⁇ l$ ), 由总产品算法( SPASP) ( 也称为信仰传播 ( BBP) 计算, 以1美元计价。 此前, 所有量子编码的BP 脱码都是在线性域中研究的。 在本文中, 我们提议对日志域内量子对GF(2 ⁇ ( 2 ⁇ l$) 的量子代码进行BP 解码算法进行BP,, 其法解码算法算算算算算法算法法算法算法法法法法方法比线性BP 也与古型研究得相同。
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