A quantum stabilizer code over GF$(q)$ corresponds to a classical additive code over GF$(q^2)$ that is self-orthogonal with respect to a symplectic inner product. We study the decoding of quantum low-density parity-check (LDPC) codes over binary finite fields GF$(q=2^l)$ by the sum-product algorithm, also known as belief propagation (BP). Conventionally, a message in a nonbinary BP for quantum codes over GF$(2^l)$ represents a probability vector over GF$(2^{2l})$, inducing high decoding complexity. In this paper, we explore the property of the symplectic inner product and show that scalar messages suffice for BP decoding of nonbinary quantum codes, rather than vector messages necessary for the conventional BP. Consequently, we propose a BP decoding algorithm for quantum codes over GF$(2^l)$ by passing scalar messages so that it has low computation complexity. The algorithm is specified in log domain by using log-likelihood ratios (LLRs) of the channel statistics to have a low implementation cost. Moreover, techniques such as message normalization or offset can be naturally applied in this algorithm to mitigate the effects of short cycles to improve BP performance. This is important for nonbinary quantum codes since they may have more short cycles compared to binary quantum codes. Several computer simulations are provided to demonstrate these advantages. The scalar-based strategy can also be used to improve the BP decoding of classical linear codes over GF$(2^l)$ with many short cycles.
翻译:GF$(q) 上的量稳定器代码(q) $ 的量级稳定器代码与GF$(q) $(q) 的经典添加码(g) =2) 对应, 它代表了对内成份的自动反向调值。 我们研究了对二进制的量低密度对等检查(LDPC)代码对二进制有限域的解码(GF$(q=2) q) 美元(q) 的量级稳定器代码(GF$ (2) 美元) 的值非二进制双倍的量级BP 代码(q=2) 。 公约中, 在非二进制的量制代码中, 表示量制量制代码( 2 ⁇ ) 的量级代码(b) 的值对量级周期(gFFGF$ (2) 的值值值值值值值值值值值比值值高, 导致高解码的复杂性值。 在本文件中, 算算算算算出的内产值产品中的数值比值值值的值值值值值值值值值值值值值值值比值, 的值比值比值比值值的值的值值值的值比值比值值值的值值值值值值值值的值值值值值值值值值值值值值。