论文标题
Quantum XYZ产品代码
Quantum XYZ Product Codes
论文作者
论文摘要
我们研究了HyperGraph产品代码构建的三个变体，与三种经典代码的标准同源产品不同。当使用3个经典的LDPC代码实例化时，此“ XYZ产品”会产生非CSS量子LDPC代码，该代码可能显示出很大的最小距离。该结构的最简单实例与3个重复代码的乘积相对应，是3维复曲面代码的非CSS变体，称为Chamon代码。一般建筑是在丹妮丝·莫里斯（Denise Maurice）的博士学位论文中引入的，但到目前为止仍然了解不足。原因是，虽然可以使用组合工具分析HyperGraph产品代码，但XYZ产品代码也至关重要地取决于三种经典代码的奇偶校验 - 检查矩阵的代数特性，从而使他们的分析更加涉及。 我们研究XYZ产品代码的主要动机是逻辑运营商的自然代表是二维对象。这与3个维度的标准超图产品代码形成鲜明对比，这些代码总是接收一维逻辑运算符。特别是，具有恒定速率的XYZ产品代码的特定实例可能显示最小距离至$θ（n^{2/3}）$。尽管我们在这里没有证明这一结果，但我们获得了一大批XYZ产品代码的尺寸，并且在限制在尺寸1的代码上时，我们减少了将最小距离计算到涉及涉及二进制3次数的更基本组合问题的最小距离。我们还详细讨论了一些XYZ产品代码的家族，这些系列可以嵌入与本地相互作用的三个维度。其中一些代码似乎与Haah的立方法共享属性，并且可能是与对数能量屏障自我校正量子记忆的有趣候选者。
We study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this "XYZ product" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Denise Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes also depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved. Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes with constant rate might display a minimum distance as large as $Θ(N^{2/3})$. While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes that can be embedded in three dimensions with local interaction. Some of these codes seem to share properties with Haah's cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier.
