二维平移不变码的物质化对称性
Materialised symmetries of 2D translationally invariant codes
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中文总结 AI 辅助
本研究针对二维平移不变码，提出将无限晶格上的对称空间分解为平移不变子空间，并给出平面波形式的显式基，进而通过限制得到矩形周期晶格基，为匹配解码器提供理论基础，并以gross码为例展示应用。
中文摘要 AI 辅助
近期，对近期的qLDPC码作为表面码和颜色码的高性能替代方案产生了显著兴趣。其中一类码是二维平移不变（TI）码，如双变量自行车码，它们与拓扑码具有相似的性质。研究此类码的基本对象是物质化对称性，可用于构建基于匹配的解码器。这些解码器推广了环面码的最小权重完美匹配解码器，并提供了类似的渐近性能保证，而启发式解码器如BP及其变体则缺乏这些保证。尽管如此，对称性的数学结构及其在平移和限制到有限尺寸晶格下的性质尚未得到充分研究。我们描述了无限晶格上二维CSS TI码的对称空间分解为平移不变子空间，这使我们能够以平面波形式写出对称性的显式基。然后，我们描述了如何调整这个无限晶格基，使得任何相应的矩形周期晶格基可以通过限制到与周期边界兼容的基元素来获得。这使我们能够描述对称空间如何随不同矩形尺寸变化。我们通过示例说明，在实践中，通常可以直接确定扭曲边界上的对称性基，而无需重新计算分解。我们评论了该框架在基于匹配的解码器中的应用，并提供了此类码的对称性示例，如gross码。
英文摘要
There has been significant recent interest in near-term qLDPC codes as high-performance alternatives to surface and color codes. One such class of codes is 2-dimensional translationally invariant (TI) codes, such as bivariate bicycle codes, which share similar properties to topological codes. Fundamental objects in the study of such codes are the materialised symmetries, which can be used for the construction of matching-based decoders. These decoders generalise the minimum-weight perfect matching decoder for toric codes and provide similar asymptotic performance guarantees that heuristic decoders such as BP and its variants lack. Despite this, the mathematical structure of symmetries along with their properties under translation and restriction to finite-sized lattices has not been well-studied. We describe a decomposition of symmetry spaces of 2D CSS TI codes on infinite lattices into translation-invariant subspaces which allow us to write an explicit basis of symmetries in a plane-wave-like form. We then describe how to adapt this infinite lattice basis so that any corresponding rectangular periodic lattice basis can be obtained by restriction to basis elements compatible with the periodic boundaries. This allows us to describe how the symmetry space varies with different rectangular dimensions. We illustrate via examples that it is often straightforward in practice to determine a basis of symmetries on twisted boundaries as well, without needing to recompute the decomposition. We comment on the applications of this framework to matching-based decoders and provide examples of symmetries of such codes as the gross code.
发表机构
- School of Physics, University of Sydney（悉尼大学物理学院）
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