具有恒定速率和多项式距离的双曲色码
Hyperbolic color codes with constant rate and polynomial distance
- The University of Tokyo（东京大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文构造了基于算术双曲流形的双曲色码，在任意维度D≥4下同时实现恒定编码速率和多项式码距，为高码率容错量子计算提供新平台。
AI中文摘要：
量子硬件的最新进展放宽了传统上对量子纠错码施加的严格几何局域性约束，激发了人们对高码率量子低密度奇偶校验（qLDPC）码的兴趣。与此同时，色码为容错量子计算提供了特别丰富的环境，支撑了诸如单次纠错和自纠正量子计算等协议。双曲色码提供了一类高码率qLDPC码，同时保留了与容错量子计算相关的色码结构特征。然而，先前构造的双曲色码最多只能达到对数级码距。在本工作中，我们通过构建在支持多项式距离双曲环面码的算术双曲流形之上，构造了同时具有恒定速率和多项式距离的双曲色码。我们的构造适用于任意维度\\(D\geq 4\\)。在偶数维度中，所得的类型-\\(D/2\\)色码具有恒定的编码速率和多项式距离，而在其他情况下，逻辑量子比特数和码距均表现出多项式缩放。我们进一步推导了作为维度和码类型函数的显式多项式下界指数。这些结果建立了一族同时实现恒定速率和多项式距离的双曲色码，并为探索结合高码率量子码与色码结构优势的容错量子计算协议提供了测试平台。
英文摘要：
Recent advances in quantum hardware relax the strict geometric-locality constraints traditionally imposed on quantum error-correcting codes, motivating interest in high-rate quantum low-density parity-check (qLDPC) codes. At the same time, color codes provide a particularly rich setting for fault-tolerant quantum computation, underlying protocols such as single-shot error correction and self-correcting quantum computation. Hyperbolic color codes provide a class of high-rate qLDPC codes that also retain the structural features of color codes relevant to fault-tolerant quantum computation. However, previous constructions of hyperbolic color codes have achieved at most logarithmic code distance. In this work, we construct hyperbolic color codes with both constant rate and polynomial distance by building on arithmetic hyperbolic manifolds that support polynomial-distance hyperbolic toric codes. Our construction applies in arbitrary dimension \(D\geq 4\). In even dimensions, the resulting type-\(D/2\) color codes have constant encoding rate and polynomial distance, while in other cases the number of logical qubits and the code distance both exhibit polynomial scaling. We further derive explicit exponents for polynomial lower bounds as functions of the dimension and code type. These results establish a family of hyperbolic color codes simultaneously achieving constant rate and polynomial distance and providing a testbed to explore fault-tolerant quantum computation protocols that combine high-rate quantum codes with the structural advantages of color codes.
