来自广义自行车码代数扩张的小型量子码
Small Quantum Codes from Algebraic Extensions of Generalized Bicycle Codes
AI总结：
本文针对小规模量子处理单元，提出基于广义自行车码奇偶校验矩阵代数操作的小型量子LDPC码构造方法，数值表明其性能与同规模表面码相当，并以非局域连通性为代价实现多逻辑量子比特编码。
摘要：
量子纠错正在迅速迎来首批实验实现，但渐近最优的纠错码与实验上可行的码之间仍存在显著差距。量子低密度奇偶校验（LDPC）码的范围从编码率趋于零的表面码，一直到具有恒定编码率和线性距离的非常有前景的码。在本工作中，受当前小规模实验量子处理单元的驱动，我们设计了受量子LDPC码的一个子集——即广义自行车（GB）码——启发的小型量子码。我们引入了一种基于对GB码奇偶校验矩阵进行代数操作而非对Tanner图进行操作的码构造方法。我们的构造产生了小规模量子LDPC码族，并且我们通过数值演示表明，在现象学噪声模型下，它们在相似规模下的性能与表面码的性能相当。我们码族的优势在于它们在一个码中编码许多逻辑量子比特，代价是非局域连通性。随后，我们探索了该码构造的三种变体，重点是通过使其更接近当前短程连通性设备的实验能力来减少长程连通性。
英文摘要：
Quantum error correction is rapidly seeing first experimental implementations, but there is a significant gap between asymptotically optimal error-correcting codes and codes that are experimentally feasible. Quantum LDPC codes range from the surface code, which has a vanishing encoding rate, to very promising codes with constant encoding rate and linear distance. In this work, motivated by current small-scale experimental quantum processing units, we devise small quantum codes that are inspired by a subset of quantum LDPC codes, known as generalized bicycle (GB) codes. We introduce a code construction based on algebraic manipulation of the parity-check matrix of GB codes, rather than manipulation of Tanner graphs. Our construction leads to families of quantum LDPC codes of small size, and we demonstrate numerically that their performance scales comparably to the performance of surface codes for similar sizes under a phenomenological noise model. The advantage of our code family is that they encode many logical qubits in one code, at the expense of non-local connectivity. We then explore three variants of the code construction focusing on reducing the long-range connectivity by bringing it closer to the current experimental capabilities of short-range connectivity devices.
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