Hypergraph product codes are a promising avenue to achieving fault-tolerant quantum computation with constant overhead. When embedding these and other constant-rate qLDPC codes into 2D, a significant number of nonlocal connections are required, posing difficulties for some quantum computing architectures. In this work, we introduce a fault-tolerance scheme that aims to alleviate the effects of implementing this nonlocality by measuring generators acting on spatially distant qubits less frequently than those which do not. We investigate the performance of a simplified version of this scheme, where the measured generators are randomly selected. When applied to hypergraph product codes and a modified small-set-flip decoding algorithm, we prove that for a sufficiently high percentage of generators being measured, a threshold still exists. We also find numerical evidence that the logical error rate is exponentially suppressed even when a large constant fraction of generators are not measured.

中文翻译：

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超图产品代码的部分综合症测量

超图产品代码是实现具有恒定开销的容错量子计算的有前途的途径。当将这些和其他恒定速率 qLDPC 码嵌入到 2D 中时，需要大量的非局部连接，这给某些量子计算架构带来了困难。在这项工作中，我们引入了一种容错方案，旨在通过测量作用于空间遥远量子位的生成器的频率低于不作用于空间遥远量子位的生成器来减轻实现这种非定域性的影响。我们研究了该方案简化版本的性能，其中测量的生成器是随机选择的。当应用于超图乘积代码和改进的小集翻转解码算法时，我们证明，对于足够高百分比的被测量生成器，阈值仍然存在。我们还发现了数值证据，表明即使未测量大量恒定比例的生成器，逻辑错误率也会呈指数级抑制。