We use the recently introduced lifted product to construct a family of quantum low density parity check codes (QLDPC codes). The codes we obtain can be viewed as stacks of surface codes that are interconnected, leading to the name lift-connected surface (LCS) codes. LCS codes offer a wide range of parameters—a particularly striking feature is that they show interesting properties that are favorable compared to the standard surface code. For example, already at moderate numbers of physical qubits in the order of tens, LCS codes of equal size have lower logical error rate or similarly, require fewer qubits for a fixed target logical error rate. We present and analyze the construction and provide numerical simulation results for the logical error rate under code capacity and phenomenological noise. These results show that LCS codes attain thresholds that are comparable to corresponding (non-connected) copies of surface codes, while the logical error rate can be orders of magnitude lower, even for representatives with the same parameters. This provides a code family showing the potential of modern product constructions at already small qubit numbers. Their amenability to 3D-local connectivity renders them particularly relevant for near-term implementations.

中文翻译：

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电梯连接表面代码


我们使用最近推出的提升产品来构建一系列量子低密度奇偶校验码（QLDPC 码）。我们获得的代码可以看作是相互连接的表面代码的堆栈，因此被称为提升连接表面（LCS）代码。 LCS 代码提供了广泛的参数，一个特别引人注目的特点是它们显示出与标准表面代码相比有利的有趣属性。例如，在物理量子位数量适中（大约为数十）时，相同大小的 LCS 代码具有较低的逻辑错误率，或者类似地，需要更少的量子位来实现固定的目标逻辑错误率。我们展示并分析了该结构，并提供了代码容量和唯象噪声下逻辑错误率的数值模拟结果。这些结果表明，LCS 代码达到的阈值与表面代码的相应（非连接）副本相当，而逻辑错误率可以低几个数量级，即使对于具有相同参数的代表也是如此。这提供了一个代码系列，显示了现代产品结构在已经很小的量子比特数上的潜力。它们对 3D 本地连接的适应性使其特别适合近期实施。