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arXiv 2103.06994quant-ph
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利用表面GKP码的低开销容错量子纠错

Low overhead fault-tolerant quantum error correction with the surface-GKP code

Kyungjoo Noh, Christopher Chamberland, Fernando G. S. L. Brandão

备注：
41 pages, 14 figures, 3 tables
发表信息：
PRX Quantum 3, 010315 (2022)
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摘要：

容错量子纠错是实现具有重要实际意义的量子算法所必需的。在这项工作中，我们提出了一个非常有效的使用表面GKP代码，即，表面代码由玻色子GKP量子比特组成，而不是裸露的二维量子比特。在我们的提议中，我们在GKP量子比特之间使用纠错的两个量子比特门，并引入最大似然解码策略来纠正两个GKP量子比特门中的移位错误。我们提出的解码降低了GKP量子比特的总CNOT失败率，例如，在$12$ dB的GKP压缩下从$0.87\%$到$0.36\%$。然后，将GKP码与表面码级联，在假设GKP态的有限压缩是主要噪声源的条件下，得到了GKP压缩的阈值为$9.9$ dB。更重要的是，我们表明，一个低的逻辑故障率$p_{L} < 10^{-7}$可以实现与适度的硬件要求，例如，在$12$ dB的GKP压缩下的$291$模式和$97$量子位，而不是在等效噪声水平下的标准旋转表面码的$1457$裸量子位（即，$p=0.36\%$）。我们的表面GKP码的这种低故障率是可能的，通过使用空时相关的边缘的匹配图的表面码解码器。此外，匹配图中的所有边权重是使用所有校正子测量轮的完整历史基于来自GKP纠错的模拟信息来动态计算的。我们还表明，在实验参数相当保守的情况下，利用耗散稳定化方法，即大-小-大方法，可以在实验上实现GKP压缩$\gtrsim 12$ dB的高压缩GKP态.最后，我们引入了一个三能级辅助方案，以减轻在GKP态制备过程中的辅助衰减误差。

英文摘要：

Fault-tolerant quantum error correction is essential for implementing quantum algorithms of significant practical importance. In this work, we propose a highly effective use of the surface-GKP code, i.e., the surface code consisting of bosonic GKP qubits instead of bare two-dimensional qubits. In our proposal, we use error-corrected two-qubit gates between GKP qubits and introduce a maximum likelihood decoding strategy for correcting shift errors in the two-GKP-qubit gates. Our proposed decoding reduces the total CNOT failure rate of the GKP qubits, e.g., from $0.87\%$ to $0.36\%$ at a GKP squeezing of $12$dB, compared to the case where the simple closest-integer decoding is used. Then, by concatenating the GKP code with the surface code, we find that the threshold GKP squeezing is given by $9.9$dB under the the assumption that finite-squeezing of the GKP states is the dominant noise source. More importantly, we show that a low logical failure rate $p_{L} < 10^{-7}$ can be achieved with moderate hardware requirements, e.g., $291$ modes and $97$ qubits at a GKP squeezing of $12$dB as opposed to $1457$ bare qubits for the standard rotated surface code at an equivalent noise level (i.e., $p=0.36\%$). Such a low failure rate of our surface-GKP code is possible through the use of space-time correlated edges in the matching graphs of the surface code decoder. Further, all edge weights in the matching graphs are computed dynamically based on analog information from the GKP error correction using the full history of all syndrome measurement rounds. We also show that a highly-squeezed GKP state of GKP squeezing $\gtrsim 12$dB can be experimentally realized by using a dissipative stabilization method, namely, the Big-small-Big method, with fairly conservative experimental parameters. Lastly, we introduce a three-level ancilla scheme to mitigate ancilla decay errors during a GKP state preparation.

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