We give a fault tolerant construction for error correction and computation using two punctured quantum Reed-Muller (PQRM) codes. In particular, we consider the $[[127,1,15]]$ self-dual doubly-even code that has transversal Clifford gates (CNOT, H, S) and the triply-even $[[127,1,7]]$ code that has transversal T and CNOT gates. We show that code switching between these codes can be accomplished using Steane error correction. For fault-tolerant ancilla preparation we utilize the low-depth hypercube encoding circuit along with different code automorphism permutations in different ancilla blocks, while decoding is handled by the high-performance classical successive cancellation list decoder. In this way, every logical operation in this universal gate set is amenable to extended rectangle analysis. The CNOT exRec has a failure rate approaching $10^{-9}$ at $10^{-3}$ circuit-level depolarizing noise. Furthermore, we map the PQRM codes to a 2D layout suitable for implementation in arrays of trapped atoms and try to reduce the circuit depth of parallel atom movements in state preparation. The resulting protocol is strictly fault-tolerant for the $[[127,1,7]]$ code and practically fault-tolerant for the $[[127,1,15]]$ code. Moreover, each patch requires a permutation consisting of $7$ sub-hypercube swaps only. These are swaps of rectangular grids in our 2D hypercube layout and can be naturally created with acousto-optic deflectors (AODs). Lastly, we show for the family of $[[2^{2r},{2r\choose r},2^r]]$ QRM codes that the entire logical Clifford group can be achieved using only permutations, transversal gates, and fold-transversal gates.

翻译：我们提出了一种利用两个截断量子Reed-Muller码进行容错纠错与计算的构造。具体而言，我们研究了具有横向Clifford门（CNOT、H、S）的$[[127,1,15]]$自对偶双偶码，以及具有横向T门和CNOT门的三偶$[[127,1,7]]$码。我们证明了通过Steane纠错方案可以实现这些码之间的码切换。对于容错辅助态制备，我们采用低深度超立方编码电路，并在不同辅助块中应用不同的码自同构置换，而解码则由高性能经典连续消除列表译码器处理。通过这种方式，该通用门集中的每个逻辑操作都适用于扩展矩形分析。在电路级去极化噪声为$10^{-3}$时，CNOT扩展矩形的失效率可接近$10^{-9}$。此外，我们将截断量子Reed-Muller码映射到适合在囚禁原子阵列中实现的二维布局，并尝试降低并行原子移动在态制备过程中的电路深度。所得协议对$[[127,1,7]]$码具有严格容错性，对$[[127,1,15]]$码具有实际容错性。此外，每个补丁仅需包含$7$次子超立方交换的置换操作。这些交换对应于我们二维超立方布局中的矩形网格交换，可自然地通过声光偏转器实现。最后，我们针对$[[2^{2r},{2r\choose r},2^r]]$量子Reed-Muller码族证明，仅通过置换操作、横向门及折叠横向门即可实现完整逻辑Clifford群。