The surface code is a leading candidate quantum error correcting code, owing to its high threshold, and compatibility with existing experimental architectures. Bravyi {et al.} [7] showed that encoding a state in the surface code using local unitary operations requires time at least linear in the lattice size $L$, however the most efficient known method for encoding an unknown state, introduced by Dennis {et al.} [18], has $O(L^2)$ time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly $2L$ time steps to encode an unknown state in a distance $L$ planar code. We further show how an $O(L)$ complexity local unitary encoder for the toric code can be found by enforcing locality in the $O(\log L)$-depth non-local renormalisation encoder. We relate these techniques by providing an $O(L)$ local unitary circuit to convert between a toric code and a planar code, and also provide optimal encoders for the rectangular, rotated and 3D surface codes. Furthermore, we show how our encoding circuit for the planar code can be used to prepare fermionic states in the compact mapping, a recently introduced fermion to qubit mapping that has a stabiliser structure similar to that of the surface code and is particularly efficient for simulating the Fermi-Hubbard model.

中文翻译：

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表面码的最优局部酉编码电路

表面码是一种领先的候选量子纠错码，因为它的阈值高，并且与现有的实验架构兼容。Bravyi {et al.} [7] 表明，使用局部幺正运算在表面代码中编码状态需要时间至少在晶格大小 $L$ 中是线性的，但是 Dennis 引入的已知最有效的未知状态编码方法{et al.} [18]，具有 $O(L^2)$ 时间复杂度。在这里，我们为平面表面代码提出了一个最优的局部单一编码电路，它使用精确的 $2L$ 时间步长对距离 $L$ 平面代码中的未知状态进行编码。我们进一步展示了如何通过在 $O(\log L)$ 深度非局部重整化编码器中强制局部性来找到复曲面代码的 $O(L)$ 复杂性局部酉编码器。我们通过提供 $O(L)$ 局部单一电路来在复曲面代码和平面代码之间进行转换来关联这些技术，并且还为矩形、旋转和 3D 表面代码提供最佳编码器。此外，我们展示了我们的平面码编码电路如何用于在紧凑映射中准备费米子态，这是最近引入的费米子到量子位映射，它具有类似于表面码的稳定器结构，对于模拟费米-哈伯德模型。