arXiv 2109.145992021-09-29quant-phcs.ITmath.IT

量子LDPC码稳定器测量电路的界限和局部实现的障碍

Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes

Nicolas Delfosse, Michael E. Beverland, Maxime A. Tremblay

备注：: 21 pages, 12 figures

摘要：

在这项工作中，我们建立的克利福德电路，衡量一个家庭的交换泡利运营商的大小上的下界。我们的边界取决于一对图之间的相互作用：测量的泡利算子集的坦纳图，以及表示实现电路所需的量子位连接的连接图。对于有望实现低开销量子纠错的本地扩展量子码，我们证明了在$N$量子位的2D方形片中使用本地Clifford门实现的任何校正子提取电路的深度至少为$Ω(n/\sqrt{N})$，其中$n$是代码长度。然后，我们提出了两个家庭的量子电路饱和这个界限。首先，我们仅使用$O(n^2)$辅助量子比特构造了具有有限深度的量子LDPC码的二维局部伴随式提取电路。其次，我们设计了一个基于深度为$O(\sqrt{n})$的$O(n)$辅助量子比特的超图乘积码的二维局部伴随式提取电路。最后，我们使用电路噪声模拟来比较超图乘积码的性能，超图乘积码使用最后一个2D伴随式提取电路家族与用全连接量子位实现的伴随式提取电路。虽然有一个阈值约为$10^{-3}$的一个完全连接的实现，我们观察到没有阈值的2D本地实现，尽管模拟的错误率低至$10^{-6}$。这表明量子LDPC码对于2D本地量子硬件是不切实际的。我们相信，我们的证明技术是独立的利益，并可以找到其他应用。我们对电路大小的界限是从电路的两个量子比特子集之间的相关量的下限和每个电路门引入的相关量的上限导出的，它们一起提供了电路大小的下限。

英文摘要：

In this work we establish lower bounds on the size of Clifford circuits that measure a family of commuting Pauli operators. Our bounds depend on the interplay between a pair of graphs: the Tanner graph of the set of measured Pauli operators, and the connectivity graph which represents the qubit connections required to implement the circuit. For local-expander quantum codes, which are promising for low-overhead quantum error correction, we prove that any syndrome extraction circuit implemented with local Clifford gates in a 2D square patch of $N$ qubits has depth at least $Ω(n/\sqrt{N})$ where $n$ is the code length. Then, we propose two families of quantum circuits saturating this bound. First, we construct 2D local syndrome extraction circuits for quantum LDPC codes with bounded depth using only $O(n^2)$ ancilla qubits. Second, we design a family of 2D local syndrome extraction circuits for hypergraph product codes using $O(n)$ ancilla qubits with depth $O(\sqrt{n})$. Finally, we use circuit noise simulations to compare the performance of a family of hypergraph product codes using this last family of 2D syndrome extraction circuits with a syndrome extraction circuit implemented with fully connected qubits. While there is a threshold of about $10^{-3}$ for a fully connected implementation, we observe no threshold for the 2D local implementation despite simulating error rates of as low as $10^{-6}$. This suggests that quantum LDPC codes are impractical with 2D local quantum hardware. We believe that our proof technique is of independent interest and could find other applications. Our bounds on circuit sizes are derived from a lower bound on the amount of correlations between two subsets of qubits of the circuit and an upper bound on the amount of correlations introduced by each circuit gate, which together provide a lower bound on the circuit size.

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摘要：

英文摘要：

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