论文标题
用低深度随机电路进行量子编码
Quantum coding with low-depth random circuits
论文作者
论文摘要
随机量子电路在确定近期量子计算机的计算优势方面发挥了核心作用。在这里，我们使用$ d \ ge 1 $空间尺寸的局部连接的低深度随机电路的合奏来生成量子错误校正代码。对于随机稳定器代码和擦除通道，我们发现有力的证据表明，深度$ O（\ log n）$随机电路是必需的，足以收敛（概率很高），以零故障概率，以低于最佳擦除阈值（由通道容量设置的最佳擦除阈值），任何$ d $。随机电路上的先前结果仅表明$ O（n^{1/d}）$深度就足够了，或者$ o（\ log^3 n）$深度足以实现全能连接（$ d \ to \ fo \ infty $）。然后，我们研究了所谓的中度偏差限制中擦除阈值的临界行为，在这种情况下，故障概率和最佳阈值的距离都以$ n $的形式收敛到零。我们发现，仅适用于$ o（\ log n）$的必要深度量表$ d \ ge 2 $，并且随机电路需要$ o（\ sqrt {n}）$ depth，$ d = 1 $。最后，我们引入了一种“消除”算法，该算法使用量子测量值来删除逻辑运算符，从而通过将其转换为其他稳定器或量规操作员，从而导致代码失败。通过这样的目标测量，我们可以在$ d \ ge 2 $的容量低于容量的情况下实现亚属性深度，而不会增加支票操作员的最大重量。我们发现，以低于容量的任何速度，具有数千个逻辑Qubits的高性能代码是可以实现的，深度为4-8的$ d = 2 $尺寸的深度为4-8。这些结果表明，有限速率量子代码实际上与近期设备相关，并且可能会大大降低资源需求，以实现近期应用的错误公差。
Random quantum circuits have played a central role in establishing the computational advantages of near-term quantum computers over their conventional counterparts. Here, we use ensembles of low-depth random circuits with local connectivity in $D\ge 1$ spatial dimensions to generate quantum error-correcting codes. For random stabilizer codes and the erasure channel, we find strong evidence that a depth $O(\log N)$ random circuit is necessary and sufficient to converge (with high probability) to zero failure probability for any finite amount below the optimal erasure threshold, set by the channel capacity, for any $D$. Previous results on random circuits have only shown that $O(N^{1/D})$ depth suffices or that $O(\log^3 N)$ depth suffices for all-to-all connectivity ($D \to \infty$). We then study the critical behavior of the erasure threshold in the so-called moderate deviation limit, where both the failure probability and the distance to the optimal threshold converge to zero with $N$. We find that the requisite depth scales like $O(\log N)$ only for dimensions $D \ge 2$, and that random circuits require $O(\sqrt{N})$ depth for $D=1$. Finally, we introduce an "expurgation" algorithm that uses quantum measurements to remove logical operators that cause the code to fail by turning them into additional stabilizers or gauge operators. With such targeted measurements, we can achieve sub-logarithmic depth in $D\ge 2$ below capacity without increasing the maximum weight of the check operators. We find that for any rate beneath the capacity, high-performing codes with thousands of logical qubits are achievable with depth 4-8 expurgated random circuits in $D=2$ dimensions. These results indicate that finite-rate quantum codes are practically relevant for near-term devices and may significantly reduce the resource requirements to achieve fault tolerance for near-term applications.
