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 either additional stabilizers or into gauge operators in a subsystem code. With such targeted measurements, we can achieve sublogarithmic depth in $D\ge 2$ spatial dimensions 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.

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具有低深度随机电路的量子编码

随机量子电路在建立近期量子计算机相对于传统计算机的计算优势方面发挥了核心作用。在这里，我们使用具有局部连接性的低深度随机电路的集合$D\ge 1$空间维度来生成量子纠错码。对于随机稳定器代码和擦除通道，我们发现强有力的证据表明深度$哦(\mathrm{日志}N)$ 对于任何低于最佳擦除阈值（由信道容量设置）的任何有限数量，随机电路是必要且足以（以高概率）收敛到零失败概率的 $D$. 先前关于随机电路的结果仅表明$哦(N^{1/D})$ 深度就足够了 $哦({\mathrm{日志}}^{3}N)$ 深度足以实现全对全连接（$D\to \infty$）。然后，我们研究擦除阈值在所谓的中等偏差极限中的临界行为，其中失败概率和到最佳阈值的距离都收敛为零$N$. 我们发现必要的深度尺度像$哦(\mathrm{日志}N)$ 仅适用于尺寸 $D\ge 2$ 并且随机电路需要 $哦(\sqrt{N})$ 深度为 $D=1$. 最后，我们引入了一种“清除”算法，该算法使用量子测量来删除导致代码失败的逻辑运算符，方法是将它们转换为额外的稳定器或子系统代码中的规范运算符。通过这种有针对性的测量，我们可以实现亚对数深度$D\ge 2$空间尺寸低于容量，而不会增加检查操作员的最大重量。我们发现，对于低于容量的任何速率，具有数千个逻辑量子位的高性能代码可以通过深度 4-8 去除随机电路来实现$D=2$方面。这些结果表明，有限速率量子代码实际上与近期设备相关，并且可以显着减少资源需求，以实现近期应用的容错。