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 $Dge 1$ 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 $Dge 2$ 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.
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