  Recent work [M. J. Gullans et al., Physical Review X, 11(3):031066 (2021)]
has shown that quantum error correcting codes defined by random Clifford
encoding circuits can achieve a non-zero encoding rate in correcting errors
even if the random circuits on $n$ qubits, embedded in one spatial dimension
(1D), have a logarithmic depth $d=\mathcal{O}(\log{n})$. However, this was
demonstrated only for a simple erasure noise model. In this work, we discover
that this desired property indeed holds for the conventional Pauli noise model.
Specifically, we numerically demonstrate that the hashing bound, i.e., a rate
known to be achieved with $d=\mathcal{O}(n)$-depth random encoding circuits,
can be attained even when the circuit depth is restricted to
$d=\mathcal{O}(\log n)$ in 1D for depolarizing noise of various strengths. This
analysis is made possible with our development of a tensor-network
maximum-likelihood decoding algorithm that works efficiently for $\log$-depth
encoding circuits in 1D.
