polylogarithmic depth.

More concretely, at the end the encoder obtains $$| l_1 \rangle | l_2 \rangle \cdots  | l_N \rangle
| h_1 h_2 \cdots h_N 0 \cdots 0 \rangle_{\rm tape}$$ in only $O ( (\log N)^a)$ depth for some positive integer $a$. Finally, the encoder truncates the message tape: He keeps only say the first $N ( \bar{L} + \delta)$ qubits in the message tape $| h_1 h_2 \cdots h_N 0 \cdots 0 \rangle_{\rm
tape} $ for some $\delta > 0$ and throws away the other qubits. This truncation minimizes the number of qubits needed. The only overhead cost compared to the classical case is the storage of the length tapes of the individual signals. This takes only $ N \lceil \log l_{\rm max} \rceil$ qubits.[^9]

Decoding
--------

Decoding can be done by adding an appropriate number of qubits in the zero state $|  0 \rangle$ behind the truncated message tape and simply running the encoding process backwards (again with only depth $O ( (\log N)^a)$).

What about fidelity? The key observation is the following:

The typical subspace $S_\delta$ is the subspace where the first $N
(\bar{L} + \delta)$ qubits are arbitrary, and any qubits beyond that are in the 