subsections.

Encoding {#ss:sequent}
--------

The encoding algorithm is similar to that of Section 3 except that the signals are encoded one-by-one. More concretely, it is done through alternating applications of the swap and shift operations. $$\begin{aligned}
&    & | h_1\rangle | l_1 \rangle
      | h_2\rangle | l_2 \rangle \cdots | h_N \rangle | l_N \rangle
      | {\bf 0} \rangle_{\rm
tape} \otimes \nonumber \\
& & | 0 \rangle_{\rm total~length} \nonumber \\
&\stackrel{\rm swap}{\longrightarrow} & | {\bf 0} \rangle | l_1 \rangle
      | h_2\rangle | l_2 \rangle \cdots | h_N \rangle | l_N \rangle
      | 0 \cdots 0 h_1\rangle_{\rm
tape} \otimes \nonumber \\
& & | 0 \rangle_{\rm total~length}
\nonumber \\
& \stackrel{\rm shift}{\longrightarrow} &  | {\bf 0} \rangle | l_1
\rangle
      | h_2\rangle | l_2 \rangle  \cdots | h_N \rangle | l_N \rangle
      | h_1  0 \cdots 0\rangle_{\rm
tape} \otimes \nonumber \\
& & | 0 \rangle_{\rm total~length}
\nonumber \\
& \stackrel{\rm add}{\longrightarrow} &  | {\bf 0} \rangle | l_1
\rangle
      | h_2\rangle | l_2 \rangle  \cdots | h_N \rangle | l_N \rangle
      | h_1  0 \cdots 0\rangle_{\rm
tape} \otimes \nonumber \\
& & | l_1 \rangle_{\rm total~length}
\nonumber \\
&\stackrel{\rm swap}{\longrightarrow} & | {\bf 0} \rangle | l_1 \rangle
      