Drawing a line through a box in tikz

^{Premise: I'm not a physicist, I do not know what you're talking about.}

You can use the option execute at end picture={...} to add some code to be executed after your tikzcd.

\tikzcdmatrixname is the name of the TikZ matrix the tikzcd is translated into.

\tikzcdmatrixname-r-c is the name of the cell of the TikZ matrix at the r^th row and c^th column.

Maybe there is a better method (without manual adjustments xshift and yshift) but I don't know it.

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{quantikz}
\begin{document}
    \begin{tikzcd}[execute at
        end picture={
            \draw[thick] ([yshift=-23.8pt, 
            xshift=39pt]\tikzcdmatrixname-1-1.north 
            west) -- 
            ([yshift=-23.8pt, xshift=-39pt]\tikzcdmatrixname-1-5.north west);
        }]
        & \gate[wires=2]{J_{12}} & \gate[wires=3]{\raisebox{4pt}{$J_{13}$}} & 
        \qw                    & \qw \\
        &                        &  & 
        \gate[wires=2]{J_{23}} & \qw \\
        & \qw                    &                        
        &                        & \qw
    \end{tikzcd}
\end{document}

Here is a pure TikZ alternative without fine-tuning with a matrix:

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{matrix, fit}
\tikzset{gate/.style={rectangle, thick, fill=white,align=center, draw, text width=1.5em}}
\begin{document}
    \begin{tikzpicture}
    \matrix[matrix of nodes,
        nodes in empty cells,
        column sep=2.5em,
        row sep=4ex] (mymatr) {
        &[-1em]&&&[-1em]\\
        &&&&\\
        &&&&\\};
    \draw[thick] (mymatr-1-1) -- (mymatr-1-5);
    \draw[thick] (mymatr-3-1) -- (mymatr-3-5);
    \node[fit=(mymatr-1-3)(mymatr-3-3), gate, text height=5ex] {$J_{13}$};
    \draw[thick] (mymatr-2-1) -- (mymatr-2-5);
    \node[fit=(mymatr-1-2)(mymatr-2-2), gate] {$J_{12}$};
    \node[fit=(mymatr-2-4)(mymatr-3-4), gate] {$J_{23}$};
    \end{tikzpicture}
\end{document}

Here is a pure TikZ alternative without fine-tuning and without any matrix:

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{positioning}
\tikzset{gate/.style={rectangle, thick, align=center, draw, text width=1.5em}}
\begin{document}
    \begin{tikzpicture}[node distance=1.5em]
    \node[gate, minimum height=18ex, text height=0ex] (j13) {$J_{13}$};
   \node[left =of j13.north west, anchor=north east, minimum height=12ex, gate] (j12) {$J_{12}$};
    \node[right =of j13.south east, anchor=south west, minimum height=12ex, gate] (j23) {$J_{23}$};
    \coordinate (start1) at ([yshift=-2ex]j12.north west);
    \coordinate (end1) at ([xshift=1em]j23.east |- start1);
    \draw[thick] (start1) -- +(-1em, 0);
    \draw[thick] (start1 -| j12.east) -- (start1 -| j13.west);
    \draw[thick] (start1 -| j13.east) -- (end1);
    \coordinate (start2) at ([yshift=2ex]j12.south west);
    \draw[thick] (start2) -- +(-1em, 0);
    \draw[thick] (start2 -| j12.east) --
    (start2 -| j23.west);
    \draw[thick] (start2 -| j23.east) --
     +(1em, 0);
     \coordinate (start3) at ([yshift=2ex]j23.south west -| start1);
    \draw[thick] (start3) +(-1em, 0) -- (start3 -| j13.west) ;
    \draw[thick] (start3 -| j13.east) --
     (start3 -| j23.west);
     \draw[thick] (start3 -| j13.east) --
     (start3 -| j23.west);
    \draw[thick] (start3 -| j23.east) --
     +(1em, 0);
    \end{tikzpicture}
\end{document}