One can go quite far without the use of tikz for schematics. However, the cost of simplicity is compromise. For example, in this MWE, I used an ellipsis ... instead of an axis-break symbol. Also, because I scaled the text-size arrows, the heads may be bigger than you like.
My basic approach was to stack the elements that you had already created, in order to squeeze out the vertical space. Also, I introduced \ffcolorbox which just kerned left the distance \fboxrule following the placement of an \fcolorbox, in order to avoid having double-thick lines between the boxes. I also placed a phantom ellipsis to the left of the boxes, so that the central row of boxes would be symmetrically placed.
\documentclass{article}
\usepackage{xcolor}
\newcommand\ffcolorbox[3]{\fcolorbox{#1}{#2}{#3}\kern-\fboxrule}
\usepackage[usestackEOL]{stackengine}
\setstackgap{S}{0pt}
\usepackage{graphicx}
\begin{document}
\centering
\fboxsep=10mm\fboxrule=0.5mm
\Shortstack{%
\fcolorbox{black}{blue!40!white}{{\bf P}}\\\scalebox{4}{$\downarrow\uparrow$}\\
\fboxsep=3mm \fboxrule=0.5mm\phantom{\scalebox{2.5}{\ldots}}
\ffcolorbox{black}{blue!40!white}{$a_1$}\ffcolorbox{black}{blue!40!white}%
{$o_1$}\ffcolorbox{black}{blue!40!white}{$a_2$}\ffcolorbox{black}{blue!40!white}%
{$o_2$}\ffcolorbox{black}{blue!40!white}{$a_3$}\fcolorbox{black}{blue!40!white}%
{$o_4$}\scalebox{2.5}{\ldots}\\
\scalebox{4}{$\downarrow\uparrow$}\\\fboxsep=10mm\fboxrule=0.5mm%
\fcolorbox{black}{blue!40!white}{{\bf Q}}%
}
\end{document}

In response to the comment, it is also possible to provide the desired overlaps, by using negative stack gaps. However, it must be assured that the arrows are stacked last, so that they end up on top. This requirement complicates the syntax somewhat, but is still possible. Each \stackon and \stackunder has as its optional argument the stacking gap. Overlap is shown with the central tape. If overlap is desired with the P and Q boxes, the [0pt] and [-1pt] gap settings should be set more negatively.
\documentclass{article}
\usepackage{xcolor}
\newcommand\ffcolorbox[3]{\fcolorbox{#1}{#2}{#3}\kern-\fboxrule}
\usepackage[usestackEOL]{stackengine}
\usepackage{graphicx}
\begin{document}
\centering
\fboxsep=10mm\fboxrule=0.5mm
\stackunder[-5pt]{\stackon[-5pt]{%
\fboxsep=3mm \fboxrule=0.5mm\phantom{~~~~~~~~.}
\ffcolorbox{black}{blue!40!white}{$a_1$}\ffcolorbox{black}{blue!40!white}%
{$o_1$}\ffcolorbox{black}{blue!40!white}{$a_2$}\ffcolorbox{black}{blue!40!white}%
{$o_2$}\ffcolorbox{black}{blue!40!white}{$a_3$}\fcolorbox{black}{blue!40!white}%
{$o_4$}\scalebox{2.5}{\ldots}%
}
{\stackunder[0pt]{\fcolorbox{black}{blue!40!white}{{\bf P}}}
{\scalebox{4}{$\downarrow\uparrow$}}}
}{%
\stackon[-1pt]{\fcolorbox{black}{blue!40!white}{{\bf Q}}}%
{\scalebox{4}{$\downarrow\uparrow$}}
}
\end{document}
