You can't really nest floating environments (it doesn't make sense logically). That's the main problem, and may stem from the fact that you think you need a table
environment in order to place a tabular
. That's not the case. You're probably after the following:
\documentclass{article}
\usepackage{braket}
\begin{document}
\begin{table}[ht]
\centering
\begin{tabular}[t]{| c | c |}
\hline
State & $1$-step \\ \hline
$\ket{n, \leftarrow, 0}$ & $\leftarrow$ \\
$\ket{n, \leftarrow, 1}$ & $\rightarrow$ \\
$\ket{n, \rightarrow, 0}$ & $\rightarrow$ \\
$\ket{n, \rightarrow, 1}$ & $\leftarrow$ \\
\hline
\end{tabular}\hfill%
\begin{tabular}[t]{| c | c |}
\hline
State & $2$-step \\ \hline
$\ket{n, \leftarrow, \leftarrow, 0}$ & $\leftarrow$ \\
$\ket{n, \leftarrow, \leftarrow, 1}$ & $\rightarrow$ \\
$\ket{n, \leftarrow, \rightarrow, 0}$ & $\rightarrow$ \\
$\ket{n, \leftarrow, \rightarrow, 1}$ & $\leftarrow$ \\
\hline
\end{tabular}\hfill%
\begin{tabular}[t]{| c | c |}
\hline
State & $3$-step \\ \hline
$\ket{n, \leftarrow, \leftarrow, \leftarrow, 0}$ & $\leftarrow$ \\
$\ket{n, \leftarrow, \leftarrow, \leftarrow, 1}$ & $\rightarrow$ \\
$\ket{n, \leftarrow, \leftarrow, \rightarrow, 0}$ & $\rightarrow$ \\
$\ket{n, \leftarrow, \leftarrow, \rightarrow, 1}$ & $\leftarrow$ \\
\hline
\end{tabular}
\caption{Directions}
\end{table}
\end{document}
You may also be interested in a booktabs
implementation:
\documentclass{article}
\usepackage{braket,booktabs}
\begin{document}
\begin{table}[ht]
\centering
\begin{tabular}[t]{c c}
\toprule
State & $1$-step \\ \midrule
$\ket{n, \leftarrow, 0}$ & $\leftarrow$ \\
$\ket{n, \leftarrow, 1}$ & $\rightarrow$ \\
$\ket{n, \rightarrow, 0}$ & $\rightarrow$ \\
$\ket{n, \rightarrow, 1}$ & $\leftarrow$ \\
\bottomrule
\end{tabular}\hfill%
\begin{tabular}[t]{c c}
\toprule
State & $2$-step \\ \midrule
$\ket{n, \leftarrow, \leftarrow, 0}$ & $\leftarrow$ \\
$\ket{n, \leftarrow, \leftarrow, 1}$ & $\rightarrow$ \\
$\ket{n, \leftarrow, \rightarrow, 0}$ & $\rightarrow$ \\
$\ket{n, \leftarrow, \rightarrow, 1}$ & $\leftarrow$ \\
\bottomrule
\end{tabular}\hfill%
\begin{tabular}[t]{c c}
\toprule
State & $3$-step \\ \midrule
$\ket{n, \leftarrow, \leftarrow, \leftarrow, 0}$ & $\leftarrow$ \\
$\ket{n, \leftarrow, \leftarrow, \leftarrow, 1}$ & $\rightarrow$ \\
$\ket{n, \leftarrow, \leftarrow, \rightarrow, 0}$ & $\rightarrow$ \\
$\ket{n, \leftarrow, \leftarrow, \rightarrow, 1}$ & $\leftarrow$ \\
\bottomrule
\end{tabular}
\caption{Directions}
\end{table}
\end{document}
As an aside: You could have placed the entire table into a single tabular
as well.
Best Answer
Ultimately,
\FloatBarrier
with\usepackage{placeins}
package worked.