Vertical alignment of tikz circle in equation

The standard way to center content in math is \vcenter which alignes a list of vertical material centered to the math axis (the invisible line the minus sign, the equal sign, fractions etc. are aligned):

\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{equation}
  A + B \Big \langle \vcenter{\hbox{\tikz \draw (0,0) circle (0.5cm);}} \Big \rangle
\end{equation}
\end{document}

Output:

It is still worth mentioning that baseline can help you in an efficient way: by defining a style vertical align that abbreviates the long baseline=... command.

\documentclass{article}
\usepackage{tikz}
\begin{document}
    \tikzset{
        vertical align/.style={
            baseline=-.5*(height("$+$")-depth("$+$"))
        }
    }
    \foreach\size in{\tiny,\footnotesize,\normalsize,\Large,\huge}{
        \size
        \begin{equation}
            A+B\Big\langle\tikz[vertical align]\draw circle(0.5);\Big\rangle^{
            A+B\Big\langle\tikz[vertical align]\draw circle(0.5);\Big\rangle^{
            A+B\Big\langle\tikz[vertical align]\draw circle(0.5);\Big\rangle}}
        \end{equation}
    }
    \end{document}

---

What can be done further is that you can apply style globally by saying

\tikzset{
    every picture/.style={vertical align}
}

For irrelevant pictures, you can cancel the effect by either grouping or saying [baseline=0pt] or just [baseline]. (Or you can define no vertical align/.style={baseline})

---

You can also encode the drawing of the circle in a .pic. For example

\tikzset{
    cir5/.pic={
        \draw circle(0.5);
    }
}

Together with the global setting of baseline, the code is now much simpler

    \begin{equation}
          A+B\Big\langle\tikz\pic{cir5};\Big\rangle
    \end{equation}

in contrast to

    \begin{equation}
        A+B\Big\langle\vcenter{\hbox{\tikz\draw circle(0.5);}}\Big\rangle
    \end{equation}

---

Of course one can do something like \def\vcentertikz#1{\vcenter{\hbox{\tikz{#1}}}}, just like one can do \def\tikzvcenterpic#1{\tikz[vcenter]{\pic{#1}}}. So as long as one does not need tricky baseline setting (such as baseline=(node.anchor)), both methods are interchangeable.

---

The following is the full code.

\documentclass{article}
\usepackage{tikz}
\begin{document}
    \tikzset{
        vertical align/.style={
            baseline=-.5*(height("$+$")-depth("$+$"))
        }
    }
    \foreach\size in{\tiny,\footnotesize,\normalsize,\Large,\huge}{
        \size
        \begin{equation}
            A+B\Big\langle\tikz[vertical align]\draw circle(0.5);\Big\rangle^{
            A+B\Big\langle\tikz[vertical align]\draw circle(0.5);\Big\rangle^{
            A+B\Big\langle\tikz[vertical align]\draw circle(0.5);\Big\rangle}}
        \end{equation}
    }

    \clearpage

    \tikzset{
        every picture/.style={vertical align}
    }
    \foreach\size in{\tiny,\footnotesize,\normalsize,\Large,\huge}{
        \size
        \begin{equation}
            A+B\Big\langle\tikz\draw circle(0.5);\Big\rangle^{
            A+B\Big\langle\tikz\draw circle(0.5);\Big\rangle^{
            A+B\Big\langle\tikz\draw circle(0.5);\Big\rangle}}
        \end{equation}
    }

    \clearpage

    \tikzset{
        cir5/.pic={
            \draw circle(0.5);
        }
    }
    \foreach\size in{\tiny,\footnotesize,\normalsize,\Large,\huge}{
        \size
        \begin{equation}
              A+B\Big\langle\tikz\pic{cir5};\Big\rangle
            ^{A+B\Big\langle\tikz\pic{cir5};\Big\rangle
            ^{A+B\Big\langle\tikz\pic{cir5};\Big\rangle}}
        \end{equation}
    }

    \clearpage

    \foreach\size in{\tiny,\footnotesize,\normalsize,\Large,\huge}{
        \size
        \begin{equation}
            A + B \Big \langle \vcenter{\hbox{\tikz \draw (0,0) circle (0.5cm);}} \Big \rangle^{
            A + B \Big \langle \vcenter{\hbox{\tikz \draw (0,0) circle (0.5cm);}} \Big \rangle^{
            A + B \Big \langle \vcenter{\hbox{\tikz \draw (0,0) circle (0.5cm);}} \Big \rangle}}
        \end{equation}
    }

\end{document}