Let $C$ be a circuit with variables $\mathcal{X}_n$. For $S \subseteq [n] \uplus [n]$, let $\mathbf{Stab}^\bullet(S) \leq \Gamma_n$ be the pointwise stabiliser of $S$. The set $S$ supports a gate $g$ if every circuit automorphism extending an element of $\mathbf{Stab}^\bullet(S)$ fixes $g$. If $C$ is rigid, then $\Gamma_n$ acts on the gates of $C$, and this says that $\mathbf{Stab}^\bullet(S)$ fixes $g$. The support size of $C$ is the least $k$ such that every gate of $C$ has a support of size at most $k$.

Write $\sup(g)$ for the intersection of all supports of $g$ of size at most the support size of $C$ — a bounded intersection: a support of unbounded size, such as the complement of a single point, trivially has a trivial pointwise stabiliser and so carries no information about $g$. If $g$ has a support $S$ with $|S_L|, |S_R| < n/2$, where $S_L$ and $S_R$ are the two sides of $S$, then $\sup(g)$ is itself a support of $g$, the unique minimal one. A wire from $h$ to $g$ is support-changing if $\sup(h) \not\subseteq \sup(g)$. If the support size of $C$ is less than $n/2$, the support depth of $C$ is the maximum number of support-changing wires on a path. Otherwise, the support depth of $C$ is its depth. In either case, the support depth is at most the depth.