  We consider the problem of $\textit{subgroup testing}$ for a quantum circuit
$C$: given access to $C$, determine whether it implements a unitary from a
subgroup $\mathcal{G}$ of the unitary group. In particular, the group
$\mathcal{G}$ can be the trivial subgroup (i.e., identity testing) or groups
such as the Pauli or Clifford groups, or even their $q$-ary extensions. We also
consider a $\textit{promise}$ version of the problem where $C$ is promised to
be in some subgroup of the unitaries that contains $\mathcal{G}$ (e.g.,
identity testing for Clifford circuits).
  We present a novel structural property of Clifford unitaries. Namely, that
their (normalized) trace is bounded by $1/\sqrt{2}$ in absolute value,
regardless of the dimension. We show a similar property for the $q$-ary
Cliffords. This allows us to analyze a very simple single-query identity test
under the Clifford promise and show that it has (at least) constant soundness.
The same test has perfect soundness under the Pauli promise.
  We use the test to show that identity/Pauli/Clifford testing (without
promise) are all computationally equivalent, thus establishing computational
hardness for Pauli and Clifford testing.
