Unitary Subgroup Testing

We consider the problem of $\textit{subgroup testing}$ for a quantum circuit $C$: given access to $C$, determine whether it implements a unitary that is $a$-close or $b$-far from a subgroup $\mathcal{G}$ of the unitary group. It encompasses the problem of exact testing, property testing and tolerant testing. In this work, we study these problems with the group $\mathcal{G}$ as the trivial subgroup (i.e.\ identity testing) or the Pauli or Clifford group and their $q$-ary extension, and a $\textit{promise}$ version of these problems where $C$ is promised to be in some subgroup of the unitaries that contains $\mathcal{G}$ (e.g. identity testing for Clifford circuits). Our main result is an equivalence between Pauli testing, Clifford testing and Identity testing. We derive the equivalence between Clifford and Identity testing by showing a structural property of the Clifford unitaries. Namely, that their (normalized) trace lies in the discrete set $\{2^{-k/2}: k \in \mathbb{N}\} \cup \{0\}$, regardless of the dimension. We also state and prove the analogous property for the $q$-ary Cliffords. This result allows us to analyze a very simple single-query identity test under the Clifford/Pauli promise. To prove the equivalence between Pauli and Identity testing, we analyze the conjugation action of a non-Pauli unitary on the Pauli group and show that its distance from the Pauli group affects the number of fixed points. We believe that these results are of interest, independent of their application to establish the equivalences. We use the equivalences to compare (and thus establish) computational hardness for the problems of Pauli and Clifford testing.