Note on quantum cellular automata and strong equivalence

In this note, we present some results on the classification of quantum cellular automata (QCA) in 1D under strong equivalence rather than stable equivalence. Under strong equivalence, we only allow adding ancillas carrying the original on-site representation of the symmetry, while under stable equivalence, we allow adding ancillas carrying any representation of the symmetry. The former may be more realistic, because in physical systems especially in AMO/quantum computing contexts, we would not expect additional spins carrying arbitrary representations of the symmetry to be present. Ref.~\onlinecite{mpu} proposed two kinds of symmetry-protected indices (SPIs) for QCA with discrete symmetries under strong equivalence. In this note, we show that the more refined of these SPIs still only has a one-to-one correspondence to equivalence classes of $\mathbb{Z}_N$ symmetric QCA when $N$ is prime. We show a counter-example for $N=4$. We show that QCA with $\mathbb{Z}_2$ symmetry under strong equivalence, for a given on-site representation, are classified by $\mathbb{Z}^{pq}$ where $p$ is the number of prime factors of the on-site Hilbert space dimension and $q$ is the number of prime factors of the trace of the nontrivial on-site $\mathbb{Z}_2$ element. Finally, we show that the GNVW index has a formulation in terms of a $\mathbb{Z}_2$ SPI in a doubled system, and we provide a direct connection between the SPI formulation of the GNVW index and a second Renyi version of the mutual information formula for the GNVW index.