  The star-triangle relation plays an important role in the realm of exactly
solvable models, offering exact results for classical two-dimensional
statistical mechanical models. In this article, we construct integrable quantum
circuits using the star-triangle relation. Our construction relies on families
of mutually commuting two-parameter transfer matrices for statistical
mechanical models solved by the star-triangle relation, and differs from
previously known constructions based on Yang-Baxter integrable vertex models.
At special value of the spectral parameter, the transfer matrices are mapped
into integrable quantum circuits, for which infinite families of local
conserved charges can be derived. We demonstrate the construction by giving two
examples of circuits acting on a chain of $Q-$state qudits: $Q$-state Potts
circuits, whose integrability has been conjectured recently by Lotkov et al.,
and $\mathbb{Z}_Q$ circuits, which are novel to our knowledge. In the first
example, we present for $Q=3$ a connection to the Zamolodchikov-Fateev
19-vertex model.
