  In this paper, we are concerned with the inversion of circulant matrices and
their quantized tensor-train (QTT) structure. In particular, we show that the
inverse of a complex circulant matrix $A$, generated by the first column of the
form $(a_0,\dots,a_{m-1},0,\dots,0,a_{-n},\dots, a_{-1})^\top$ admits a QTT
representation with the QTT ranks bounded by $(m+n)$. Under certain assumptions
on the entries of $A$, we also derive an explicit QTT representation of
$A^{-1}$. The latter can be used, for instance, to overcome stability issues
arising when numerically solving differential equations with periodic boundary
conditions in the QTT format.
