Tensor rank bounds and explicit QTT representations for the inverses of circulant matrices

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), ... ,a(m-1),0, ... ,0,a-n, ... ,a-1)T 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.