  Circulant matrices are an important family of operators, which have a wide
range of applications in science and engineering related fields. They are in
general non-sparse and non-unitary. In this paper, we present efficient quantum
circuits to implement circulant operators using fewer resources and with lower
complexity than existing methods. Moreover, our quantum circuits can be readily
extended to the implementation of Toeplitz, Hankel, and block circulant
matrices. Efficient quantum algorithms to implement the inverses and products
of circulant operators are also provided.
