Quantum Algorithms for the Multiplication of Circulant Matrices and Vectors

This article presents two quantum algorithms for computing the product of a circulant matrix and a vector. The arithmetic complexity of the first algorithm is O(Nlog2N) in most cases. For the second algorithm, when the entries in the circulant matrix and the vector take values in C or R, the complexity is O(Nlog2N) in most cases. However, when these entries take values from positive real numbers, the complexity is reduced to O(log3N) in most cases, which presents an exponential speedup compared to the classical complexity of O(NlogN) for computing the product of a circulant matrix and vector. We apply this algorithm to the convolution calculation in quantum convolutional neural networks, which effectively accelerates the computation of convolutions. Additionally, we present a concrete quantum circuit structure for quantum convolutional neural networks.