  We present efficient methods to interpolate data with a quantum computer that
complement uploading techniques and quantum post-processing. The quantum
algorithms are supported by the efficient Quantum Fourier Transform (QFT) and
classical signal and imaging processing techniques, and open the door of
quantum advantage to relevant families of data. We showcase a QFT interpolation
method, a Quantum Cosine Transform (QCT) interpolation geared towards natural
data, and we improve upon them by utilizing a quantum circuit's capabilities of
processing data in superposition. A novel circuit for the QCT is presented. We
demonstrate the methods on probability distributions and quantum encoded
images, and discuss the precision of the resulting interpolations.
