The quantum fractional Fourier transform

The Fourier transform (FT) is ubiquitous in signal processing, as it can be used to filter noise. The digital version, often named the discrete Fourier transform, when formulated on a basis of quantum states, is the quantum Fourier transform (QFT). The efficiency in the implementation of the QFT is the main reason for several quantum speedups, including the one for factoring and the one in phase estimation at the Heisenberg limit. The fractional FT (frFT) is a generalization of the FT. The frFT has recently gained attention in signal analysis as it can filter noise in scenarios where the FT is not useful. Quantum frFTs (QfrFTs), however, have never been proposed, constructed, or applied. In this work we propose a QfrFT and show that a good approximation of this transformation can be implemented on a quantum computer with exponentially less resources than those required for its conventional implementation. We then analyze some problems in signal analysis for which our defined QfrFT is useful.