  It is an open question how fast information processing can be performed and
whether quantum effects can speed up the best existing solutions. Signal
extraction, analysis and compression in diagnostics, astronomy, chemistry and
broadcasting builds on the discrete Fourier transform. It is implemented with
the Fast Fourier Transform (FFT) algorithm that assumes a periodic input of
specific lengths, which rarely holds true. A less-known transform, the
Kravchuk-Fourier (KT), allows one to operate on finite strings of arbitrary
length. It is of high demand in digital image processing and computer vision,
but features a prohibitive runtime. Here, we report a one-step computation of a
fractional quantum KT. A quantum $d$-nary (qudit) architecture we use comprises
only one gate and offers processing time independent of the input size. The
gate may employ a multiphoton Hong-Ou-Mandel effect. Existing quantum
technologies may scale it up towards diverse applications.
