Quantum Signal Processing

In signal processing, the Wigner distribution is used as a quadratic time‐frequency representation derived from the notion of autocorrelation. Wigner distribution was introduced in the context of quantum physics, to introduce quantum corrections to statistical physics. It shows a remarkable property, which is that it can be defined in an equivalent manner to the frequency version of the signal, obtained by the Fourier transform (TF). Gauss sums are similar to Fourier sums with the clear difference that the index of summation appears in the quadratic phase in a location in a linear manner. This distribution can be interpreted as the signal's power spectral density. This chapter discusses the factorization properties of different Gauss sums. Heisenberg's view is the following: physical quantities in quantum mechanics can only be processed with a non‐commutative algebra. This leads to envisage that quantum mechanics can only be understood, from a geometrical point of view, via a non‐commutative geometry.