Quantum wavelet transform for finite momentum width

Construct a quantum wavelet transform whose target states have finite width in momentum space and are partly localized in real space, thereby potentially providing a smooth tradeoff between momentum precision and noise sensitivity.

Background

The proposed smooth modification of the FFT motivates a transform intermediate between a fully global Fourier transform and a local real-space circuit. The paper identifies a quantum wavelet transform as a possible realization because wavelet-like states could possess finite momentum width while retaining partial real-space localization. This is presented as a potential, unresolved direction rather than as a developed construction.

References

A "quantum wavelet transform", where the target states have a finite width in momentum space and are partly localized in space could potentially fulfill this objective.

— Less precise but less noisy: local circuits for momentum-space state preparation and measurement  (2610.01704 - Granet et al., 1 Oct 2026) in Section Discussion and outlook