Three-dimensional quantum wavelet transforms

Wavelet transform is being widely used in the field of information processing. One-dimension and two-dimension quantum wavelet transforms have been investigated. However, three-dimensional quantum wavelet transforms have not been reported. To solve the problems, a research team led by Hai-Sheng LI published their new research on 15 Oct 2023 in Frontiers of Computer Science co-published by Higher Education Press and Springer Nature.
The team proposes a multi-level three-dimensional quantum wavelet transform theory to implement the wavelet transform for quantum videos. Then, the team constructs the iterative formulas for the multi-level three-dimensional Haar and Daubechies D4 quantum wavelet transforms, respectively. Next, quantum circuits of the two wavelet transforms using iterative methods are designed. Finally, the proposed quantum wavelet transforms are selected to realize quantum video compression as a primary application.
In the research, they analyze complexities of the proposed 3D quantum wavelet transforms. Complexity analysis shows that the proposed wavelet transforms offer exponential speed-up over their classical counterparts. Furthermore, simulation results reveal that the proposed wavelet transforms have better compression performance for quantum videos than two-dimension quantum wavelet transforms.
For future works, the team will extend the results to implement quantum versions of other wavelet transforms such as Daubechies D8 wavelet transform \cite{bibitem39} and cross wavelet transform.