27640 Quantum computing: with applications to image processing

Probleemstelling:

Quantum information processing, which exploits quantum-mechanical phenomena such as quantum superpositions and quantum entanglement, allows one to overcome the limitations of classical computation and reaches higher computational speed for certain problems like factoring large numbers, searching an unsorted database, quantum simulation, solving linear systems of equations, and machine learning. These unique quantum properties, such as quantum superposition and quantum parallelism, may also be used to speed up signal and data processing. Recently, it has been established that existing quantum algorithms are applicable to image processing tasks allowing quantum informational models of classical image processing. Quantum mechanics have targeted many image processing tasks, such as denoising, edge detection, image storage, retrieval, and compression. Although it has been shown how  quantum computing can be superior to classical computers in image processing, there are still certain complexities in transitioning from the classical to the quantum domain (see Fig. 1).


Figure 1: Comparison of image processing by classical and quantum computers (see [2]).

The continuous Heisenberg-Weyl groups have a long history in physics and signal processing. However, their discrete variants haven’t received the deserved attention. When a signal is transformed by an element of the (much larger) discrete symplectic group, autocorrelations, which are mathematical tool for finding repeating patterns in a signal, can be calculated from trace inner products of covariance matrices with signed permutation matrices from the discrete Heisenberg-Weyl group. The mapping between a signal and its autocorrelation coefficients based on the Heisenberg-Weyl group is called the Weyl transform. This instance of the Weyl transform is a special case of a general framework for representation of operators in harmonic analysis. The theory of the finite Heisenberg-Weyl group in relation to the development of adaptive radar and to the construction of spreading sequences and error-correcting codes in communications has been well investigated [1]. Recently it has been shown how beautiful properties of the Weyl transform can be used for image classification (see [3] and [4]).


Doelstelling:

Student should first familiar himself with the quantum image processing and the existing work in this field. Fourier transform, the Hadamard, and the Haar wavelet transform, are usually included as subroutines in many complicated tasks of image processing. The main goal of the thesis is to consider the discrete Heisenberg-Weyl group and the corresponding Weyl transform for image classification, but on the quantum level. Existing relations between the Weyl transform and the other well known transforms for image processing tasks should be better investigated. The student will be provided with the available data as well as the existing code in the research group GAIM. 

References:

  1. S. D. Howard, A. R. Calderbank, W. Moran. The finite Heisenberg-Weyl groups in radar and communications, 2006.
  2. X. Yao et al. Quantum image processing and its application to edge detection: theory and experiment, 2018.
  3. Q. Qiu, A. Thompson, R. Calderbank, G. Sapiro. Data representation using the Weyl transform, 2015.
  4. T. Zhao, G. O. A. Montereale Gavazzi, S. Lazendic, Y. Zhao, A. Pizurica. Acoustic seafloor classification using the Weyl transform of multibeam backscatter data, 2020.