User:X-MAN109/sandbox

Humans obtain most of their information through their eyes. Accordingly, the analysis of visual data is one of the most important functions of our brain and it has evolved high efficiency in processing visual data. Currently, visual information like images and videos constitutes the largest part of data traffic in the internet. Processing of this information requires ever larger computational power. The laws of quantum mechanics allow one to reduce the required resources for some tasks by many orders of magnitude if the image data are encoded in the quantum state of a suitable physical system.

Quantum image transform
By encoding and processing the image information in quantum-mechanical systems, a framework of quantum image processing is presented, where a pure quantum state encodes the image information: to encode the pixel values in the probability amplitudes and the pixel positions in the computational basis states. Given a image $$F=(F_{i,j})_{M \times L}$$, where $$F_{i,j}$$ represents the pixel value at position $$(i,j)$$ with $$i = 1,\dots,M$$ and $$j = 1,\dots,L$$, a vector $$\vec{f}$$ with $$ML$$ elements can be formed by letting the first $$M$$ elements of $$\vec{f}$$ be the first column of $$F$$, the next $$M$$ elements the second column, etc.

A large class of image operations is linear, e.g., unitary transformations, convolutions, and linear filtering. In the quantum computing, the linear transformation can be represented as $$|g\rangle =\hat{U} |f\rangle $$ with the input image state $$|f\rangle $$ and the output image state $$|g\rangle $$. A unitary transformation can be implemented as a unitary evolution. Some basic and commonly used image transforms (e.g., the Fourier, Hadamard, and Haar wavelet transforms) can be expressed in the form $$G=PFQ$$, with the resulting image $$G$$ and a row (column) transform matrix $$ P (Q)$$. The corresponding unitary operator $$\hat{U}$$ can then be written as $$ \hat{U}={Q}^T \otimes {P}$$. Several commonly used two-dimensional image transforms, such as the Haar wavelet, Fourier, and Hadamard transforms, are experimentally demonstrated on a quantum computer, with exponential speedup over their classical counterparts. In addition, a novel highly efficient quantum algorithm is proposed and experimentally implemented for detecting the boundary between different regions of a picture: It requires only one single-qubit gate in the processing stage, independent of the size of the picture.