Multi-dimensional color image storage and retrieval for a normal arbitrary quantum superposition state

Multi-dimensional color image processing has two difficulties: One is that a large number of bits are needed to store multi-dimensional color images, such as, a three-dimensional color image of $1024 \times 1024 \times 1024$ needs $1024 \times 1024 \times 1024 \times 24$  bits. The other one is that the efficiency or accuracy of image segmentation is not high enough for some images to be used in content-based image search. In order to solve the above problems, this paper proposes a new representation for multi-dimensional color image, called a $(n\,+\,1)$ -qubit normal arbitrary quantum superposition state (NAQSS), where $n$ qubits represent colors and coordinates of ${2^n}$ pixels (e.g., represent a three-dimensional color image of $1024 \times 1024 \times 1024$ only using 30 qubits), and the remaining 1 qubit represents an image segmentation information to improve the accuracy of image segmentation. And then we design a general quantum circuit to create the NAQSS state in order to store a multi-dimensional color image in a quantum system and propose a quantum circuit simplification algorithm to reduce the number of the quantum gates of the general quantum circuit. Finally, different strategies to retrieve a whole image or the target sub-image of an image from a quantum system are studied, including Monte Carlo sampling and improved Grover’s algorithm which can search out a coordinate of a target sub-image only running in $O(\sqrt{N/r} )$ where $N$ and $r$ are the numbers of pixels of an image and a target sub-image, respectively.