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Edge detection is a basic problem in digital image processing. Its purpose is to detect the pixels whose gray level changes obviously in the neighborhood. The Canny edge detector is currently the most popular edge detection tool. This paper studies the specific implementation of Canny detector in the quantum computing paradigm. For Gaussian smoothing filtering and Sobel sharpening operators, we have designed a new method called "Translation, Stacking and Weighted Summation", which can make full use of the parallelism of quantum computing to accelerate its classical counterpart and avoid convolution operation. For the gradient and angle calculations required for edge detection, we design new operators such as addition, multiplication and division of signed number by introducing the binary complement description of gray-scale value. For the non-maximum suppression and double threshold processing required for edge detection, we have separately designed the quantum circuits that implement these tasks by introducing quantum complement comparators. Complexity analysis shows that the quantum Canny edge detector has exponential speedup compared to its classical counterpart. The implementation of the Canny edge detector based on quantum computing is the main contribution of this paper. The simulation on the classical computer verifies the correctness of the quantum Canny edge detection results. [ABSTRACT FROM AUTHOR]