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 <math>|g\rangle =\hat{U} |f\rangle </math> with the input image state <math>|f\rangle </math> and the output image state <math>|g\rangle </math>. 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 <math>G=PFQ</math>, with the resulting image <math>G</math> and a row (column) transform matrix <math> P (Q)</math>. The corresponding unitary operator <math>\hat{U}</math> can then be written as <math> \hat{U}={Q}^T \otimes {P}</math>. Several forcommonly quantumused imagetwo-dimensional transforms, such as the Haar wavelet, Fourier, and Hadamard transforms,are impleexperimentally demonstrated on a quantum computer<ref>{{cite journal|last1=Yao|first1=Xi-Wei|last2=Wang|first2=Hengyan|last3=Liao|first3=Zeyang|last4=Chen|first4=Ming-Cheng|last5=Pan|first5=Jian|last6=Li|first6=Jun|last7=Zhang|first7=Kechao|last8=Lin|first8=Xingcheng|last9=Wang|first9=Zhehui|last10=Luo|first10=Zhihuang|last11=Zheng|first11=Wenqiang|last12=Li|first12=Jianzhong|last13=Zhao|first13=Meisheng|last14=Peng|first14=Xinhua|last15=Suter|first15=Dieter|title=Quantum Image Processing and Its Application to Edge Detection: Theory and Experiment|journal=Physical Review X|date=11 September 2017|volume=7|issue=3|page=031041|doi=10.1103/PhysRevX.7.031041|url=https://journals.aps.org/prx/abstract/10.1103/PhysRevX.7.031041}}</ref>, 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.
Several commonly used two-dimensional transforms, such as the Haar wavelet, Fourier, and Hadamard transforms,are experimentally demonstrated on a quantum computer<ref>{{cite journal|last1=Yao|first1=Xi-Wei|last2=Wang|first2=Hengyan|last3=Liao|first3=Zeyang|last4=Chen|first4=Ming-Cheng|last5=Pan|first5=Jian|last6=Li|first6=Jun|last7=Zhang|first7=Kechao|last8=Lin|first8=Xingcheng|last9=Wang|first9=Zhehui|last10=Luo|first10=Zhihuang|last11=Zheng|first11=Wenqiang|last12=Li|first12=Jianzhong|last13=Zhao|first13=Meisheng|last14=Peng|first14=Xinhua|last15=Suter|first15=Dieter|title=Quantum Image Processing and Its Application to Edge Detection: Theory and Experiment|journal=Physical Review X|date=11 September 2017|volume=7|issue=3|page=031041|doi=10.1103/PhysRevX.7.031041|url=https://journals.aps.org/prx/abstract/10.1103/PhysRevX.7.031041}}</ref>, 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.
