digital data processors, but others remain time-consuming. In particular, the rapidly increasing volume of image data as well as increasingly challenging computational tasks have become important driving forces for further improving the efficiency of image processing and analysis.Quantum information processing (QIP), which exploits quantum-mechanical phenomena such as quantum superpositions and quantum entanglement [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], allows one to overcome the limitations of classical computation and reaches higher computational speed for certain problems like factoring large numbers [24,25] , searching an unsorted database [26], boson sampling [27][28][29][30][31][32], quantum simulation [33-40], solving linear systems of equations [41][42][43][44][45], and machine learning [46][47][48]. These unique quantum properties, such as quantum superposition and quantum parallelism, may also be used to speed up signal and data processing [49,50]. For quantum image processing, quantum image representation (QImR) plays a key role, which substantively determines the kinds of processing tasks and how well they can be performed. A number of QImRs [51-54] have been discussed.In this article, we demonstrate the basic framework of quan-arXiv:1801.01465v1 [quant-ph]

Quantum computers have the potential of solving certain problems exponentially faster than classical computers. Recently, Harrow, Hassidim and Lloyd proposed a quantum algorithm for solving linear systems of equations: given an N × N matrix A and a vector b, find the vector x that satisfies A x = b. It has been shown that using the algorithm one could obtain the solution encoded in a quantum state |x using O(log N ) quantum operations, while classical algorithms require at least O(N ) steps. If one is not interested in the solution x itself but certain statistical feature of the solution x|M |x (M is some quantum mechanical operator), the quantum algorithm will be able to achieve exponential speedup over the best classical algorithm as N grows. Here we report a proof-of-concept experimental demonstration of the quantum algorithm using a 4-qubit nuclear magnetic resonance (NMR) quantum information processor. For all the three sets of experiments with different choices of b, we obtain the solutions with over 96% fidelity. This experiment is a first implementation of the algorithm. Because solving linear systems is a common problem in nearly all fields of science and engineering, we will also discuss the implication of our results on the potential of using quantum computers for solving practical linear systems.

Experimental investigation of subspace quantum process tomography in three-spin system was implemented via nuclear magnetic resonance. A quantum process was characterized by measuring a complete set of input states and corresponding outputs. The method using ancillary qubit remarkably reduces the number of the initial input states. And the pulse sequences used in this paper have fewer J-coupling evolutions. The experiment time was shortened and quantum decoherence of the system was weakened efficiently.