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codes\BoolExp.py (8643, 2018-01-31)
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codes\FROI\FRQI.py (5582, 2018-01-31)
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# Quantum-Data-Compression ### 1. Introduction. Quantum data compression, in which the data could be a list of number, a picture, or a video. The target of quantum data compression is to find a feasible way to encode classical data into quantum state. Such encoding options can be vary. Take the representation of pictures as an example, many quantum image representation(QImR) has been discussed as follows. [A Survey of Quantum Image Representations]https://doi.org/10.1007/s11128-015-1195-6
[Processing and Retrieving an Image Using Quantum Mechanics]https://doi.org/10.1117/12.485960
[A Novel Enhanced Quantum Representation of Digital Images]https://doi.org/10.1007/s11128-013-0567-z
[A Flexible Representation of Quantum Images for Polynomial Preparation, Image Compression, and Processing Operations] https://doi.org/10.1007/s11128-010-0177-y
The following discussion is based on the paper [[A Flexible Representation of Quantum Images for Polynomial Preparation, Image Compression, and Processing Operations]]. ### 2. Quantum image representation. #### 2.1 Brief Introduction of QImR. A classical image which sized 2^n * 2^n in terms of pixel can be stored in only 1 + 2n qubits, where the first qubit used to control the color information, while the other 2n qubit stores the corresponding position of the image. Thus a coded qubits having the wavefunction:  Given a simple image with only four positions:  For every position, which has its index i, the corresponding color was stored in θ. To achieve this, a straightforward way is to control every position individually. In this way, a totally 2^n * 2^n nqubits controlled gates are required. Since nqubits gate requires exponential single and CNot gates, the total gates required for the encoding quantum circuits would expand exponentially. Thus, it is not efficient to implement such method. #### 2.2 Improved QImR. As discussed above, the main reason why this method is not efficient is that the large amount of high dimensional controlled gates. To reduce this, it is worthy to notice that during the controlled operation, same operation is implemented to different positions. For example, if position |00> and |01> have the same color, we could only control the first qubit, let alone the second one. Hence, the number of controlled gates reduce from two to one, and the number of controlled qubits in the controlled gate also has a reducing. In generally, to implement the above improved methods, it is necessary to dividing different position into groups where they has the same color. In every group, calculate the Boolean expression and try to __minimize__ it. #### 2.3 Example of improved QImR. A 2^3 * 2^3 two-color image, which requires (1 + 2*3) qubits for the representation. See the following picture:  If we use the classical method, we would need *** six-qubit Controlled operation, which is a waste of resource. Here we noticed that there are only two colors, so it is easy to divide the positions into two groups, blue and red, represented by the instinct angle theta1 and theta2, respectively. Next, encoding the position into binary index, so the positions of blue block could be written as follows:  In the end, to represent the blue color, only one three-qubit controlled gate are required, which is a huge reduction compared with eight six-qubit controlled gate. ### 3. Quantum Data Storing: a dimension-reduced application of FRQI. Suppose we storing a list of data instead of a picture, the main difference is that the former has less color dimension. A binary number has only two 'colors', black and white. This greatly simplifies the quantum representation of binary number, as well as the quantum circuits design. Suppose there is a four bit classical number, say 0101, the representation procedure is listed as follows. To store a number with 2^1 * 2^1 bits, we could use 1 + 2*1 qubits to store it. In the first place, create three qubits with initial state |0>. Next, to the qubits which represent the position of the data, implementing a Hadamard operation. After that, the 2 qubits have wave function (1/2)(|00> + |01> + |10> + |11>), four component represent four positions of the binary number. The following is to control the bit representing the 'color' using control gate. For our example, the first bit is linked to the position |00>, and it has the 'color' 0, so implementing C(I, 2), where I is the unit matrix, and this operation equals no operation. For the second, using |01> to represent its position, and implementing C(X, 2), where the flip the 'color' index from |0> to |1>. The same for other positions. By doing those operations, the final wavefunction becomes (1/2)(|0>|00> + |1>|01> + |0>|10> + |1>|11>). To extract message from the wavefunction, first measuring the position part, and then the 'color' part. ### NEQR: a Now let extent our model described in __3__, and use the similar model to store more complicated information such as a picture. Consider a gary-scale picture with 256 different possible color in one position, we would need extra qubits to describe the color information. The scheme is similar as in __3__, with enough qubits representing the different positions, but there are 8 qubits, which could expanded to 256 subspace, to store the color information.

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