Using quantum computing to realize the Fourier Transform in computer vision applications

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2 In this work, we want to show the computational advantages (in terms o computational cost) to change the FFT and IFFT by the QFT and IQFT respectively.. Image processing Sometimes, when we acquire an image, we must perorm image processing techniques to enhance the output image, which serves as input image to the next step. This next step can be a recognition process to emphasize or to extract some objects rom the scene. So, beore the next step, we must ensure that the image doesn t have any unnecessary inormation. To do this, we usually transorm the image to the requency domain where we can perorm the operations to remove noise and distortions, enhance the distinctness etc. To convert the image rom space domain to requency domain we use the Discrete Fourier Transorm (DFT). The DFT is given by F( u) = ( x)exp[ j πux / ] (eq. ) and its inverse, which restore the original image is given by = π (eq. ) u= ( x) F( u)exp[ j ux / ] These transormation operations are o high computational cost, o O( ), where is the input number. The next section presents another algorithm that has a better computational cost.. The FFT algorithm The computers spent a big time to perorm the DFT. In 965, Cooley and Tukey made an algorithm that enables a computer perorm the DFT eiciently, called Fast Fourier Transorm (FFT), which has a better computation cost, O( log ) []. First, consider the equation where ux F( u) = ( x) W (eq. 3) W = exp[ j π / ]. (eq. 4) Since = n, thus can be write as = M. So, substituting in the equation 3, we obtain M F( u) = ( x) W M ux M = + + M (eq. 5) M M u( x) u(x+ ) ( x) W M (x ) W M M Simpliying the equation 5, we obtain M M ux ux u F( u) = ( x) WM (x ) WM W M + + M M (eq. 6) Analyzing the equation 6, we notice that the transormation is computed in two halves. As this algorithm has a reduced computation cost, the FFT algorithm is requently used in image processing to transorm the images rom space to requency domain. In section 3. we will see a Quantum Fourier Transorm (perormed using quantum computation), the analogous o FFT. 3. Quantum Computation Quantum computing makes use o the quantum mechanics principles to do the processing. Quantum computation gives us some phenomena not present in classical computation, such as superposition and entanglement. Superposition will be explained below. Entanglement is a phenomenon that enables the data to be saely transmitted. It is oten used in quantum inormation theory. The small inormation unit in quantum computation is a qubit (quantum bit). It is likely a bit, however, it has a dierence: a qubit can be in both possible states ( and ) at the same time, dierently o a bit, which can be in only one o the two states ( or ) at the same time. It is known, in quantum mechanics, as superposition principle. The next equation shows this principle. ψ = α + β (eq. 7) The state ψ is an arbitrary state. The components α and β are complex numbers that are proportional to the probability o the result or when a measurement is made, obeying the constraint measurement, we have α + β =. So, at the α probability to obtain and β probability to obtain. The states and orm the computational basis in Hilbert complex vector space, meaning binary values and, in Dirac notation.

3 The above states and have a matrix representation, = and = (eq. 8) To act the qubits, we use unitary operators U. They are deined as U U = U U = I. Any unitary matrix can be an operator. The most common operators are Pauli operators X, Y, Z, and Hadamard H, which have the ollowing matrix representation: X i Y= i Z H (eq. 9) The X gate perorms the same action o NOT classical gate. Hadamard gate is used to put one state in a superposition with the same probabilities, such as: H = ( + ) H = ( ) (eq. ) As we saw, the gates in quantum computation are unitary operators represented by its matrix. Quantum circuits have the same representation o classical circuits. Figure shows a general quantum circuit diagram. Figure. Quantum circuit diagram. In this circuit diagram, the state x contains the input data in a superposition state. The state y is an ancilla state, generally been in state. We will consider U as a black box that perorms the desired operation. Thus, with x been in superposition, we can apply U over all values in x simultaneously: U x, y = x, y ( x) (eq. ) During the processing, the probability o each state is changed according to the applied operations, which correspond to matrix multiplications. When the algorithm is inished, it is made a measurement that results in the major probability state. For example, consider x and y been in state. Applying one Hadamard gate on the state x, we will put it in a superposition state, like in equation: + U = ( U + U ), () +, () = (eq. ) The equation shows the processing using the superposition principle, so that the calculus o the () and () are made at the same time. Apparently, this sounds great, however, the quantum mechanics tell us that it is not possible to obtain all values o superposition. So, we can say that one qubit can be in superposition state in computation time, and when we made a measurement, the superposition state collapses to state that has the biggest probability. Thus, we can not obtain all possible output values, so, limiting the power o quantum computers, but there is some techniques to solve this problem (this will not be covered in this paper). To conclude, the main advantage o quantum computers is the possibility o solving some problems that in classical computing would spend an impracticable time, such as actorization, inormation retrieval in not ordered database, and so on. The next section describes the Quantum Fourier Transorm, the analogous o DFT in classical computation. 3. Quantum Fourier Transorm At the digital image processing using quantum computing, it is used the QFT, instead o FFT. Here, it is described the QFT algorithm, according with [3], [4] and [5]. The QFT algorithm receives as input a vector o complex numbers x, x, x,..., x N- with length N, and results at the output, another complex numbers vector, deined by equation 3: y k π i j k / x je j= = (eq. 3) A QFT can be described as FFT, with some dierent notations. Thus, a QFT in the computational base {,..., } is deined as a linear operator, which operates at the base as equation 4:

4 y> e k > (eq. 4) k= π i j k / and the action to an arbitrary state is given as equation 5: x j> yk k > (eq. 5) j j= k= where each amplitude y k is a discrete transorm o the amplitudes x j. The inverse transorm, denoted IQFT, is a linear operator whose action is given by equation 6: k > e j> (eq. 6) j= π i j k / The algorithm starts with one Hadamard operation, and n- conditional rotations at the irst qubit j, totalizing n operations. The conditions are controlled by other input qubits j,..., j n. It is ollowed by the application o another Hadamard operation, and n- rotations, to the second qubit, j, totalizing (n-) operations. Thus, it is noted that n + (n-) = n + (n-)/ operations are needed, and at most, n/ operations to change the bit order, each one constructed by CNOT gate. It is noted that the CNOT gate can be described by the diagram o Figure 4, Another useul QFT notation is the product representation, given by equation 7: j... j > π.. ( i j n π n )( i j ) j n >+ e > >+ e > n n/ (eq. 7) where the state j is described in binary notation n n j= j + j j n, and the notation.j = j l j l+... j m represent the binary raction j = j l / + j l+ / j m / m-l+. The product representation makes easier the quantum circuit construction. Figure 3 shows an eicient QFT quantum circuit. At Figure 3 the R k gate represents the operation described by the matrix given by: = R k k i / e π (eq. 8) and H gate represents and operation known as Hadamard, given by: Figure 4. CNOT gate whose operation can be described, with the inputs α e β at let, and the outputs at right. The upper output is a copy o the corresponding upper input, and the lower output is controlled by the upper input, resulting in exclusive-or operation o the inputs. As inal results, considering = n, the QFT circuit has a computational cost o O((log ) ), that is exponentially aster than the FFT algorithm, that has a computational cost o O( log ). 4. Results These results are given in terms o computational cost. The FFT algorithm has the cost o O ( log ), and the QFT has the cost o O((log ) ). Table shows the comparison o the computational cost between FFT and QFT algorithms. H (eq. 9) Figure 3. QFT quantum circuit.

5 Table. FFT, and QFT algorithms costs. = n FFT log QFT (log ) Computational advantage FFT/QFT 4 8 4, 8 4 9, , , , , , ,89.4.4, , , , ,8 At the last row, we have a inputs. We can think this as been an image with 4x4 pixels. Also, this table shows that the advantage o QFT over FFT increases when the input number increases. 4. Conclusion The proposal o this work is to explore the possibilities o the use o QFT in image processing, considering that QFT has a better computational cost than FFT. This work is just theoretical. We still don t have quantum computers available commercially, but it is expected that in near uture. There are some theoretical works in quantum computation that shows algorithms with better perormance than its analogous in classical computation. Currently, quantum computers are desired to solve problems in quantum mechanics and a ew problems in computer science. So, we tried, in this work, to show a possible real world application where quantum computers may be used. As uture work, we will try to use quantum computer simulators to simulate the QFT applied in an image transorm. 5. Reerences [] SHAPIRO, L. G.; STOCKMAN, G. C. Computer Vision. Prentice Hall,. [] GONZALEZ, R.; WOODS. R. Processamento de imagens digitais. São Paulo: Edgar Blücher,. [3] IMRE, S.; BALÁZS, F. Quantum Computing and Communications: an engineering approach. Chichester: John Wiley & Sons, 5. [4] KAYE, P.; LAFLAMME, R.; MOSCA, M. An Introduction to Quantum Computing. New York: Oxord University Press, 7. [5] NIELSEN, M. A,; CHUANG, I. L. Quantum computation and Quantum inormation. Cambridge, Cambridge University Press,. [6] McMAHON, D. Quantum computing explained. Hoboken, John Wiley & Sons, 8. [7] BELLAC, M. L. A short introduction to quantum inormation and quantum computation. Cambridge: Cambridge University Press, 6. [8] BENETI, G.; CASATI, G.; STRINI, G. Principles o quantum computation and inormation. Volume I: basic concepts. New Jersey: World Scientiic Publishing Company, 4. [9] CHEN, G.; CHURCH, D. A.; et. al. Quantum computing devices: principles, designs and analysis. Boca Raton: CRC Press, 7. [] KAYE, P.; LAFLAMME, R.; MOSCA, M. An Introduction to Quantum Computing. New York: Oxord University Press, 7. [] JOZSA, R. Quantum algorithms and the Fourier transorm. Available in: < proceedings/josza.ps>. Last accessed: jul. 7. [] MERMIN, N. D. Quantum computer science: an introduction. Cambridge: Cambridge University Press, 7. [3] NAKAHARA, M.; OHMI, T. Quantum computing: rom linear algebra to physical realizations. Boca Raton: CRC Press, 8. [4] PERRY, R. T. The temple o quantum computation. Available in: < Last accessed: out. 6. [5] PORTUGAL, R. et. al. Uma introdução à computação quântica. São Carlos: Sociedade Brasileira de Matemática Aplicada e Computacional, 4. [6] PRESKILL, J. Lecture otes or Physics 9: quantum inormation and computation. Caliornia Institute o Technology, 998. Available in: < Last accessed: 9 ago. 7.

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