Quantum Classification Algorithm Based on Competitive Learning Neural Network and Entanglement Measure
Abstract
:1. Introduction
2. Quantum Competitive Learning
3. Qubits and Quantum Gates
3.1. Qubit
3.2. Quantum Gates
4. Methodology
 Prepare two copies of the twoqubit state given by Equation (2) as follows:$${\eta}_{0}\rangle =\psi \rangle ({\sigma}_{y}\otimes {\sigma}_{y}\psi \rangle ).$$
 $CNOT$ gate is applied between the second and the forth qubits, respectively, followed by the rotation R gate as follows:$${\eta}_{1}\rangle =R\phantom{\rule{4pt}{0ex}}CNO{T}_{{\eta}_{{0}_{2}}{\eta}_{{0}_{4}}}{\eta}_{0}\rangle ,$$$$R0\rangle =\frac{0\rangle 1\rangle}{\sqrt{2}},\phantom{\rule{1.em}{0ex}}R1\rangle =\frac{0\rangle +1\rangle}{\sqrt{2}}.$$$$\begin{array}{ccc}\hfill {\eta}_{1}\rangle =\frac{1}{\sqrt{2}}& \{& (\beta \gamma \alpha \delta )0000\rangle +(\beta \gamma +\alpha \delta )0100\rangle +(\alpha \gamma \beta \delta )0001\rangle (\alpha \gamma +\beta \delta )0101\rangle +2\gamma \delta 1100\rangle \hfill \\ & & 2\alpha \beta 0110\rangle +({\beta}^{2}{\alpha}^{2})0011\rangle +({\beta}^{2}+{\alpha}^{2})0111\rangle +({\gamma}^{2}{\delta}^{2})1001\rangle ({\gamma}^{2}+{\delta}^{2})1101\rangle \hfill \\ & & +(\beta \gamma \alpha \delta )1010\rangle (\beta \gamma +\alpha \delta )1110\rangle +(\alpha \gamma +\beta \delta )1111\rangle (\alpha \gamma \beta \delta )1011\rangle \}.\hfill \end{array}$$$$C=2\sqrt{2{P}_{0000}}\phantom{\rule{1.em}{0ex}}or\phantom{\rule{1.em}{0ex}}C=2\sqrt{2{P}_{1010}}\phantom{\rule{4pt}{0ex}},$$
5. The proposed Quantum Classification Algorithm Based on Competitive Learning and Entanglement Measure: Case Study
Algorithm 1 The proposed Quantum Classification Algorithm based on Competitive Learning and Entanglement Measure (QCPNN). 

5.1. Case Study
5.1.1. QuantumStoring Layer Using Zhou’s Storage Model
 Step 1: The quantum system is initialized by the three registers $p\rangle $, $qn\rangle $ and $c\rangle $ as ${\psi}_{0}\rangle $= $p,qn,c\rangle $. Assuming that the input state is given by $p=11101$, where the first pattern in Equation (9) is considered, so the initial state can be described as ${\psi}_{0}\rangle $= $11101,00000,01\rangle $.
 Step 2: ${\psi}_{1}\rangle $ = ${\prod}_{i=1}^{n=5}{T}_{{p}_{i}{c}_{2}q{n}_{i}}^{2}{\psi}_{0}\rangle =11101,11101,01\rangle ,$ where ${T}^{2}$ is the toffli gate (Equation (1)).
 Step 3: ${\psi}_{2}\rangle $ = ${\prod}_{i=1}^{n=5}NO{T}_{q{n}_{i}}XO{R}_{{p}_{i}q{n}_{i}}{\psi}_{1}\rangle =11101,11111,01\rangle .$
 Step 4: ${\psi}_{3}\rangle $ = ${T}_{q{n}_{1}q{n}_{2}q{n}_{3}q{n}_{4}q{n}_{5}{c}_{1}}^{n=5}{\psi}_{2}\rangle =11101,11111,11\rangle .$
 Step 5: ${\psi}_{4}\rangle $ = ${S}_{{c}_{1}{c}_{2}}^{6}{\psi}_{3}\rangle =\frac{1}{\sqrt{6}}11101,11111,10\rangle +\sqrt{\frac{5}{6}}11101,11111,11\rangle $, where S is a Venture and Martinez’s gate operator [10,20] that is defined as follows:$${S}^{J}=\left(\right)open="["\; close="]">\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \sqrt{\frac{J1}{J}}& \frac{1}{\sqrt{J}}\\ 0& 0& \frac{1}{\sqrt{J}}& \sqrt{\frac{J1}{J}}\end{array}$$
 Step 6: ${\psi}_{5}\rangle $ = ${T}_{q{n}_{1}q{n}_{2}q{n}_{3}q{n}_{4}q{n}_{5}{c}_{1}}^{n=5}{\psi}_{4}\rangle =\frac{1}{\sqrt{6}}11101,11111,00\rangle +\sqrt{\frac{5}{6}}11101,11111,01\rangle .$
 Step 7: ${\psi}_{6}\rangle $ = ${\prod}_{i=1}^{n=5}XO{R}_{{p}_{i}q{n}_{i}}NO{T}_{q{n}_{i}}{\psi}_{7}=\frac{1}{\sqrt{6}}11101,11101,00\rangle +\sqrt{\frac{5}{6}}11101,11101,01\rangle .$
 Step 8: ${\psi}_{7}\rangle $ = ${\prod}_{i=1}^{n=5}{T}_{{p}_{i}{c}_{2}q{n}_{i}}^{2}{\psi}_{6}\rangle =\frac{1}{\sqrt{6}}11101,11101,00\rangle +\sqrt{\frac{5}{6}}11101,00000,01\rangle .$
5.1.2. Classification an Input Using the Proposed Algorithm
 Initialization Step: ${\psi}_{0}\rangle =inp,qn,uv\rangle .$Here, the input register is $inp\rangle =1?0?\rangle $, $qn\rangle $ is the memory register that holds the prototypes patterns and its state is given by Equation (11), and $uv\rangle $ is initialized by the state $00\rangle $. Due to the input, test, pattern $inp\rangle =1?0?\rangle $ has two well known values in the first and third qubits, so $h=\{1,3\}$. Therefore, the state of the system is described as follows:${\psi}_{0}\rangle =\frac{1}{\sqrt{6}}(1?0?,11101,00\rangle +1?0?,11001,00\rangle +1?0?,10111,00\rangle +1?0?,01000,00\rangle +1?0?,00100,00\rangle +1?0?,00010,00\rangle ).$
 Apply the competitive detection operator between the input register $inp\rangle $ and the prototype register $qn\rangle $ as ${\psi}_{1}\rangle ={\prod}_{i\in h=\{1,3\}}{X}_{q{n}_{i}}CNO{T}_{in{p}_{i}q{n}_{i}}{\psi}_{0}\rangle $.${\psi}_{1}\rangle =\frac{1}{\sqrt{6}}(1?0?,11001,00\rangle +1?0?,11101,00\rangle +1?0?,10011,00\rangle +1?0?,01100,00\rangle +1?0?,00000,00\rangle +1?0?,00110,00\rangle ).$
 Apply the Toffoligate between $j+1$ qubits of the register $qn\rangle $ and the qubit $u\rangle $ as control qubits and target qubit, respectively.$${\psi}_{2}\rangle ={T}_{({\prod}_{i\in h=\{1,3\}}q{n}_{i}q{n}_{n+1}u)}^{j+1}{\psi}_{1}\rangle ={T}_{(q{n}_{1}q{n}_{3}q{n}_{5}u)}^{3}{\psi}_{1}\rangle .$$${\psi}_{2}\rangle =\frac{1}{\sqrt{6}}(1?0?,11001,00\rangle +1?0?,11101,10\rangle +1?0?,10011,00\rangle +1?0?,01100,00\rangle +1?0?,00000,00\rangle +1?0?,00110,00\rangle ).$ Hence, the state of the twoqubit system $uv\rangle $ is$$uv\rangle =\sqrt{\frac{5}{6}}00\rangle +\frac{1}{\sqrt{6}}10\rangle .$$
 Repeat the steps 1, 2 and 3 to get another decoupled copy of the state $uv\rangle $.
 Apply the operator ${M}_{z}$ on the state $uv\rangle \otimes uv\rangle $ yields the state:$$\begin{array}{c}\hfill \frac{1}{\sqrt{2}}\{\frac{\sqrt{5}}{6}0000\rangle +\frac{\sqrt{5}}{6}0100\rangle \frac{5}{6}0011\rangle \frac{\sqrt{5}}{6}1010\rangle +\frac{5}{6}0111\rangle \frac{1}{6}1001\rangle \frac{1}{6}1101\rangle \frac{\sqrt{5}}{6}1110\rangle \}.\end{array}$$Here, it is obvious that the probability of the state $0000\rangle $, $0100\rangle $,$1010\rangle $ or $1110\rangle $ is nonzero, so according to Equation (7) the concurrence value $C>0$. Then, the test pattern $inp=1?0?$ belongs to the class label “1”.
6. Application
7. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
References
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Alarm Signal  Description 

${s}_{1}$  Bearing flow low 
${s}_{2}$  Thermal barrier flow low 
${s}_{3}$  No.1 seal differential pressure low 
${s}_{4}$  Standpipe level low 
${s}_{5}$  Charging pump flow low 
${s}_{6}$  No.1 seal leak off flow low 
${s}_{7}$  Bearing temperature high 
${s}_{8}$  Seal injection flow low 
${s}_{9}$  No.1 Seal leak off flow high 
${s}_{10}$  Seal injection filter differential pressure high 
${s}_{11}$  Standpipe level high 
${s}_{12}$  Thermal barrier temperature high 
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Zidan, M.; AbdelAty, A.H.; Elshafei, M.; Feraig, M.; AlSbou, Y.; Eleuch, H.; AbdelAty, M. Quantum Classification Algorithm Based on Competitive Learning Neural Network and Entanglement Measure. Appl. Sci. 2019, 9, 1277. https://doi.org/10.3390/app9071277
Zidan M, AbdelAty AH, Elshafei M, Feraig M, AlSbou Y, Eleuch H, AbdelAty M. Quantum Classification Algorithm Based on Competitive Learning Neural Network and Entanglement Measure. Applied Sciences. 2019; 9(7):1277. https://doi.org/10.3390/app9071277
Chicago/Turabian StyleZidan, Mohammed, AbdelHaleem AbdelAty, Mahmoud Elshafei, Marwa Feraig, Yazeed AlSbou, Hichem Eleuch, and Mahmoud AbdelAty. 2019. "Quantum Classification Algorithm Based on Competitive Learning Neural Network and Entanglement Measure" Applied Sciences 9, no. 7: 1277. https://doi.org/10.3390/app9071277