Quantum speedup for solving the minimum vertex cover problem based on Grover search algorithm

Abstract

The electronic computer possesses powerful computing capabilities, but it is still limited by Moore’s law. Quantum computers are thus found capable of not only overcoming the constraints of Moore’s law but also taking advantage of the quantum effect to increase their arithmetic power for many problems. The minimum vertex cover problem is a classical NP-complete problem, and its traditional algorithms usually require exponential computational complexity. Herein, two algorithms are proposed to solve the minimum vertex cover problem. The first algorithm achieves its classical solution, while the second algorithm is based on the Grover search algorithm to obtain the solution to the quantum computing algorithm of the problem and verify its feasibility via quantum circuit experiments. Comparing the two algorithms, it can be concluded that the quantum algorithm can improve the operational speed of the minimum vertex cover problem by order of magnitude of the square root.

Appendices

Appendix A: The example of graph $G_{1}$ with 4 vetices

To enhance the convincingness, we take the graph $G_2$ of Section 1 as an example to solve the MVCP problem. In the first step, the initial quantum state is manufactured to get the initial quantum state $\left. \vert {\mathop {\phi }\nolimits _1 } \right\rangle = \frac{1}{{\sqrt{16} }}\sum \limits _{X = 0}^{15} {\left. \vert X \right\rangle }\otimes \left. \vert {\mathop {l}\nolimits _1^0 \mathop {l}\nolimits _2^0 \mathop {l}\nolimits _3^0 \mathop {l}\nolimits _4^0 } \right\rangle \otimes \left. \vert {\mathop {o}\nolimits _5^0 } \right\rangle \otimes \left. \vert {\mathop {w}\nolimits _{1,0}^0 \mathop {w}\nolimits _{1,1}^0 \mathop {w}\nolimits _{2,0}^0...\mathop {w}\nolimits _{4,3}^0 \mathop {w}\nolimits _{4,4}^0 } \right\rangle \otimes \frac{{\left. \vert 0 \right\rangle - \left. \vert 1 \right\rangle }}{2}$. In the second step, in the second step, the target quantum state is screened to obtain the quantum superposition state $\left. \vert {\mathop {\phi }\nolimits _1 } \right\rangle ={\left( { - 1} \right) ^{f(x)}}\frac{1}{{\sqrt{16} }}\sum \limits _{X = 0}^{15} {\left. \vert X \right\rangle } \otimes \left. \vert {\mathop {l}\nolimits _1 \mathop {l}\nolimits _2 \mathop {l}\nolimits _3 \mathop {l}\nolimits _4 } \right\rangle \otimes \left. \vert {\mathop {o}\nolimits _3 } \right\rangle \otimes \left. \vert {\mathop {w}\nolimits _{1,0} \mathop {w}\nolimits _{1,1} \mathop {w}\nolimits _{2,0}...\mathop {w}\nolimits _{4,3} \mathop {w}\nolimits _{4,4} } \right\rangle \otimes \frac{{\left. \vert 0 \right\rangle - \left. \vert 1 \right\rangle }}{2}$. In the third step, the expanded target quantum state using Grover iteration is used to find the optimal solution.

There are four groups for the experiments. The fourth group builds quantum circuit diagrams to measure qubit $\left. \vert w_{4,1}\right\rangle$ that have a feasible solution with a single vertex, the fifth group measures qubit $\left. \vert w_{4,2}\right\rangle$ with two vertices, the sixth group measures $\left. \vert w_{4,3}\right\rangle$ with three vertices, and the seventh group measures $\left. \vert w_{4,4}\right\rangle$ with four vertices. Then, the number of feasible solution subsets $m_{i}$ is calculated from the results of the amplitude of the first Grover iteration in each group. The number of iterations $k=\frac{\pi }{4}\sqrt{\frac{N}{{{m_i}}}}$ is deduced from the Grover algorithm. Finally, the results of each set of experiments are obtained.

Fig. 22

Circuit with one Grover iteration implemented for the fifth group

Fig. 23

Circuit with two Grover iterations implemented for the fifth group

Fig. 24

Circuit with three Grover iterations implemented for the fifth group

Fig. 25

Probability diagram of the results of implementing one Grover iteration (a), two Grover iterations (b) and three Grover iterations (c) for the fifth group

Fig. 26

Circuit with one Grover iteration implemented for the sixth group

Fig. 27

Circuit with two Grover iterations implemented for the sixth group

Fig. 28

Probability diagram of the results of implementing one Grover iteration (a) and two Grover iterations (b) for the sixth group

Fig. 29

Circuit with one Grover iteration implemented for the seventh group

Fig. 30

Circuit with two Grover iterations implemented for the seventh group

Fig. 31

Circuit with three Grover iterations implemented for the seventh group

Fig. 32

Circuit with fourth Grover iteration implemented for the seventh group

Fig. 33

Probability diagram of the results of implementing one Grover iteration (a), two Grover iterations (b), three Grover iterations (c) and four Grover iterations (d) for the seventh group

As shown in Figs. 22, 23, 24 and 25, it is the circuit and experimental results probability diagram for the fifth group. As shown in Figs. 26, Figure.4w3sps2 and 28, it is the circuit and experimental results probability diagram for the sixth group. As shown in Figs. 29, 30, 31, 32 and 33, it is the circuit and experimental results probability diagram for the seventh group. The circuit and experimental results probability diagram for the fourth group is provided because a feasible solution could not be found.

Table 6 The experimental results of the fifth group

Table 7 The experimental results of the sixth group

Table 8 The experimental results of the seventh group

As shown in Tables 6, 7 and 8, the probability of finding the initial quantum states for the three sets of experiments is 0.125, 0.375 and 0.125, respectively, and the corresponding numbers of feasible solutions are $1(m_5)$, $3(m_6)$ and $1(m_7)$, as deduced from the results of the first iteration of the experiments. The number of iterations k for the three sets of experiments is found to be 1.721, 1.071 and 2.642 based on the iteration formula. In Tables 6, 7 and 8, the angle of the quantum state amplitude increases with the number of iterations. The fifth group of experiments achieves the optimal solution with the highest probability after two iterations, the sixth group of experiments achieves the optimal solution with the highest probability after one iteration, and the seventh group of experiments achieves the optimal solution after three iterations. According to the MVCP definition, the optimal solution is achieved after two iterations in the fifth group. At this time, the probability of obtaining the optimal solution is 0.9434, which corresponds to the sets of vertex solution $\{v_{1},v_{3}\}$ and $\{v_{2},v_{4}\}$. Therefore, the minimum vertex subset of MVCP of graph $G_{1}$ is $\{v_{1},v_{3}\}$ and $\{v_{2},v_{4}\}$, which has a size of 2. In Tables 6, 7 and 8, it can be seen that the relative error between the experimental results and the probability of finding is small, which further proves the reliability of the experimental data.

Appendix B: The quantum programming code of application example

Code 1. Code with one Grover iteration implemented for the first group.
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TOFFOLI   q[1],q[0],q[3]
TOFFOLI   q[2],q[1],q[4]
TOFFOLI   q[4],q[3],q[5]
TOFFOLI   q[5],q[0],q[6]
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TOFFOLI   q[5],q[0],q[7]
TOFFOLI   q[6],q[1],q[8]
TOFFOLI   q[7],q[1],q[9]
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TOFFOLI   q[6],q[1],q[9]
TOFFOLI   q[7],q[1],q[10]
TOFFOLI   q[8],q[2],q[11]

Code 1. Code with one Grover iteration implemented for the first group.
TOFFOLI   q[9],q[2],q[12]
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TOFFOLI   q[8],q[2],q[12]
TOFFOLI   q[9],q[2],q[13]
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TOFFOLI   q[10],q[2],q[13]
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TOFFOLI   q[2],q[10],q[14]
CNOT   q[12],q[15]
TOFFOLI   q[2],q[10],q[14]
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TOFFOLI   q[10],q[2],q[13]
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TOFFOLI   q[9],q[2],q[13]
TOFFOLI   q[8],q[2],q[12]
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TOFFOLI   q[9],q[2],q[12]
TOFFOLI   q[8],q[2],q[11]
TOFFOLI   q[7],q[1],q[10]
TOFFOLI   q[6],q[1],q[9]
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TOFFOLI   q[7],q[1],q[9]
TOFFOLI   q[6],q[1],q[8]
TOFFOLI   q[5],q[0],q[7]
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TOFFOLI   q[5],q[0],q[6]
TOFFOLI   q[4],q[3],q[5]
TOFFOLI   q[2],q[1],q[4]
TOFFOLI   q[1],q[0],q[3]
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H   q[2]
MEASURE   q[0],c[0]
MEASURE   q[1],c[1]
MEASURE   q[2],c[2]

Code 2. Code with two Grover iterations implemented for the first group.
H q[0]
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TOFFOLI q[1],q[0],q[3]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[0],q[5],q[6]
X q[0]
TOFFOLI q[5],q[0],q[7]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[7],q[1],q[9]
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TOFFOLI q[6],q[1],q[9]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[9],q[2],q[12]
X q[2]
TOFFOLI q[8],q[2],q[12]
TOFFOLI q[9],q[2],q[13]
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TOFFOLI q[10],q[2],q[13]
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TOFFOLI q[10],q[2],q[14]
CNOT q[12],q[15]
TOFFOLI q[10],q[2],q[14]
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TOFFOLI q[10],q[2],q[13]
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TOFFOLI q[9],q[2],q[13]
TOFFOLI q[8],q[2],q[12]
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TOFFOLI q[9],q[2],q[12]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[6],q[1],q[9]
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TOFFOLI q[7],q[1],q[9]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[5],q[0],q[7]
X q[0]
TOFFOLI q[0],q[5],q[6]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[1],q[0],q[3]
X q[4]

Code 2. Code with two Grover iterations implemented for the first group.
X q[0]
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TOFFOLI q[1],q[0],q[2]
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TOFFOLI q[1],q[0],q[3]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[0],q[5],q[6]
X q[0]
TOFFOLI q[5],q[0],q[7]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[7],q[1],q[9]
X q[1]
TOFFOLI q[6],q[1],q[9]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[9],q[2],q[12]
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TOFFOLI q[8],q[2],q[12]
TOFFOLI q[9],q[2],q[13]
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TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[10],q[2],q[14]
CNOT q[12],q[15]
TOFFOLI q[10],q[2],q[14]
X q[2]
TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[9],q[2],q[13]
TOFFOLI q[8],q[2],q[12]
X q[2]

Code 2. Code with two Grover iterations implemented for the first group.
TOFFOLI q[9],q[2],q[12]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[6],q[1],q[9]
X q[1]
TOFFOLI q[7],q[1],q[9]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[5],q[0],q[7]
X q[0]
TOFFOLI q[0],q[5],q[6]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[1],q[0],q[3]
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TOFFOLI q[1],q[0],q[2]
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MEASURE q[0],c[0]
MEASURE q[1],c[1]
MEASURE q[2],c[2]

Code 3. Code with three Grover iterations implemented for the first group.
H q[0]
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TOFFOLI q[1],q[0],q[3]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[0],q[5],q[6]
X q[0]
TOFFOLI q[5],q[0],q[7]
TOFFOLI q[1],q[6],q[8]
TOFFOLI q[7],q[1],q[9]
X q[1]
TOFFOLI q[6],q[1],q[9]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[2],q[8],q[11]
TOFFOLI q[9],q[2],q[12]
X q[2]
TOFFOLI q[8],q[2],q[12]
TOFFOLI q[9],q[2],q[13]
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TOFFOLI q[2],q[10],q[13]
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TOFFOLI q[10],q[2],q[14]
CNOT q[12],q[15]
TOFFOLI q[10],q[2],q[14]
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TOFFOLI q[2],q[10],q[13]
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TOFFOLI q[9],q[2],q[13]
TOFFOLI q[8],q[2],q[12]
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TOFFOLI q[9],q[2],q[12]
TOFFOLI q[2],q[8],q[11]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[6],q[1],q[9]
X q[1]
TOFFOLI q[7],q[1],q[9]
TOFFOLI q[1],q[6],q[8]
TOFFOLI q[5],q[0],q[7]
X q[0]
TOFFOLI q[0],q[5],q[6]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[1],q[0],q[3]

Code 3. Code with three Grover iterations implemented for the first group.
X q[4]
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TOFFOLI q[1],q[2],q[4]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[0],q[5],q[6]
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TOFFOLI q[5],q[0],q[7]
TOFFOLI q[1],q[6],q[8]
TOFFOLI q[7],q[1],q[9]
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TOFFOLI q[6],q[1],q[9]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[2],q[8],q[11]
TOFFOLI q[9],q[2],q[12]
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TOFFOLI q[8],q[2],q[12]
TOFFOLI q[9],q[2],q[13]
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TOFFOLI q[2],q[10],q[13]
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TOFFOLI q[10],q[2],q[14]
CNOT q[12],q[15]
TOFFOLI q[10],q[2],q[14]
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TOFFOLI q[2],q[10],q[13]
X q[2]
TOFFOLI q[9],q[2],q[13]
TOFFOLI q[8],q[2],q[12]

Code 3. Code with three Grover iterations implemented for the first group.
X q[2]
TOFFOLI q[9],q[2],q[12]
TOFFOLI q[2],q[8],q[11]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[6],q[1],q[9]
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TOFFOLI q[7],q[1],q[9]
TOFFOLI q[1],q[6],q[8]
TOFFOLI q[5],q[0],q[7]
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TOFFOLI q[0],q[5],q[6]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[1],q[0],q[3]
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TOFFOLI q[1],q[0],q[3]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[0],q[5],q[6]
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TOFFOLI q[5],q[0],q[7]
TOFFOLI q[1],q[6],q[8]
TOFFOLI q[7],q[1],q[9]
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TOFFOLI q[6],q[1],q[9]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[2],q[8],q[11]
TOFFOLI q[9],q[2],q[12]

Code 3. Code with three Grover iterations implemented for the first group.
X q[2]
TOFFOLI q[8],q[2],q[12]
TOFFOLI q[9],q[2],q[13]
X q[2]
TOFFOLI q[2],q[10],q[13]
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TOFFOLI q[10],q[2],q[14]
CNOT q[12],q[15]
TOFFOLI q[10],q[2],q[14]
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TOFFOLI q[2],q[10],q[13]
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TOFFOLI q[9],q[2],q[13]
TOFFOLI q[8],q[2],q[12]
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TOFFOLI q[9],q[2],q[12]
TOFFOLI q[2],q[8],q[11]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[6],q[1],q[9]
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TOFFOLI q[7],q[1],q[9]
TOFFOLI q[1],q[6],q[8]
TOFFOLI q[5],q[0],q[7]
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TOFFOLI q[0],q[5],q[6]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[1],q[0],q[3]
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H q[1]
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H q[2]
MEASURE q[0],c[0]
MEASURE q[1],c[1]
MEASURE q[2],c[2]

Code 4. Code with one Grover iteration implemented for the second group.
H q[0]
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TOFFOLI q[6],q[1],q[8]
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TOFFOLI q[7],q[1],q[10]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[9],q[2],q[12]
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TOFFOLI q[8],q[2],q[12]
TOFFOLI q[9],q[2],q[13]
X q[2]
TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[10],q[2],q[14]
CNOT q[12],q[15]
TOFFOLI q[10],q[2],q[14]
X q[2]
TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[9],q[2],q[13]
TOFFOLI q[8],q[2],q[12]
X q[2]
TOFFOLI q[9],q[2],q[12]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[6],q[1],q[9]
X q[1]
TOFFOLI q[7],q[1],q[9]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[5],q[0],q[7]
X q[0]
TOFFOLI q[0],q[5],q[6]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[1],q[0],q[3]

Code 4. Code with one Grover iteration implemented for the second group.
X q[4]
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TOFFOLI q[1],q[0],q[2]
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TOFFOLI q[1],q[2],q[4]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[0],q[5],q[6]
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TOFFOLI q[5],q[0],q[7]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[7],q[1],q[9]
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TOFFOLI q[6],q[1],q[9]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[9],q[2],q[12]
X q[2]
TOFFOLI q[8],q[2],q[12]
TOFFOLI q[9],q[2],q[13]
X q[2]

Code 4. Code with one Grover iteration implemented for the second group.
TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[10],q[2],q[14]
CNOT q[12],q[15]
TOFFOLI q[10],q[2],q[14]
X q[2]
TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[9],q[2],q[13]
TOFFOLI q[8],q[2],q[12]
X q[2]
TOFFOLI q[9],q[2],q[12]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[6],q[1],q[9]
X q[1]
TOFFOLI q[7],q[1],q[9]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[5],q[0],q[7]
X q[0]
TOFFOLI q[0],q[5],q[6]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[1],q[0],q[3]
X q[4]
X q[0]
X q[1]
X q[2]
X q[3]
H q[0]
H q[1]
H q[2]
X q[0]
X q[1]
X q[2]
H q[2]
TOFFOLI q[1],q[0],q[2]
X q[0]
X q[1]
H q[2]
H q[0]
H q[1]
X q[2]
H q[2]
MEASURE q[0],c[0]
MEASURE q[1],c[1]
MEASURE q[2],c[2]

Code 5. Code with two Grover iterations implemented for the second group.
H q[0]
H q[1]
H q[2]
X q[3]
X q[4]
X q[15]
X q[0]
X q[1]
X q[2]
H q[15]
TOFFOLI q[1],q[0],q[3]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[0],q[5],q[6]
X q[0]
TOFFOLI q[5],q[0],q[7]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[7],q[1],q[9]
X q[1]
TOFFOLI q[6],q[1],q[9]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[9],q[2],q[12]
X q[2]
TOFFOLI q[8],q[2],q[12]
TOFFOLI q[9],q[2],q[13]
X q[2]
TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[10],q[2],q[14]
CNOT q[13],q[15]
TOFFOLI q[10],q[2],q[14]
X q[2]
TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[9],q[2],q[13]
TOFFOLI q[8],q[2],q[12]
X q[2]
TOFFOLI q[9],q[2],q[12]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[6],q[1],q[9]
X q[1]
TOFFOLI q[7],q[1],q[9]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[5],q[0],q[7]
X q[0]
TOFFOLI q[0],q[5],q[6]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[1],q[0],q[3]
X q[4]

Code 5. Code with two Grover iterations implemented for the second group.
X q[0]
X q[1]
X q[2]
X q[3]
X q[4]
H q[0]
H q[1]
H q[2]
X q[3]
X q[0]
X q[1]
X q[2]
H q[2]
TOFFOLI q[1],q[0],q[2]
X q[0]
X q[1]
H q[2]
H q[0]
H q[1]
X q[2]
X q[0]
X q[1]
H q[2]
X q[2]
TOFFOLI q[1],q[0],q[3]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[0],q[5],q[6]
X q[0]
TOFFOLI q[5],q[0],q[7]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[7],q[1],q[9]
X q[1]
TOFFOLI q[6],q[1],q[9]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[9],q[2],q[12]
X q[2]
TOFFOLI q[8],q[2],q[12]
TOFFOLI q[9],q[2],q[13]
X q[2]
TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[10],q[2],q[14]
CNOT q[13],q[15]

Code 5. Code with two Grover iterations implemented for the second group.
TOFFOLI q[10],q[2],q[14]
X q[2]
TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[9],q[2],q[13]
TOFFOLI q[8],q[2],q[12]
X q[2]
TOFFOLI q[9],q[2],q[12]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[6],q[1],q[9]
X q[1]
TOFFOLI q[7],q[1],q[9]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[5],q[0],q[7]
X q[0]
TOFFOLI q[0],q[5],q[6]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[1],q[0],q[3]
X q[4]
X q[0]
X q[1]
X q[2]
X q[3]
H q[0]
H q[1]
H q[2]
X q[0]
X q[1]
X q[2]
H q[2]
TOFFOLI q[1],q[0],q[2]
X q[0]
X q[1]
H q[2]
H q[0]
H q[1]
X q[2]
H q[2]
MEASURE q[0],c[0]
MEASURE q[1],c[1]
MEASURE q[2],c[2]

Code 6. Code with one Grover iteration implemented for the third group.
H q[0]
H q[1]
H q[2]
X q[3]
X q[4]
X q[15]
X q[0]
X q[1]
X q[2]
H q[15]
TOFFOLI q[1],q[0],q[3]
TOFFOLI q[2],q[1],q[4]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[5],q[0],q[6]
X q[0]
TOFFOLI q[5],q[0],q[7]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[7],q[1],q[9]
X q[1]
TOFFOLI q[6],q[1],q[9]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[9],q[2],q[12]
X q[2]
TOFFOLI q[8],q[2],q[12]
TOFFOLI q[9],q[2],q[13]
X q[2]
TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[2],q[10],q[14]
CNOT q[14],q[15]
TOFFOLI q[2],q[10],q[14]
X q[2]
TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[9],q[2],q[13]
TOFFOLI q[8],q[2],q[12]
X q[2]

Code 6. Code with one Grover iteration implemented for the third group.
TOFFOLI q[9],q[2],q[12]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[6],q[1],q[9]
X q[1]
TOFFOLI q[7],q[1],q[9]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[5],q[0],q[7]
X q[0]
TOFFOLI q[5],q[0],q[6]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[2],q[1],q[4]
TOFFOLI q[1],q[0],q[3]
X q[4]
X q[0]
X q[1]
X q[2]
X q[3]
H q[0]
H q[1]
H q[2]
X q[0]
X q[1]
X q[2]
H q[2]
TOFFOLI q[1],q[0],q[2]
X q[0]
X q[1]
H q[2]
H q[0]
H q[1]
X q[2]
H q[2]
MEASURE q[0],c[0]
MEASURE q[1],c[1]
MEASURE q[2],c[2]

Code 7. Code with two Grover iterations implemented for the third group.
H q[0]
H q[1]
H q[2]
X q[3]
X q[4]
X q[15]
X q[0]
X q[1]
X q[2]
H q[15]
TOFFOLI q[1],q[0],q[3]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[0],q[5],q[6]
X q[0]
TOFFOLI q[5],q[0],q[7]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[7],q[1],q[9]
X q[1]
TOFFOLI q[6],q[1],q[9]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[9],q[2],q[12]
X q[2]
TOFFOLI q[8],q[2],q[12]
TOFFOLI q[9],q[2],q[13]
X q[2]
TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[10],q[2],q[14]
CNOT q[14],q[15]
TOFFOLI q[10],q[2],q[14]
X q[2]
TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[9],q[2],q[13]
TOFFOLI q[8],q[2],q[12]
X q[2]
TOFFOLI q[9],q[2],q[12]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[6],q[1],q[9]
X q[1]
TOFFOLI q[7],q[1],q[9]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[5],q[0],q[7]
X q[0]

Code 7. Code with two Grover iterations implemented for the third group.
TOFFOLI q[0],q[5],q[6]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[1],q[0],q[3]
X q[4]
X q[0]
X q[1]
X q[2]
X q[3]
X q[4]
H q[0]
H q[1]
H q[2]
X q[3]
X q[0]
X q[1]
X q[2]
H q[2]
TOFFOLI q[1],q[0],q[2]
X q[0]
X q[1]
H q[2]
H q[0]
H q[1]
X q[2]
X q[0]
X q[1]
H q[2]
X q[2]
TOFFOLI q[1],q[0],q[3]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[0],q[5],q[6]
X q[0]
TOFFOLI q[5],q[0],q[7]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[7],q[1],q[9]
X q[1]
TOFFOLI q[6],q[1],q[9]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[9],q[2],q[12]
X q[2]

Code 7. Code with two Grover iterations implemented for the third group.
TOFFOLI q[8],q[2],q[12]
TOFFOLI q[9],q[2],q[13]
X q[2]
TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[10],q[2],q[14]
CNOT q[14],q[15]
TOFFOLI q[10],q[2],q[14]
X q[2]
TOFFOLI q[10],q[2],q[13]
X q[2]
TOFFOLI q[9],q[2],q[13]
TOFFOLI q[8],q[2],q[12]
X q[2]
TOFFOLI q[9],q[2],q[12]
TOFFOLI q[8],q[2],q[11]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[6],q[1],q[9]
X q[1]
TOFFOLI q[7],q[1],q[9]
TOFFOLI q[6],q[1],q[8]
TOFFOLI q[5],q[0],q[7]
X q[0]
TOFFOLI q[0],q[5],q[6]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[1],q[0],q[3]
X q[4]
X q[0]
X q[1]
X q[2]
X q[3]
H q[0]
H q[1]
H q[2]
X q[0]
X q[1]
X q[2]
H q[2]
TOFFOLI q[1],q[0],q[2]
X q[0]
X q[1]
H q[2]
H q[0]
H q[1]
X q[2]
H q[2]
MEASURE q[0],c[0]
MEASURE q[1],c[1]
MEASURE q[2],c[2]

Code 8. Code with three Grover iterations implemented for the third group.
H q[0]
H q[1]
H q[2]
X q[3]
X q[4]
X q[15]
X q[0]
X q[1]
X q[2]
H q[15]
TOFFOLI q[1],q[0],q[3]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[0],q[5],q[6]
X q[0]
TOFFOLI q[5],q[0],q[7]
TOFFOLI q[1],q[6],q[8]
TOFFOLI q[7],q[1],q[9]
X q[1]
TOFFOLI q[6],q[1],q[9]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[2],q[8],q[11]
TOFFOLI q[9],q[2],q[12]
X q[2]
TOFFOLI q[8],q[2],q[12]
TOFFOLI q[9],q[2],q[13]
X q[2]
TOFFOLI q[2],q[10],q[13]
X q[2]
TOFFOLI q[10],q[2],q[14]
CNOT q[14],q[15]
TOFFOLI q[10],q[2],q[14]
X q[2]
TOFFOLI q[2],q[10],q[13]
X q[2]
TOFFOLI q[9],q[2],q[13]
TOFFOLI q[8],q[2],q[12]

Code 8. Code with three Grover iterations implemented for the third group.
X q[2]
TOFFOLI q[9],q[2],q[12]
TOFFOLI q[2],q[8],q[11]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[6],q[1],q[9]
X q[1]
TOFFOLI q[7],q[1],q[9]
TOFFOLI q[1],q[6],q[8]
TOFFOLI q[5],q[0],q[7]
X q[0]
TOFFOLI q[0],q[5],q[6]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[1],q[0],q[3]
X q[4]
X q[0]
X q[1]
X q[2]
X q[3]
X q[4]
H q[0]
H q[1]
H q[2]
X q[3]
X q[0]
X q[1]
X q[2]
H q[2]
TOFFOLI q[1],q[0],q[2]
X q[0]
X q[1]
H q[2]
H q[0]
H q[1]
X q[2]
X q[0]
X q[1]
H q[2]
X q[2]
TOFFOLI q[1],q[0],q[3]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[0],q[5],q[6]
X q[0]
TOFFOLI q[5],q[0],q[7]
TOFFOLI q[1],q[6],q[8]
TOFFOLI q[7],q[1],q[9]
X q[1]
TOFFOLI q[6],q[1],q[9]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[2],q[8],q[11]
TOFFOLI q[9],q[2],q[12]

Code 8. Code with three Grover iterations implemented for the third group.
X q[2]
TOFFOLI q[8],q[2],q[12]
TOFFOLI q[9],q[2],q[13]
X q[2]
TOFFOLI q[2],q[10],q[13]
X q[2]
TOFFOLI q[10],q[2],q[14]
CNOT q[14],q[15]
TOFFOLI q[10],q[2],q[14]
X q[2]
TOFFOLI q[2],q[10],q[13]
X q[2]
TOFFOLI q[9],q[2],q[13]
TOFFOLI q[8],q[2],q[12]
X q[2]
TOFFOLI q[9],q[2],q[12]
TOFFOLI q[2],q[8],q[11]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[6],q[1],q[9]
X q[1]
TOFFOLI q[7],q[1],q[9]
TOFFOLI q[1],q[6],q[8]
TOFFOLI q[5],q[0],q[7]
X q[0]
TOFFOLI q[0],q[5],q[6]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[1],q[0],q[3]
X q[4]
X q[0]
X q[1]
X q[2]
X q[3]
X q[4]
H q[0]
H q[1]
H q[2]
X q[3]
X q[0]
X q[1]
X q[2]
H q[2]
TOFFOLI q[1],q[0],q[2]
X q[0]
X q[1]
H q[2]
H q[0]
H q[1]
X q[2]
X q[0]
X q[1]
H q[2]
X q[2]

Code 8. Code with three Grover iterations implemented for the third group.
TOFFOLI q[1],q[0],q[3]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[0],q[5],q[6]
X q[0]
TOFFOLI q[5],q[0],q[7]
TOFFOLI q[1],q[6],q[8]
TOFFOLI q[7],q[1],q[9]
X q[1]
TOFFOLI q[6],q[1],q[9]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[2],q[8],q[11]
TOFFOLI q[9],q[2],q[12]
X q[2]
TOFFOLI q[8],q[2],q[12]
TOFFOLI q[9],q[2],q[13]
X q[2]
TOFFOLI q[2],q[10],q[13]
X q[2]
TOFFOLI q[10],q[2],q[14]
CNOT q[14],q[15]
TOFFOLI q[10],q[2],q[14]
X q[2]
TOFFOLI q[2],q[10],q[13]
X q[2]
TOFFOLI q[9],q[2],q[13]
TOFFOLI q[8],q[2],q[12]
X q[2]
TOFFOLI q[9],q[2],q[12]
TOFFOLI q[2],q[8],q[11]
TOFFOLI q[7],q[1],q[10]
TOFFOLI q[6],q[1],q[9]
X q[1]
TOFFOLI q[7],q[1],q[9]
TOFFOLI q[1],q[6],q[8]
TOFFOLI q[5],q[0],q[7]
X q[0]
TOFFOLI q[0],q[5],q[6]
TOFFOLI q[4],q[3],q[5]
TOFFOLI q[1],q[2],q[4]
TOFFOLI q[1],q[0],q[3]
X q[4]
X q[0]
X q[1]
X q[2]
X q[3]
H q[0]
H q[1]
H q[2]
X q[0]

Code 8. Code with three Grover iterations implemented for the third group.
X q[1]
X q[2]
H q[2]
TOFFOLI q[1],q[0],q[2]
X q[0]
X q[1]
H q[2]
H q[0]
H q[1]
X q[2]
H q[2]
MEASURE q[0],c[0]
MEASURE q[1],c[1]
MEASURE q[2],c[2]

Appendix C: The code of the classical solution

The code of screening the feasible solutions.

The code of counting the number of vertices.

The code of finding the optimal solution of the vertex cover problem.