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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 <img src="../../../../images/qiskit-heading.gif" alt="Note
Step1:  In this section, we first judge the version of Python and import the packages of qiskit, math to implement the following code. We show our algorithm on the ibm_qasm_simulator, if you need to run it on the real quantum conputer, please remove the "#" in frint of "import Qconfig".
Step2:  Here we define the number pi in the math lib, because we need to use u3 gate. And we also define a list about the parameter theta which we need to use in the u3 gate. As the same above, if you want to implement on the real quantum comnputer, please remove the symbol "#" and configure your local Qconfig.py file.
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Python Code:
# import math lib
from math import pi
# import Qiskit
from qiskit import Aer, IBMQ, execute
from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister
# import basic plot tools
from qiskit.tools.visualization import plot_histogram
# To use local qasm simulator
backend = Aer.get_backend('qasm_simulator')
Explanation: <img src="../../../../images/qiskit-heading.gif" alt="Note: In order for images to show up in this jupyter notebook you need to select File => Trusted Notebook" width="500 px" align="left">
Quantum K-Means algorithm
The latest version of this notebook is available on https://github.com/qiskit/qiskit-tutorial.
Contributors
Shan Jin, Xi He, Xiaokai Hou, Li Sun, Dingding Wen, Shaojun Wu and Xiaoting Wang$^{1}$
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China，Chengdu, China，610051
Introduction
Clustering algorithm is a typical unsupervised learning algorithm, which is mainly used to automatically classify similar samples into one category.In the clustering algorithm, according to the similarity between the samples, the samples are divided into different categories. For different similarity calculation methods, different clustering results will be obtained. The commonly used similarity calculation method is the Euclidean distance method.
What we want to show is the quantum K-Means algorithm. The K-Means algorithm is a distance-based clustering algorithm that uses distance as an evaluation index for similarity, that is, the closer the distance between two objects is, the greater the similarity. The algorithm considers the cluster to be composed of objects that are close together, so the compact and independent cluster is the ultimate target.
Experiment design
The implementation of the quantum K-Means algorithm mainly uses the swap test to compare the distances among the input data points. Select K points randomly from N data points as centroids, measure the distance from each point to each centroid, and assign it to the nearest centroid- class, recalculate centroids of each class that has been obtained, and iterate 2 to 3 steps until the new centroid is equal to or less than the specified threshold, and the algorithm ends. In our example, we selected 6 data points, 2 centroids, and used the swap test circuit to calculate the distance. Finally, we obtained two clusters of data points.
$|0\rangle$ is an auxiliary qubit, through left $H$ gate, it will be changed to $\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$. Then under the control of $|1\rangle$, the circuit will swap two vectors $|x\rangle$ and $|y\rangle$. Finally, we get the result at the right end of the circuit:
$$|0_{anc}\rangle |x\rangle |y\rangle \rightarrow \frac{1}{2}|0_{anc}\rangle(|xy\rangle + |yx\rangle) + \frac{1}{2}|1_{anc}\rangle(|xy\rangle - |yx\rangle)$$
If we measure auxiliary qubit alone, then the probability of final state in the ground state $|1\rangle$ is:
$$P(|1_{anc}\rangle) = \frac{1}{2} - \frac{1}{2}|\langle x | y \rangle|^2$$
If we measure auxiliary qubit alone, then the probability of final state in the ground state $|1\rangle$ is:
$$Euclidean \ distance = \sqrt{(2 - 2|\langle x | y \rangle|)}$$
So, we can see that the probability of measuring $|1\rangle$ has positive correlation with the Euclidean distance.
The schematic diagram of quantum K-Means is as the follow picture.[1] 
<img src="../images/k_means_circuit.png">
To make our algorithm can be run using qiskit, we design a more detailed circuit to achieve our algorithm. 
|
Quantum K-Means circuit
<img src="../images/k_means.png">
Data points
<table border="1">
<tr>
<td>point num</td>
<td>theta</td>
<td>phi</td>
<td>lam</td>
<td>x</td>
<td>y</td>
</tr>
<tr>
<td>1</td>
<td>0.01</td>
<td>pi</td>
<td>pi</td>
<td>0.710633</td>
<td>0.703562</td>
</tr>
<tr>
<td>2</td>
<td>0.02</td>
<td>pi</td>
<td>pi</td>
<td>0.714142</td>
<td>0.7</td>
</tr>
<tr>
<td>3</td>
<td>0.03</td>
<td>pi</td>
<td>pi</td>
<td>0.717633</td>
<td>0.696421</td>
</tr>
<tr>
<td>4</td>
<td>0.04</td>
<td>pi</td>
<td>pi</td>
<td>0.721107</td>
<td>0.692824</td>
</tr>
<tr>
<td>5</td>
<td>0.05</td>
<td>pi</td>
<td>pi</td>
<td>0.724562</td>
<td>0.68921</td>
</tr>
<tr>
<td>6</td>
<td>1.31</td>
<td>pi</td>
<td>pi</td>
<td>0.886811</td>
<td>0.462132</td>
</tr>
<tr>
<td>7</td>
<td>1.32</td>
<td>pi</td>
<td>pi</td>
<td>0.889111</td>
<td>0.457692</td>
</tr>
<tr>
<td>8</td>
<td>1.33</td>
<td>pi</td>
<td>pi</td>
<td>0.891388</td>
<td>0.453241</td>
</tr>
<tr>
<td>9</td>
<td>1.34</td>
<td>pi</td>
<td>pi</td>
<td>0.893643</td>
<td>0.448779</td>
</tr>
<tr>
<td>10</td>
<td>1.35</td>
<td>pi</td>
<td>pi</td>
<td>0.895876</td>
<td>0.444305</td>
</tr>
## Quantum K-Means algorithm program
End of explanation
theta_list = [0.01, 0.02, 0.03, 0.04, 0.05, 1.31, 1.32, 1.33, 1.34, 1.35]
Explanation: In this section, we first judge the version of Python and import the packages of qiskit, math to implement the following code. We show our algorithm on the ibm_qasm_simulator, if you need to run it on the real quantum conputer, please remove the "#" in frint of "import Qconfig".
End of explanation
# create Quantum Register called "qr" with 5 qubits
qr = QuantumRegister(5, name="qr")
# create Classical Register called "cr" with 5 bits
cr = ClassicalRegister(5, name="cr")
    
# Creating Quantum Circuit called "qc" involving your Quantum Register "qr"
# and your Classical Register "cr"
qc = QuantumCircuit(qr, cr, name="k_means")
    
#Define a loop to compute the distance between each pair of points
for i in range(9):
    for j in range(1,10-i):
        # Set the parament theta about different point
        theta_1 = theta_list[i]
        theta_2 = theta_list[i+j]
        #Achieve the quantum circuit via qiskit
        qc.h(qr[2])
        qc.h(qr[1])
        qc.h(qr[4])
        qc.u3(theta_1, pi, pi, qr[1])
        qc.u3(theta_2, pi, pi, qr[4])
        qc.cswap(qr[2], qr[1], qr[4])
        qc.h(qr[2])
                
        qc.measure(qr[2], cr[2])
        qc.reset(qr)
            
        job = execute(qc,   backend=backend, shots=1024)
        result = job.result()   
        print(result)
        print('theta_1:' + str(theta_1))
        print('theta_2:' + str(theta_2))
#         print( result.get_data(qc))
        plot_histogram(result.get_counts())
Explanation: Here we define the number pi in the math lib, because we need to use u3 gate. And we also define a list about the parameter theta which we need to use in the u3 gate. As the same above, if you want to implement on the real quantum comnputer, please remove the symbol "#" and configure your local Qconfig.py file.
End of explanation
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