Full explanation of these examples https://programmingaltanai.wordpress.com/2019/12/22/quantum-computing/

Qubits

Current computers manipulate individual bits as binary 0 and 1 states but Quantum computers leverage quantum mechanical phenomena to manipulate information, relying on quantum bits, or qubits.

Quantum Registers

A Quantum computer stores its data in quantum registers and they can store data in quantum superposition.

These examples of Quantum computing are reused with some modification from IBM's Qiskit https://github.com/Qiskit . Qiskit is an open-source framework for working with quantum computers at the level of circuits, pulses, and algorithms.

Circuit

quantum circuit usually starts with the qubits in the |0,…,0> state and evolve with results of operation.

Gates

A quantum gate is a basic quantum circuit operating on a small number of qubits. Number of qubits in the input and output of the gate must be equal

gate rotates the states |0⟩ and |1⟩ to |+⟩ and |−⟩ . It acts on a single qubit. It maps the basis state.

|0} to (|0}+|1}) / $latex \sqrt{2}  $ and |1} to (|0}-|1}) / $latex \sqrt{2}  $

This means that measurement will have equal probabilities to become 0 or 1 thereby creating a state of superimposition.

controlled-X gate also called  controlled-NOT acts on a pair of qubits, with one acting as ‘control’ and the other as ‘target. whenever the control is in state |1⟩ 

rotating the qubit state around the x , y or z axis by the given angle. On the Bloch sphere

• Rx gate - corresponds to rotating the qubit state around the x axis by the given angle.
• Ry Gate - corresponds to rotating the qubit state around the y axis by the given angle.
• Rz Gate - corresponds to rotating the qubit state around the z axis by the given angle.

Acts on a single qubit. It equates to a rotation around the X, Y and Z-axis  by  radians

• Pauli-X maps |0} to |1} and |1} to |0} . Thus also called bit flip
• Pauli-Y maps |0} to i|1} and |1} to -i|0} . Thus also called bit flip
• Pauli-Z matrix leaves the basis state |0} unchanged and maps |1} to -|1} . It is also called phase-flip

• U1 - Equivalent to Rz.
• U2 - two parameters control two different rotations within the gate. Has a duration of one unit of gate time.
• U3 -three parameters allow the construction of any single qubit gate, Has a duration of one unit of gate time.

Swaps the state of 2 qubits 

performs half-way of a two-qubit swap

Measurement

Circuit representation of measurement. The two lines on the right hand side represents a classical bit, the single line on the left hand side represents a qubit.

The two lines on the right hand side represents a classical bit, the single line on the left hand side represents a qubit.

Following program creates a quantum register Circuit to perform operation and a classical Register to store the information once a result has been obtained.

Some simple programming constructs examples:

Initialising with 2 qubits in the zero state

circuit = QuantumCircuit(2, 2)

or Quantum Circuit acting on a quantum register of three qubits where by default, each qubit in the register is initialized to |0⟩

circ = QuantumCircuit(3)

A Hadamard gate H on qubit 0, which puts it into the superposition state

circ.h(0)

Execute the circuit based on some simulator and give no of shots

job = execute(circuit, simulator, shots=100)

Lets try some programs 

qr = QuantumRegister(2, name="q")
cr = ClassicalRegister(1, name="result")
circuit = QuantumCircuit(qr, cr)

# Inverse qr state , since it was initially 0 by default it becomes 1 
circuit.x(qr)

# measure qbit 0 
circuit.measure(qr[0], cr)
circuit.draw()

job = execute(circuit, backend=local_simulator)
job.result().get_counts()

output for flipped state of X is 1 with maximum probability {'1': 1024}

To demonstrate the CX gate which flips the value of target qubit only when control qubits state is 1 , we need following steps

step 1  - declare 2 Quantum registers ( target i1 and control i2) and declare 1 classical register for result

i1 = QuantumRegister(1, name='input1')
i2 = QuantumRegister(1, name='input2')
cr = ClassicalRegister(1, name="result")

step 2 - First invert control state i1 , that is set i1 to 1 . Then apply controlled x on target i1 when i2 is control is 1

circuit = QuantumCircuit(i1, i2, cr)
circuit.x(i1) 
circuit.cx(i1, i2) 

Measure i2 and get count of probability

circuit.measure(i2, cr)
job = execute(circuit, backend=local_simulator)
job.result().get_counts()

Output {'1': 1024}

Transforms two qubits, both of which started off in the state |0⟩, into the state |00⟩ + |11⟩√2

Bell states - 50-50 chance of measuring 0 or 1 on any one of the 2 qubits.

Imports statements from Qiskit

import numpy as np
from qiskit import(  QuantumCircuit,  execute,  Aer)
from qiskit.visualization import plot_histogram

Program to add Gates to the circuit one-by-one to form the Bell state |𝜓⟩=(|00⟩+|11⟩)/ √2

Use Aer's qasm_simulator

simulator = Aer.get_backend('qasm_simulator')
circuit = QuantumCircuit(2, 2)

Add a H gate on qubit 0 , making it go in superimposiion

circuit.h(0)

Add a CX (CNOT) gate on control qubit 0 and target qubit 1 making them go into state of entanglement

circuit.cx(0, 1)

Measure qbit 0 and 1 into classical 0 and 1

circuit.measure([0,1], [0,1])

Execute the circuit on the qasm simulator and grab results from the job

job = execute(circuit, simulator, shots=1000)
result = job.result()

Return counts for the occurrence of states 00 and 11 

counts = result.get_counts(circuit)
print("\nTotal count for 00 and 11 are:",counts)

Under ideal conditions ( simulator ) it may be equal  Total count for 00 and 11 are: {'00': 500, '11': 500}

However with actual super computers , results will differ by a small margin  Total count for 00 and 11 are: {'00': 553, '11': 447}

Draw the circuit

circuit.draw()