Lab02_Single_Qubit_Gates.ipynb

Kernel: Python 3

Lab 2 Single Qubit Gates

Prerequisite Ch.1.3 Representing Qubit States Ch.1.4 Single Qubit Gates

Other relevant materials Grokking the Bloch Sphere

import numpy as np

# Importing standard Qiskit libraries
from qiskit import QuantumCircuit, transpile, Aer, IBMQ, execute
from qiskit.tools.jupyter import *
from qiskit.visualization import *
from ibm_quantum_widgets import *
from qiskit.providers.aer import QasmSimulator

# Loading your IBM Quantum account(s)
provider = IBMQ.load_account()
backend = Aer.get_backend('statevector_simulator')

Part 1 - Effect of Single-Qubit Gates on state |0>

Goal

Create quantum circuits to apply various single qubit gates on state |0> and understand the change in state and phase of the qubit.

To see the effect of each of the gates, we will take a single circuit with 4 qubits and apply a different gate to each of the qubits to plot the values of each of those qubits on the blocsphere.

qc1 = QuantumCircuit(4)

# perform gate operations on individual qubits
qc1.x(0)
qc1.y(1)
qc1.z(2)
qc1.s(3)

# Draw circuit
qc1.draw()

# Plot blochshere
out1 = execute(qc1,backend).result().get_statevector()
plot_bloch_multivector(out1)

/opt/conda/lib/python3.8/site-packages/qiskit/visualization/bloch.py:69: MatplotlibDeprecationWarning: 
The M attribute was deprecated in Matplotlib 3.4 and will be removed two minor releases later. Use self.axes.M instead.
  x_s, y_s, _ = proj3d.proj_transform(xs3d, ys3d, zs3d, renderer.M)

Effect on Qubit on application of Gate	Satevector	QSphere Plot	Statevector (Post Measurement)
Input State = $\begin{pmatrix}1+0j & 0+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0+0j & 1+0j\end{pmatrix}$ - qubit has probability 1 of being in state ‘1’ Post measurement, quit state is ‘1’ with phase 0
Input State = $\begin{pmatrix}1+0j & 0+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0+0j & 0+1j\end{pmatrix}$ - qubit has probability 1 of being in state ‘1’ Post measurement, quit state is ‘1’ with phase pi/2
Input State = $\begin{pmatrix}1+0j & 0+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}1+0j & 0+0j\end{pmatrix}$ - qubit has probability 1 of being in state ‘0’ Post measurement, quit state is ‘0’ with phase 0
Input State = $\begin{pmatrix}1+0j & 0+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}1+0j & 0+0j\end{pmatrix}$ - qubit has probability 1 of being in state ‘0’ Post measurement, quit state is ‘0’ with phase 0

Part 2 - Effect of Single-Qubit Gates on state |1>

Goal

Create quantum circuits to apply various single qubit gates on state |1> and understand the change in state and phase of the qubit.

To see the effect of each of the gates, we will take a single circuit with 4 qubits and apply a different gate to each of the qubits to plot the values of each of those qubits on the blocsphere.

qc2 = QuantumCircuit(4)

# initialize qubits
qc2.x(range(4))

# perform gate operations on individual qubits
qc2.x(0)
qc2.y(1)
qc2.z(2)
qc2.s(3)

# Draw circuit
qc2.draw()

# Plot blochshere
out2 = execute(qc2,backend).result().get_statevector()
plot_bloch_multivector(out2)

Effect on Qubit on application of Gate	Satevector	QSphere Plot	Statevector (Post Measurement)
Input State = $\begin{pmatrix}0+0j & 1+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}1+0j & 0+0j\end{pmatrix}$ - qubit has probability 1 of being in state ‘0’ Post measurement, quit state is ‘0’ with phase 0
Input State = $\begin{pmatrix}0+0j & 1+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0-1j & 0+0j\end{pmatrix}$ - qubit has probability 1 of being in state ‘0’ Post measurement, quit state is ‘0’ with phase 3pi/2
Input State = $\begin{pmatrix}0+0j & 1+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0+0j & -1+0j\end{pmatrix}$ - qubit has probability 1 of being in state ‘1’ Post measurement, quit state is ‘1’ with phase pi
Input State = $\begin{pmatrix}0+0j & 1+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0+0j & 0+1j\end{pmatrix}$ - qubit has probability 1 of being in state ‘1’ Post measurement, quit state is ‘1’ with phase pi/2

Part 3 - Effect of Single-Qubit Gates on state |+>

Goal

Create quantum circuits to apply various single qubit gates on state |+> and understand the change in state and phase of the qubit.

To see the effect of each of the gates, we will take a single circuit with 4 qubits and apply a different gate to each of the qubits to plot the values of each of those qubits on the blocsphere.

qc3 = QuantumCircuit(4)

# initialize qubits
qc3.h(range(4))

# perform gate operations on individual qubits
qc3.x(0)
qc3.y(1)
qc3.z(2)
qc3.s(3)

# Draw circuit
qc3.draw()

# Plot blochshere
out3 = execute(qc3,backend).result().get_statevector()
plot_bloch_multivector(out3)

Effect on Qubit on application of Gate	Satevector	QSphere Plot	Probability (Histogram)
Input State = $\begin{pmatrix}0.707+0j & 0.707+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0.707+0j & 0.707+0j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’
Input State = $\begin{pmatrix}0.707+0j & 0.707+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0-0.707j & 0+0.707j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’
Input State = $\begin{pmatrix}0.707+0j & 0.707+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0.707+0j & -0.707+0j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’
Input State = $\begin{pmatrix}0.707+0j & 0.707+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0.707+0j & 0+0.707j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’

Part 4 - Effect of Single-Qubit Gates on state |->

Goal

Create quantum circuits to apply various single qubit gates on state |-> and understand the change in state and phase of the qubit.

To see the effect of each of the gates, we will take a single circuit with 4 qubits and apply a different gate to each of the qubits to plot the values of each of those qubits on the blocsphere.

qc4 = QuantumCircuit(4)

# initialize qubits
qc4.x(range(4))
qc4.h(range(4))

# perform gate operations on individual qubits
qc4.x(0)
qc4.y(1)
qc4.z(2)
qc4.s(3)

# Draw circuit
qc4.draw()

# Plot blochshere
out4 = execute(qc4,backend).result().get_statevector()
plot_bloch_multivector(out4)

Effect on Qubit on application of Gate	Satevector	QSphere Plot	Probability (Histogram)
Input State = $\begin{pmatrix}0.707+0j & -0.707+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}-0.707+0j & 0.707+0j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’
Input State = $\begin{pmatrix}0.707+0j & -0.707+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0+0.707j & 0+0.707j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’
Input State = $\begin{pmatrix}0.707+0j & -0.707+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0.707+0j & 0.707+0j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’
Input State = $\begin{pmatrix}0.707+0j & -0.707+0j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0.707+0j & 0-0.707j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’

Part 5 - Effect of Single-Qubit Gates on state |i>

Goal

Create quantum circuits to apply various single qubit gates on state |i> and understand the change in state and phase of the qubit.

To see the effect of each of the gates, we will take a single circuit with 4 qubits and apply a different gate to each of the qubits to plot the values of each of those qubits on the blocsphere.

qc5 = QuantumCircuit(4)

# initialize qubits
qc5.h(range(4))
qc5.s(range(4))

# perform gate operations on individual qubits
qc5.x(0)
qc5.y(1)
qc5.z(2)
qc5.s(3)

# Draw circuit
qc5.draw()

# Plot blochshere
out5 = execute(qc5,backend).result().get_statevector()
plot_bloch_multivector(out5)

Effect on Qubit on application of Gate	Satevector	QSphere Plot	Probability (Histogram)
Input State = $\begin{pmatrix}0.707+0j & 0+0.707j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0+0.707j & 0.707+0j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’
Input State = $\begin{pmatrix}0.707+0j & 0+0.707j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0.707+0j & 0+0.707j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’
Input State = $\begin{pmatrix}0.707+0j & 0+0.707j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0.707+0j & 0-0.707j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’
Input State = $\begin{pmatrix}0.707+0j & 0+0.707j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0.707+0j & -0.707+0j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’

Part 6 - Effect of Single-Qubit Gates on state |-i>

Goal

Create quantum circuits to apply various single qubit gates on state |-i> and understand the change in state and phase of the qubit.

To see the effect of each of the gates, we will take a single circuit with 4 qubits and apply a different gate to each of the qubits to plot the values of each of those qubits on the blocsphere.

qc6 = QuantumCircuit(4)

# initialize qubits
qc6.x(range(4))
qc6.h(range(4))
qc6.s(range(4))

# perform gate operations on individual qubits
qc6.x(0)
qc6.y(1)
qc6.z(2)
qc6.s(3)

# Draw circuit
qc6.draw()

# Plot blochshere
out6 = execute(qc6,backend).result().get_statevector()
plot_bloch_multivector(out6)

Effect on Qubit on application of Gate	Satevector	QSphere Plot	Probability (Histogram)
Input State = $\begin{pmatrix}0.707+0j & 0-0.707j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0-0.707j & 0.707+0j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’
Input State = $\begin{pmatrix}0.707+0j & 0-0.707j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}-0.707+0j & 0+0.707j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’
Input State = $\begin{pmatrix}0.707+0j & 0-0.707j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0.707+0j & 0+0.707j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’
Input State = $\begin{pmatrix}0.707+0j & 0-0.707j\end{pmatrix}$ Before measurement, - qubit state = $\begin{pmatrix}0.707+0j & 0.707+0j\end{pmatrix}$ - qubit has probability 0.5 of being in each of the states ‘0’ and ‘1’

import qiskit
qiskit.__qiskit_version__