Qiskit

Introduction

https://quantum-computing.ibm.com/

https://quantum-computing.ibm.com/challenges/fall-2020

https://quantum-computing.ibm.com/jupyter/user/IBMQuantumChallenge2020/week-1/ex_1a_en.ipynb

Theory

Installation

Installation https://qiskit.org/documentation/install.html

conda create -n qiskit python=3
conda activate qiskit
pip install qiskit _OR_ pip install qiskit[visualization]

Setting Up Qiskit

https://qiskit.org/textbook/ch-states/representing-qubit-states.html

qc = QuantumCircuit(1)  # Create a quantum circuit with one qubit
initial_state = [0,1]   # Define initial_state as |1>
qc.initialize(initial_state, 0) # Apply initialisation operation to the 0th qubit
qc.draw('text')  # Let's view our circuit (text drawing is required for the 'Initialize' gate due to a known bug in qiskit)
result = execute(qc,backend).result() # Do the simulation, returning the result
out_state = result.get_statevector()
print(out_state) # Display the output state vector

qc.measure_all()
qc.draw()
result = execute(qc,backend).result()
counts = result.get_counts()
plot_histogram(counts)

Take superposition as initial state

initial_state = [1/sqrt(2), 1j/sqrt(2)]  # Define state |q>

The Bloch Sphere

from qiskit_textbook.widgets import plot_bloch_vector_spherical
coords = [pi/2,0,1] # [Theta, Phi, Radius]
plot_bloch_vector_spherical(coords) # Bloch Vector with spherical coordinates

Qiskit allows measuring in the Z-basis, only.

Quantum Gates

Identity gate.

Pauli X gate. $\sigma _{x}=X={\begin{bmatrix}0&1\\1&0\end{bmatrix}}=|0\rangle \langle 1|+|1\rangle \langle 0|$

Pauli Y gate $\sigma _{y}=Y={\begin{bmatrix}0&-i\\i&0\end{bmatrix}}=-i|0\rangle \langle 1|+i|1\rangle \langle 0|$

Pauli Z gate $\sigma _{z}=Z={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}=|0\rangle \langle 0|-|1\rangle \langle 1|$

Hadamard gate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \tfrac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} = |+\rangle\langle0| + |-\rangle\langle1| }

R gate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_\phi = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\phi} \end{bmatrix}}

S gate or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt Z} gate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = \begin{bmatrix} 1 & 0 \\ 0 & e^{\frac{i\pi}{2}} \end{bmatrix}}

T gate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T = \begin{bmatrix} 1 & 0 \\ 0 & e^{\frac{i\pi}{4}} \end{bmatrix}}

U1 gate: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_1 = U_3(0, 0, \lambda) = U_1 = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\lambda}\\ \end{bmatrix} }

U2 gate: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_2 = U_3(\tfrac{\pi}{2}, \phi, \lambda) = \tfrac{1}{\sqrt{2}}\begin{bmatrix} 1 & -e^{i\lambda} \\ e^{i\phi} & e^{i\lambda+i\phi} \end{bmatrix} }

qc = QuantumCircuit(1)
qc.x(0)
#qc.y(0) # Y-gate on qubit 0
#qc.z(0) # Z-gate on qubit 0
#qc.rz(pi/4, 0)
#qc.s(0)   # Apply S-gate to qubit 0
#qc.sdg(0) # Apply Sdg-gate to qubit 0
qc.t(0)   # Apply T-gate to qubit 0
qc.tdg(0) # Apply Tdg-gate to qubit 0

qc.draw('mpl')
# Let's see the result
backend = Aer.get_backend('statevector_simulator')
out = execute(qc,backend).result().get_statevector()
plot_bloch_multivector(out)

Multiple Qubits

Eg. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\otimes H = \begin{bmatrix} 0 &H\\H&0\end{bmatrix}} .

backend = Aer.get_backend('unitary_simulator')
unitary = execute(qc,backend).result().get_unitary()
#
# In Jupyter Notebooks we can display this nicely using Latex.
# If not using Jupyter Notebooks you may need to remove the 
# array_to_latex function and use print(unitary) instead.
from qiskit_textbook.tools import array_to_latex
array_to_latex(unitary, pretext="\\text{Circuit = }\n")

Eg. CNOT is a conditional gate that performs an X-gate on the second qubit, if the state of the first qubit (control) is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1\rangle} . Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle CNOT= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ \end{bmatrix} } . This matrix swaps the amplitudes of |01⟩ and |11⟩ in the statevector.

CNOT if a control qubit is on the superposition: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{CNOT}|0{+}\rangle = \tfrac{1}{\sqrt{2}}(|00\rangle + |11\rangle)} , which is Bell State. Entanglement, but no-communication theorem.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{CNOT}|++\rangle = |++\rangle)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{CNOT}|--\rangle = = \tfrac{1}{2}(|00\rangle - |01\rangle - |10\rangle + |11\rangle) = |{-}{-}\rangle }

qc = QuantumCircuit(2)
# Apply H-gate to the first:
qc.h(0)
# Apply a CNOT:
qc.cx(0,1)
qc.draw()
#
# Let's see the result:
backend = Aer.get_backend('statevector_simulator')
final_state = execute(qc,backend).result().get_statevector()
# Print the statevector neatly:
array_to_latex(final_state, pretext="\\text{Statevector = }")
#
results = execute(qc,backend).result().get_counts()
plot_histogram(results)

Entanglement.

Exercises

Week 1

Week 2

Week 3