One Qubit Gates - Clifford family¶

In [1]:

from qiskit import *
from math import pi
import numpy as np
from qiskit.visualization import *
import matplotlib.pyplot as plt

Clifford gates¶

Hadamard gate¶

$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix}= u2(0,\pi)$

In [2]:

qc = QuantumCircuit(1)
qc.h(0)
qc.draw(output='mpl')

Out[2]:

In [8]:

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

[0.70710678+0.j 0.70710678+0.j]

Out[8]:

In [9]:

# Run the quantum circuit on a unitary simulator backend
backend = Aer.get_backend('unitary_simulator')
job = execute(qc, backend)
result = job.result()

# Show the results
print(result.get_unitary(qc, decimals=3))

[[ 0.707+0.j  0.707-0.j]
 [ 0.707+0.j -0.707+0.j]]

Experiment 1:¶

In [10]:

qc = QuantumCircuit(2)
qc.h(0)
qc.x(1)
qc.barrier()
qc.h(1)
qc.draw(output='mpl')

Out[10]:

In [11]:

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

[ 0.5-6.123234e-17j  0.5-6.123234e-17j -0.5+6.123234e-17j
 -0.5+6.123234e-17j]

Out[11]:

$(H \otimes I) \times ( X\otimes H)$

In [12]:

# Run the quantum circuit on a unitary simulator backend
backend = Aer.get_backend('unitary_simulator')
job = execute(qc, backend)
result = job.result()

# Show the results
print(result.get_unitary(qc, decimals=3))

[[ 0.5-0.j  0.5-0.j  0.5+0.j  0.5-0.j]
 [ 0.5-0.j -0.5+0.j  0.5+0.j -0.5+0.j]
 [-0.5+0.j -0.5+0.j  0.5+0.j  0.5-0.j]
 [-0.5+0.j  0.5-0.j  0.5+0.j -0.5+0.j]]

Experiment 2:¶

In [13]:

qc = QuantumCircuit(3)
qc.h(0)
qc.h(1)
qc.h(2)
qc.barrier()
qc1 = qc.copy()
qc.x(0)
qc.y(1)
qc.z(2)
qc.barrier()
qc2 = qc.copy()
qc.x(0)
qc.y(1)
qc.z(2)
qc.barrier()
qc3 = qc.copy()
qc.h(0)
qc.h(1)
qc.h(2)
qc.draw(output='mpl')

Out[13]:

In [14]:

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

[ 1.00000000e+00-1.83697020e-16j  7.49879891e-33+6.12323400e-17j
  3.74939946e-33+6.12323400e-17j -3.74939946e-33+2.29584502e-49j
  0.00000000e+00+6.12323400e-17j -3.74939946e-33+0.00000000e+00j
 -3.74939946e-33+0.00000000e+00j  0.00000000e+00-2.29584502e-49j]

Out[14]:

In [15]:

# Run the quantum circuit on a unitary simulator backend
backend = Aer.get_backend('unitary_simulator')
job = execute(qc, backend)
result = job.result()

# Show the results
print(result.get_unitary(qc, decimals=3))

[[ 1.-0.j  0.+0.j  0.+0.j -0.+0.j  0.+0.j -0.+0.j -0.+0.j  0.-0.j]
 [ 0.+0.j  1.-0.j -0.+0.j  0.+0.j -0.+0.j  0.+0.j  0.-0.j -0.+0.j]
 [ 0.+0.j -0.+0.j  1.-0.j  0.+0.j -0.+0.j  0.-0.j  0.+0.j -0.+0.j]
 [-0.+0.j  0.+0.j  0.+0.j  1.-0.j  0.-0.j -0.+0.j -0.+0.j  0.+0.j]
 [ 0.+0.j -0.+0.j -0.+0.j  0.-0.j  1.-0.j  0.+0.j  0.+0.j -0.+0.j]
 [-0.+0.j  0.+0.j  0.-0.j -0.+0.j  0.+0.j  1.-0.j -0.+0.j  0.+0.j]
 [-0.+0.j  0.-0.j  0.+0.j -0.+0.j  0.+0.j -0.+0.j  1.-0.j  0.+0.j]
 [ 0.-0.j -0.+0.j -0.+0.j  0.+0.j -0.+0.j  0.+0.j  0.+0.j  1.-0.j]]

$\psi_{1}$ at first barrier¶

In [16]:

# Let's see the result
backend = Aer.get_backend('statevector_simulator')
out = execute(qc1,backend).result().get_statevector()
print(out)
plot_bloch_multivector(out)

[0.35355339+0.j 0.35355339+0.j 0.35355339+0.j 0.35355339+0.j
 0.35355339+0.j 0.35355339+0.j 0.35355339+0.j 0.35355339+0.j]

Out[16]:

$\psi_{2}$ at second barrier¶

In [17]:

# Let's see the result
backend = Aer.get_backend('statevector_simulator')
out = execute(qc2,backend).result().get_statevector()
print(out)
plot_bloch_multivector(out)

[ 0.-0.35355339j  0.-0.35355339j  0.+0.35355339j  0.+0.35355339j
  0.+0.35355339j  0.+0.35355339j -0.-0.35355339j -0.-0.35355339j]

Out[17]:

$\psi_{3}$ at third barrier¶

In [18]:

# Let's see the result
backend = Aer.get_backend('statevector_simulator')
out = execute(qc3,backend).result().get_statevector()
print(out)
plot_bloch_multivector(out)

[0.35355339+0.j 0.35355339+0.j 0.35355339+0.j 0.35355339+0.j
 0.35355339+0.j 0.35355339+0.j 0.35355339+0.j 0.35355339+0.j]

Out[18]:

S and T Gates¶

$S$ (or, $\sqrt{Z}$ phase) gate¶

$S = \begin{pmatrix} 1 & 0\\ 0 & i \end{pmatrix}= u1(\pi/2)$

In [19]:

qc = QuantumCircuit(1)
qc.s(0)
qc.draw(output='mpl')

Out[19]:

$S^{\dagger}$ (or, conjugate of $\sqrt{Z}$ phase) gate¶

$S^{\dagger} = \begin{pmatrix} 1 & 0\\ 0 & -i \end{pmatrix}= u1(-\pi/2)$

In [20]:

qc = QuantumCircuit(1)
qc.sdg(0)
qc.draw(output='mpl')

Out[20]:

$T$ (or, $\sqrt{S}$ phase) gate¶

$T = \begin{pmatrix} 1 & 0\\ 0 & e^{i \pi/4} \end{pmatrix}= u1(\pi/4)$

In [21]:

qc = QuantumCircuit(1)
qc.t(0)
qc.draw(output='mpl')

Out[21]:

$T^{\dagger}$ (or, conjugate of $\sqrt{S}$ phase) gate¶

$T^{\dagger} = \begin{pmatrix} 1 & 0\\ 0 & e^{-i \pi/4} \end{pmatrix}= u1(-pi/4)$

They can be added as below.

In [22]:

qc = QuantumCircuit(1)
qc.tdg(0)
qc.draw(output='mpl')

Out[22]:

Lets Play a Quantum Game!¶

Task: Start from |0> and visit all states shown above:

In [27]:

qc = QuantumCircuit(1)
qc.x(0)
qc.barrier()
qc1 = qc.copy()
qc.h(0)
qc.barrier()
qc2 = qc.copy()
qc.s(0)
qc.barrier()
qc3 = qc.copy()
qc.s(0)
qc.barrier()
qc4 = qc.copy()
qc.s(0)
qc.barrier()
qc5 = qc.copy()
qc.sdg(0)
qc.h(0)
qc.barrier()
qc6 = qc.copy()
qc.draw(output='mpl')

Out[27]:

In [28]:

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

[1.-1.8369702e-16j 0.+6.1232340e-17j]

Out[28]:

In [30]:

# Let's see the result
backend = Aer.get_backend('statevector_simulator')
for qc in [qc1,qc2,qc3,qc4,qc5,qc6]:
    out = execute(qc,backend).result().get_statevector()
    print(out)

[0.+0.j 1.+0.j]
[ 0.70710678-8.65956056e-17j -0.70710678+8.65956056e-17j]
[ 7.07106781e-01-8.65956056e-17j -8.65956056e-17-7.07106781e-01j]
[0.70710678-8.65956056e-17j 0.70710678-8.65956056e-17j]
[7.07106781e-01-8.65956056e-17j 8.65956056e-17+7.07106781e-01j]
[1.-1.8369702e-16j 0.+6.1232340e-17j]

In [ ]:

One Qubit Gates - Clifford family¶

Clifford gates¶

Hadamard gate¶

Experiment 1:¶

Experiment 2:¶

ψ1\psi_{1} at first barrier¶

ψ2\psi_{2} at second barrier¶

ψ3\psi_{3} at third barrier¶

S and T Gates¶

SS (or, √Z\sqrt{Z} phase) gate¶

S†S^{\dagger} (or, conjugate of √Z\sqrt{Z} phase) gate¶

TT (or, √S\sqrt{S} phase) gate¶

T†T^{\dagger} (or, conjugate of √S\sqrt{S} phase) gate¶