CNOT Gate

CNOT gate

Multiqubit gate requires multiple qubit circuit.

%matplotlib inline
import numpy as np
import IPython
import matplotlib.pyplot as plt
from qiskit import QuantumCircuit
from qiskit import BasicAer
from qiskit.tools.jupyter import *
from qiskit.visualization import *
import seaborn as sns
sns.set()

from helper import *
import os
import glob
import moviepy.editor as mpy

CNOT gate

This gate is a conditional gate that performs an X-gate on the second qubit (target), if the state of the first qubit (control) is |1⟩ .

qc = QuantumCircuit(2)
qc.cx(0,1)
style = {'backgroundcolor': 'lavender'}
qc.draw(output='mpl', style = style)

Let's see the matrix element of the CNOT gate.

getMatrix(qc)

matrix([[1.+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, 1.+0.j, 0.+0.j],
        [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]])

CNOT gate has two representation based on which qubit is target.

$$\text{CNOT} = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ \end{array} \right) , \quad \text{CNOT} = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{array} \right)$$

SWAP gate

• What will happen if we apply 3 CNOT gates as shown below?

qc = QuantumCircuit(2)
qc.h(0)
qc.barrier()
qc.cx(0,1)
qc.cx(1,0)
qc.cx(0,1)
qc.barrier()
qc.h(1)
print(getStateVector(qc))
style = {'backgroundcolor': 'lavender'}
qc.draw(output='mpl', style = style)

[1.-6.123234e-17j 0.+0.000000e+00j 0.+0.000000e+00j 0.+0.000000e+00j]

In fact, these three CNOT gates are creating SWAP gate. Let's check.

qc = QuantumCircuit(2)
qc.h(0)
qc.barrier()
qc.swap(0,1)
qc.barrier()
qc.h(1)
print(getStateVector(qc))
style = {'backgroundcolor': 'lavender'}
qc.draw(output='mpl', style = style)

[1.-6.123234e-17j 0.+0.000000e+00j 0.+0.000000e+00j 0.+0.000000e+00j]

Circuit Identity

|+> and |-> state are Hadamard basis. Lets see the effect of this basis change.

qc = QuantumCircuit(2)
qc.h(0)
qc.h(1)
qc.barrier()
qc.cx(0,1)
qc.barrier()
qc.h(0)
qc.h(1)
print(getStateVector(qc))
style = {'backgroundcolor': 'lavender'}
qc.draw(output='mpl', style = style)

[1.-1.2246468e-16j 0.+0.0000000e+00j 0.+0.0000000e+00j 0.+0.0000000e+00j]

The above circuit is equivalent to the following circuit.

qc = QuantumCircuit(2)
qc.cx(1,0)
print(getStateVector(qc))
style = {'backgroundcolor': 'lavender'}
qc.draw(output='mpl', style = style)

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

Toffoli Gate

The Toffoli gate is a three-qubit gate with two controls and one target. It performs an X on the target only if both controls are in the state |1⟩. A Toffoli can also be thought of as a controlled-controlled-NOT, and is also called the CCX gate.

qc = QuantumCircuit(3)
qc.ccx(0,1,2)
print(getStateVector(qc))
style = {'backgroundcolor': 'lavender'}
qc.draw(output='mpl', style = style)

[1.-5.55111512e-17j 0.+0.00000000e+00j 0.+0.00000000e+00j
 0.+0.00000000e+00j 0.+0.00000000e+00j 0.+0.00000000e+00j
 0.+0.00000000e+00j 0.+0.00000000e+00j]

Lets see the matrix form of Tiffoli gate.

getMatrix(qc)

matrix([[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, 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, 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, 1.-0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]])