One Qubit Gates - Rotation¶
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from qiskit import *
from math import pi
from qiskit.visualization import plot_bloch_multivector
General Unitary Gate¶
$U_3(\theta, \phi, \lambda) = \begin{bmatrix} \cos(\theta/2) & -e^{i\lambda}\sin(\theta/2) \\ e^{i\phi}\sin(\theta/2) & e^{i\lambda+i\phi}\cos(\theta/2) \end{bmatrix}$
$\begin{aligned} U_3(\tfrac{\pi}{2}, \phi, \lambda) = U_2 = \tfrac{1}{\sqrt{2}}\begin{bmatrix} 1 & -e^{i\lambda} \\ e^{i\phi} & e^{i\lambda+i\phi} \end{bmatrix} & \quad & U_3(0, 0, \lambda) = U_1 = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\lambda}\\ \end{bmatrix} \end{aligned}$
U3 gate¶
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qc = QuantumCircuit(1)
qc.u3(pi/4,pi/4,pi/4,0)
qc.draw(output='mpl')
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# Let's see the result
backend = Aer.get_backend('statevector_simulator')
out = execute(qc,backend).result().get_statevector()
plot_bloch_multivector(out)
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U2 Gate¶
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qc = QuantumCircuit(1)
qc.u2(pi/4,pi/4,0)
qc.draw(output='mpl')
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# Let's see the result
backend = Aer.get_backend('statevector_simulator')
out = execute(qc,backend).result().get_statevector()
plot_bloch_multivector(out)
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U1 Gate¶
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qc = QuantumCircuit(1)
qc.x(0)
qc.u1(pi/4,0)
qc.draw(output='mpl')
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# 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. +0.j 0.70710678+0.70710678j]
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Standard Rotations¶
The standard rotation gates are those that define rotations around the Paulis $P=\{X,Y,Z\}$. They are defined as
$$ R_P(\theta) = \exp(-i \theta P/2) = \cos(\theta/2)I -i \sin(\theta/2)P$$Rotation around X-axis¶
$$ R_x(\theta) = \begin{pmatrix} \cos(\theta/2) & -i\sin(\theta/2)\\ -i\sin(\theta/2) & \cos(\theta/2) \end{pmatrix} = u3(\theta, -\pi/2,\pi/2) $$In [32]:
qc = QuantumCircuit(1)
qc.rx(pi/2,0)
qc.draw(output='mpl')
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# Let's see the result
backend = Aer.get_backend('statevector_simulator')
out = execute(qc,backend).result().get_statevector()
plot_bloch_multivector(out)
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Rotation around Y-axis¶
$$ R_y(\theta) = \begin{pmatrix} \cos(\theta/2) & - \sin(\theta/2)\\ \sin(\theta/2) & \cos(\theta/2). \end{pmatrix} =u3(\theta,0,0) $$In [34]:
qc = QuantumCircuit(1)
qc.ry(pi/2,0)
qc.draw(output='mpl')
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# 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]
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Rotation around Z-axis¶
$$ R_z(\phi) = \begin{pmatrix} e^{-i \phi/2} & 0 \\ 0 & e^{i \phi/2} \end{pmatrix}\equiv u1(\phi) $$Note here we have used an equivalent as is different to u1 by global phase $e^{-i \phi/2}$.
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qc = QuantumCircuit(1)
qc.rz(pi/2,0)
qc.draw(output='mpl')
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# 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.+0.j 0.+0.j]
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Note this is different due only to a global phase
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