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import mimiqcircuits.operations.gates.gate as mcg
from mimiqcircuits.operations.gates.standard.u import GateU
from mimiqcircuits.matrices import pmatrix, umatrix
from symengine import pi
[docs]
class GateU1(mcg.Gate):
r"""Single qubit generic unitary gate :math:`{U_1}`.
Equivalent to :func:`GateP`
**Matrix representation:**
.. math::
\operatorname{U1}(\lambda) = \begin{pmatrix}
1 & 0 \\
0 & e^{i\lambda}
\end{pmatrix}
Parameters:
lambda (float): Euler angle 3 in radians.
Examples:
>>> from mimiqcircuits import *
>>> from symengine import *
>>> lmbda = Symbol('lambda')
>>> GateU1(lmbda)
U1(lambda)
>>> GateU1(lmbda).matrix()
[1.0, 0]
[0, exp(I*lambda)]
<BLANKLINE>
>>> c = Circuit().push(GateU1(lmbda), 0)
>>> c
1-qubit circuit with 1 instructions:
└── U1(lambda) @ q[0]
<BLANKLINE>
>>> GateU1(lmbda).power(2), GateU1(lmbda).inverse()
(U1(2*lambda), U1(-lambda))
>>> GateU1(lmbda).decompose()
1-qubit circuit with 1 instructions:
└── U(0, 0, lambda, 0.0) @ q[0]
<BLANKLINE>
"""
_name = "U1"
_num_qubits = 1
_qregsizes = [1]
_parnames = ("lmbda",)
def __init__(self, lmbda):
self.lmbda = lmbda
def _matrix(self):
return pmatrix(self.lmbda)
[docs]
def inverse(self):
return GateU1(-self.lmbda)
def _power(self, pwr):
return GateU1(pwr * self.lmbda)
def _decompose(self, circ, qubits, bits):
q = qubits[0]
circ.push(GateU(0, 0, self.lmbda), q)
return circ
[docs]
class GateU2(mcg.Gate):
r"""Single qubit generic unitary gate :math:`{U_2}`.
**Matrix representation:**
.. math::
\operatorname{U2}(\phi,\lambda) = \frac{1}{\sqrt{2}}e^{-(\phi+\lambda)/2}\begin{pmatrix}
1 & -e^{i\lambda} \\
e^{i\phi} & e^{i(\phi+\lambda)}
\end{pmatrix}
Parameters:
phi: Euler angle in radians.
lambda: Euler angle in radians.
Examples:
>>> from mimiqcircuits import *
>>> from symengine import *
>>> phi, lmbda = symbols('phi lambda')
>>> GateU2(phi, lmbda)
U2(phi, lambda)
>>> GateU2(phi, lmbda).matrix()
[0.707106781186548, -0.707106781186548*exp(I*lambda)]
[0.707106781186548*exp(I*phi), 0.707106781186548*exp(I*(lambda + phi))]
<BLANKLINE>
>>> c = Circuit().push(GateU2(phi, lmbda), 0)
>>> c
1-qubit circuit with 1 instructions:
└── U2(phi, lambda) @ q[0]
<BLANKLINE>
>>> GateU2(phi, lmbda).power(2), GateU2(phi, lmbda).inverse()
(U2(phi, lambda)**2, U2(-lambda - pi, -phi + pi))
>>> GateU2(phi, lmbda).decompose()
1-qubit circuit with 1 instructions:
└── U((1/2)*pi, phi, lambda, 0.0) @ q[0]
<BLANKLINE>
"""
_name = "U2"
_num_qubits = 1
_qregsizes = [1]
_parnames = ("phi", "lmbda")
def __init__(self, phi, lmbda):
self.phi = phi
self.lmbda = lmbda
def _matrix(self):
return umatrix(pi / 2, self.phi, self.lmbda)
[docs]
def inverse(self):
return GateU2(-self.lmbda - pi, -self.phi + pi)
def _decompose(self, circ, qubits, bits):
q = qubits[0]
circ.push(GateU(pi / 2, self.phi, self.lmbda), q)
return circ
[docs]
class GateU3(mcg.Gate):
r"""Single qubit generic unitary gate :math:`{U_3}`.
This gate is equivalent to :func:`GateU` up to a global phase, :math:`\operatorname{U3}(\theta,\phi,\lambda) = e^{-i(\phi + \lambda + \theta)/2} \operatorname{U}(\theta,\phi,\lambda)`
**Matrix representation:**
.. math::
\operatorname{U3}(\theta,\phi,\lambda) = \frac{1}{2}e^{-i(\phi + \lambda + \theta)/2}
\begin{pmatrix}
1 + e^{i\theta} & -i e^{i\lambda}(1 - e^{i\theta}) \\
i e^{i\phi}(1 - e^{i\theta}) & e^{i(\phi + \lambda)}(1 + e^{i\theta})
\end{pmatrix}
Parameters:
theta: Euler angle 1 in radians.
phi: Euler angle 2 in radians.
lambda: Euler angle 3 in radians.
Examples:
>>> from mimiqcircuits import *
>>> from symengine import *
>>> theta, phi, lmbda = symbols('theta phi lambda')
>>> GateU3(theta, phi, lmbda)
U3(theta, phi, lambda)
>>> GateU3(theta, phi, lmbda).matrix()
[1.0*cos((1/2)*theta), -exp(I*lambda)*sin((1/2)*theta)]
[exp(I*phi)*sin((1/2)*theta), exp(I*(lambda + phi))*cos((1/2)*theta)]
<BLANKLINE>
>>> c = Circuit().push(GateU3(theta, phi, lmbda), 0)
>>> GateU3(theta, phi, lmbda).power(2), GateU3(theta, phi, lmbda).inverse()
(U3(theta, phi, lambda)**2, U3(-theta, -lambda, -phi))
>>> GateU3(theta, phi, lmbda).decompose()
1-qubit circuit with 1 instructions:
└── U(theta, phi, lambda, 0.0) @ q[0]
<BLANKLINE>
"""
_name = "U3"
_num_qubits = 1
_qregsizes = [1]
_parnames = ("theta", "phi", "lmbda")
def __init__(self, theta, phi, lmbda):
self.theta = theta
self.phi = phi
self.lmbda = lmbda
def _matrix(self):
return umatrix(self.theta, self.phi, self.lmbda)
[docs]
def inverse(self):
return GateU3(-self.theta, -self.lmbda, -self.phi)
def _decompose(self, circ, qubits, bits):
q = qubits[0]
circ.push(GateU(self.theta, self.phi, self.lmbda), q)
return circ
