# This code is part of Qiskit.
#
# (C) Copyright IBM 2017.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.
"""U1 Gate."""
from __future__ import annotations
from cmath import exp
import numpy
from qiskit.circuit.controlledgate import ControlledGate
from qiskit.circuit.gate import Gate
from qiskit.circuit.parameterexpression import ParameterValueType
from qiskit.circuit.quantumregister import QuantumRegister
from qiskit.circuit._utils import _ctrl_state_to_int
[docs]class U1Gate(Gate):
r"""Single-qubit rotation about the Z axis.
This is a diagonal gate. It can be implemented virtually in hardware
via framechanges (i.e. at zero error and duration).
.. warning::
This gate is deprecated. Instead, the following replacements should be used
.. math::
U1(\lambda) = P(\lambda)= U(0,0,\lambda)
.. code-block:: python
circuit = QuantumCircuit(1)
circuit.p(lambda, 0) # or circuit.u(0, 0, lambda)
**Circuit symbol:**
.. parsed-literal::
┌───────┐
q_0: ┤ U1(λ) ├
└───────┘
**Matrix Representation:**
.. math::
U1(\lambda) =
\begin{pmatrix}
1 & 0 \\
0 & e^{i\lambda}
\end{pmatrix}
**Examples:**
.. math::
U1(\lambda = \pi) = Z
.. math::
U1(\lambda = \pi/2) = S
.. math::
U1(\lambda = \pi/4) = T
.. seealso::
:class:`~qiskit.circuit.library.standard_gates.RZGate`:
This gate is equivalent to RZ up to a phase factor.
.. math::
U1(\lambda) = e^{i{\lambda}/2} RZ(\lambda)
:class:`~qiskit.circuit.library.standard_gates.U3Gate`:
U3 is a generalization of U2 that covers all single-qubit rotations,
using two X90 pulses.
Reference for virtual Z gate implementation:
`1612.00858 <https://arxiv.org/abs/1612.00858>`_
"""
def __init__(
self, theta: ParameterValueType, label: str | None = None, *, duration=None, unit="dt"
):
"""Create new U1 gate."""
super().__init__("u1", 1, [theta], label=label, duration=duration, unit=unit)
def _define(self):
# pylint: disable=cyclic-import
from qiskit.circuit.quantumcircuit import QuantumCircuit
from .u3 import U3Gate # pylint: disable=cyclic-import
q = QuantumRegister(1, "q")
qc = QuantumCircuit(q, name=self.name)
rules = [(U3Gate(0, 0, self.params[0]), [q[0]], [])]
for instr, qargs, cargs in rules:
qc._append(instr, qargs, cargs)
self.definition = qc
[docs] def control(
self,
num_ctrl_qubits: int = 1,
label: str | None = None,
ctrl_state: str | int | None = None,
):
"""Return a (multi-)controlled-U1 gate.
Args:
num_ctrl_qubits (int): number of control qubits.
label (str or None): An optional label for the gate [Default: None]
ctrl_state (int or str or None): control state expressed as integer,
string (e.g. '110'), or None. If None, use all 1s.
Returns:
ControlledGate: controlled version of this gate.
"""
if num_ctrl_qubits == 1:
gate = CU1Gate(self.params[0], label=label, ctrl_state=ctrl_state)
elif ctrl_state is None and num_ctrl_qubits > 1:
gate = MCU1Gate(self.params[0], num_ctrl_qubits, label=label)
else:
return super().control(
num_ctrl_qubits=num_ctrl_qubits, label=label, ctrl_state=ctrl_state
)
gate.base_gate.label = self.label
return gate
[docs] def inverse(self):
r"""Return inverted U1 gate (:math:`U1(\lambda)^{\dagger} = U1(-\lambda)`)"""
return U1Gate(-self.params[0])
def __array__(self, dtype=None):
"""Return a numpy.array for the U1 gate."""
lam = float(self.params[0])
return numpy.array([[1, 0], [0, numpy.exp(1j * lam)]], dtype=dtype)
[docs]class CU1Gate(ControlledGate):
r"""Controlled-U1 gate.
This is a diagonal and symmetric gate that induces a
phase on the state of the target qubit, depending on the control state.
**Circuit symbol:**
.. parsed-literal::
q_0: ─■──
│λ
q_1: ─■──
**Matrix representation:**
.. math::
CU1(\lambda) =
I \otimes |0\rangle\langle 0| + U1 \otimes |1\rangle\langle 1| =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & e^{i\lambda}
\end{pmatrix}
.. seealso::
:class:`~qiskit.circuit.library.standard_gates.CRZGate`:
Due to the global phase difference in the matrix definitions
of U1 and RZ, CU1 and CRZ are different gates with a relative
phase difference.
"""
def __init__(
self,
theta: ParameterValueType,
label: str | None = None,
ctrl_state: str | int | None = None,
*,
duration=None,
unit="dt",
_base_label=None,
):
"""Create new CU1 gate."""
super().__init__(
"cu1",
2,
[theta],
num_ctrl_qubits=1,
label=label,
ctrl_state=ctrl_state,
base_gate=U1Gate(theta, label=_base_label),
duration=duration,
unit=unit,
)
def _define(self):
"""
gate cu1(lambda) a,b
{ u1(lambda/2) a; cx a,b;
u1(-lambda/2) b; cx a,b;
u1(lambda/2) b;
}
"""
# pylint: disable=cyclic-import
from qiskit.circuit.quantumcircuit import QuantumCircuit
from .x import CXGate # pylint: disable=cyclic-import
# ┌─────────┐
# q_0: ┤ U1(λ/2) ├──■────────────────■─────────────
# └─────────┘┌─┴─┐┌──────────┐┌─┴─┐┌─────────┐
# q_1: ───────────┤ X ├┤ U1(-λ/2) ├┤ X ├┤ U1(λ/2) ├
# └───┘└──────────┘└───┘└─────────┘
q = QuantumRegister(2, "q")
qc = QuantumCircuit(q, name=self.name)
rules = [
(U1Gate(self.params[0] / 2), [q[0]], []),
(CXGate(), [q[0], q[1]], []),
(U1Gate(-self.params[0] / 2), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(U1Gate(self.params[0] / 2), [q[1]], []),
]
for instr, qargs, cargs in rules:
qc._append(instr, qargs, cargs)
self.definition = qc
[docs] def control(
self,
num_ctrl_qubits: int = 1,
label: str | None = None,
ctrl_state: str | int | None = None,
):
"""Controlled version of this gate.
Args:
num_ctrl_qubits (int): number of control qubits.
label (str or None): An optional label for the gate [Default: None]
ctrl_state (int or str or None): control state expressed as integer,
string (e.g. '110'), or None. If None, use all 1s.
Returns:
ControlledGate: controlled version of this gate.
"""
if ctrl_state is None:
gate = MCU1Gate(self.params[0], num_ctrl_qubits=num_ctrl_qubits + 1, label=label)
gate.base_gate.label = self.label
return gate
return super().control(num_ctrl_qubits=num_ctrl_qubits, label=label, ctrl_state=ctrl_state)
[docs] def inverse(self):
r"""Return inverted CU1 gate (:math:`CU1(\lambda)^{\dagger} = CU1(-\lambda)`)"""
return CU1Gate(-self.params[0], ctrl_state=self.ctrl_state)
def __array__(self, dtype=None):
"""Return a numpy.array for the CU1 gate."""
eith = exp(1j * float(self.params[0]))
if self.ctrl_state:
return numpy.array(
[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, eith]], dtype=dtype
)
else:
return numpy.array(
[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, eith, 0], [0, 0, 0, 1]], dtype=dtype
)
class MCU1Gate(ControlledGate):
r"""Multi-controlled-U1 gate.
This is a diagonal and symmetric gate that induces a
phase on the state of the target qubit, depending on the state of the control qubits.
**Circuit symbol:**
.. parsed-literal::
q_0: ────■────
│
.
│
q_(n-1): ────■────
┌───┴───┐
q_n: ┤ U1(λ) ├
└───────┘
.. seealso::
:class:`~qiskit.circuit.library.standard_gates.CU1Gate`:
The singly-controlled-version of this gate.
"""
def __init__(
self,
lam: ParameterValueType,
num_ctrl_qubits: int,
label: str | None = None,
ctrl_state: str | int | None = None,
*,
duration=None,
unit="dt",
_base_label=None,
):
"""Create new MCU1 gate."""
super().__init__(
"mcu1",
num_ctrl_qubits + 1,
[lam],
num_ctrl_qubits=num_ctrl_qubits,
label=label,
ctrl_state=ctrl_state,
base_gate=U1Gate(lam, label=_base_label),
duration=duration,
unit=unit,
)
def _define(self):
# pylint: disable=cyclic-import
from qiskit.circuit.quantumcircuit import QuantumCircuit
q = QuantumRegister(self.num_qubits, "q")
qc = QuantumCircuit(q, name=self.name)
if self.num_ctrl_qubits == 0:
definition = U1Gate(self.params[0]).definition
if self.num_ctrl_qubits == 1:
definition = CU1Gate(self.params[0]).definition
else:
from .u3 import _gray_code_chain
scaled_lam = self.params[0] / (2 ** (self.num_ctrl_qubits - 1))
bottom_gate = CU1Gate(scaled_lam)
definition = _gray_code_chain(q, self.num_ctrl_qubits, bottom_gate)
for instr, qargs, cargs in definition:
qc._append(instr, qargs, cargs)
self.definition = qc
def control(
self,
num_ctrl_qubits: int = 1,
label: str | None = None,
ctrl_state: str | int | None = None,
):
"""Controlled version of this gate.
Args:
num_ctrl_qubits (int): number of control qubits.
label (str or None): An optional label for the gate [Default: None]
ctrl_state (int or str or None): control state expressed as integer,
string (e.g. '110'), or None. If None, use all 1s.
Returns:
ControlledGate: controlled version of this gate.
"""
ctrl_state = _ctrl_state_to_int(ctrl_state, num_ctrl_qubits)
new_ctrl_state = (self.ctrl_state << num_ctrl_qubits) | ctrl_state
gate = MCU1Gate(
self.params[0],
num_ctrl_qubits=num_ctrl_qubits + self.num_ctrl_qubits,
label=label,
ctrl_state=new_ctrl_state,
)
gate.base_gate.label = self.label
return gate
def inverse(self):
r"""Return inverted MCU1 gate (:math:`MCU1(\lambda)^{\dagger} = MCU1(-\lambda)`)"""
return MCU1Gate(-self.params[0], self.num_ctrl_qubits)