# Copyright 2018 The Cirq Developers
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""Quantum channels that are commonly used in the literature."""
from typing import Iterable, Optional, Union
import numpy as np
from cirq import protocols, value
from cirq.ops import common_gates, raw_types
[docs]@value.value_equality
class AsymmetricDepolarizingChannel(raw_types.Gate):
"""A channel that depolarizes asymmetrically along different directions."""
[docs] def __init__(self, p_x: float, p_y: float, p_z: float) -> None:
r"""The asymmetric depolarizing channel.
This channel applies one of four disjoint possibilities: nothing (the
identity channel) or one of the three pauli gates. The disjoint
probabilities of the three gates are p_x, p_y, and p_z and the
identity is done with probability 1 - p_x - p_y - p_z. The supplied
probabilities must be valid probabilities and the sum p_x + p_y + p_z
must be a valid probability or else this constructor will raise a
ValueError.
This channel evolves a density matrix via
\rho -> (1 -p_x + p_y + p_z) \rho
+ p_x X \rho X + p_y Y \rho Y + p_z Z \rho Z
Args:
p_x: The probability that a Pauli X and no other gate occurs.
p_y: The probability that a Pauli Y and no other gate occurs.
p_z: The probability that a Pauli Z and no other gate occurs.
Raises:
ValueError: if the args or the sum of args are not probabilities.
"""
def validate_probability(p, p_str):
if p < 0:
raise ValueError('{} was less than 0.'.format(p_str))
elif p > 1:
raise ValueError('{} was greater than 1.'.format(p_str))
return p
self._p_x = validate_probability(p_x, 'p_x')
self._p_y = validate_probability(p_y, 'p_y')
self._p_z = validate_probability(p_z, 'p_z')
self._p_i = 1 - validate_probability(p_x + p_y + p_z, 'p_x + p_y + p_z')
def _channel_(self) -> Iterable[np.ndarray]:
return (
np.sqrt(self._p_i) * np.eye(2),
np.sqrt(self._p_x) * np.array([[0, 1], [1, 0]]),
np.sqrt(self._p_y) * np.array([[0, -1j], [1j, 0]]),
np.sqrt(self._p_z) * np.array([[1, 0], [0, -1]]),
)
def _value_equality_values_(self):
return self._p_x, self._p_y, self._p_z
def __repr__(self) -> str:
return 'cirq.asymmetric_depolarize(p_x={!r},p_y={!r},p_z={!r})'.format(
self._p_x, self._p_y, self._p_z
)
def __str__(self) -> str:
return 'asymmetric_depolarize(p_x={!r},p_y={!r},p_z={!r})'.format(
self._p_x, self._p_y, self._p_z
)
def _circuit_diagram_info_(
self, args: protocols.CircuitDiagramInfoArgs
) -> str:
return 'A({!r},{!r},{!r})'.format(self._p_x, self._p_y, self._p_z)
[docs]def asymmetric_depolarize(
p_x: float, p_y: float, p_z: float
) -> AsymmetricDepolarizingChannel:
r"""Returns a AsymmetricDepolarizingChannel with given parameter.
This channel evolves a density matrix via
\rho -> (1 -p_x + p_y + p_z) \rho
+ p_x X \rho X + p_y Y \rho Y + p_z Z \rho Z
Args:
p_x: The probability that a Pauli X and no other gate occurs.
p_y: The probability that a Pauli Y and no other gate occurs.
p_z: The probability that a Pauli Z and no other gate occurs.
Raises:
ValueError: if the args or the sum of the args are not probabilities.
"""
return AsymmetricDepolarizingChannel(p_x, p_y, p_z)
[docs]@value.value_equality
class DepolarizingChannel(raw_types.Gate):
"""A channel that depolarizes a qubit."""
[docs] def __init__(self, p) -> None:
r"""The symmetric depolarizing channel.
This channel applies one of four disjoint possibilities: nothing (the
identity channel) or one of the three pauli gates. The disjoint
probabilities of the three gates are all the same, p / 3, and the
identity is done with probability 1 - p. The supplied probability
must be a valid probability or else this constructor will raise a
ValueError.
This channel evolves a density matrix via
\rho -> (1 - p) \rho
+ (p / 3) X \rho X + (p / 3) Y \rho Y + (p / 3) Z \rho Z
Args:
p: The probability that one of the Pauli gates is applied. Each of
the Pauli gates is applied independently with probability p / 3.
Raises:
ValueError: if p is not a valid probability.
"""
self._p = p
self._delegate = AsymmetricDepolarizingChannel(p / 3, p / 3, p / 3)
def _channel_(self) -> Iterable[np.ndarray]:
return self._delegate._channel_()
def _value_equality_values_(self):
return self._p
def __repr__(self) -> str:
return 'cirq.depolarize(p={!r})'.format(self._p)
def __str__(self) -> str:
return 'depolarize(p={!r})'.format(self._p)
def _circuit_diagram_info_(
self, args: protocols.CircuitDiagramInfoArgs
) -> str:
return 'D({!r})'.format(self._p)
[docs]def depolarize(p: float) -> DepolarizingChannel:
r"""Returns a DepolarizingChannel with given probability of error.
This channel applies one of four disjoint possibilities: nothing (the
identity channel) or one of the three pauli gates. The disjoint
probabilities of the three gates are all the same, p / 3, and the
identity is done with probability 1 - p. The supplied probability
must be a valid probability or else this constructor will raise a
ValueError.
This channel evolves a density matrix via
\rho -> (1 - p) \rho
+ (p / 3) X \rho X + (p / 3) Y \rho Y + (p / 3) Z \rho Z
Args:
p: The probability that one of the Pauli gates is applied. Each of
the Pauli gates is applied independently with probability p / 3.
Raises:
ValueError: if p is not a valid probability.
"""
return DepolarizingChannel(p)
[docs]@value.value_equality
class GeneralizedAmplitudeDampingChannel(raw_types.Gate):
"""Dampen qubit amplitudes through non ideal dissipation.
This channel models the effect of energy dissipation into the environment
as well as the environment depositing energy into the system.
"""
[docs] def __init__(self, p: float, gamma: float) -> None:
r"""The generalized amplitude damping channel.
Construct a channel to model energy dissipation into the environment
as well as the environment depositing energy into the system. The
probabilities with which the energy exchange occur are given by gamma,
and the probability of the environment being not excited is given by
`p`.
The stationary state of this channel is the diagonal density matrix
with probability `p` of being |0⟩ and probability `1-p` of being |1⟩.
This channel evolves a density matrix via
$$
\rho \rightarrow M_0 \rho M_0^\dagger
+ M_1 \rho M_1^\dagger
+ M_2 \rho M_2^\dagger
+ M_3 \rho M_3^\dagger
$$
With
$$
\begin{align}
M_0 &= \sqrt{p} \begin{bmatrix}
1 & 0 \\
0 & \sqrt{1 - \gamma}
\end{bmatrix}
\\
M_1 &= \sqrt{p} \begin{bmatrix}
0 & \sqrt{\gamma} \\
0 & 0
\end{bmatrix}
\\
M_2 &= \sqrt{1-p} \begin{bmatrix}
\sqrt{1-\gamma} & 0 \\
0 & 1
\end{bmatrix}
\\
M_3 &= \sqrt{1-p} \begin{bmatrix}
0 & 0 \\
\sqrt{\gamma} & 0
\end{bmatrix}
\end{align}
$$
Args:
gamma: the probability of the interaction being dissipative.
p: the probability of the qubit and environment exchanging energy.
Raises:
ValueError: if gamma or p is not a valid probability.
"""
def validate_probability(p, p_str):
if p < 0:
raise ValueError('{} was less than 0.'.format(p_str))
elif p > 1:
raise ValueError('{} was greater than 1.'.format(p_str))
return p
self._gamma = validate_probability(gamma, 'gamma')
self._p = validate_probability(p, 'p')
def _channel_(self) -> Iterable[np.ndarray]:
p0 = np.sqrt(self._p)
p1 = np.sqrt(1. - self._p)
sqrt_g = np.sqrt(self._gamma)
sqrt_g1 = np.sqrt(1. - self._gamma)
return (
p0 * np.array([[1., 0.], [0., sqrt_g1]]),
p0 * np.array([[0., sqrt_g], [0., 0.]]),
p1 * np.array([[sqrt_g1, 0.], [0., 1.]]),
p1 * np.array([[0., 0.], [sqrt_g, 0.]]),
)
def _value_equality_values_(self):
return self._p, self._gamma
def __repr__(self) -> str:
return 'cirq.generalized_amplitude_damp(p={!r},gamma={!r})'.format(
self._p, self._gamma
)
def __str__(self) -> str:
return 'generalized_amplitude_damp(p={!r},gamma={!r})'.format(
self._p, self._gamma
)
def _circuit_diagram_info_(
self, args: protocols.CircuitDiagramInfoArgs
) -> str:
return 'GAD({!r},{!r})'.format(self._p, self._gamma)
[docs]def generalized_amplitude_damp(
p: float, gamma: float
) -> GeneralizedAmplitudeDampingChannel:
r"""
Returns a GeneralizedAmplitudeDampingChannel with the given
probabilities gamma and p.
This channel evolves a density matrix via:
\rho -> M_0 \rho M_0^\dagger + M_1 \rho M_1^\dagger
+ M_2 \rho M_2^\dagger + M_3 \rho M_3^\dagger
With:
M_0 = \sqrt{p} \begin{bmatrix}
1 & 0 \\
0 & \sqrt{1 - \gamma}
\end{bmatrix}
M_1 = \sqrt{p} \begin{bmatrix}
0 & \sqrt{\gamma} \\
0 & 0
\end{bmatrix}
M_2 = \sqrt{1-p} \begin{bmatrix}
\sqrt{1-\gamma} & 0 \\
0 & 1
\end{bmatrix}
M_3 = \sqrt{1-p} \begin{bmatrix}
0 & 0 \\
\sqrt{gamma} & 0
\end{bmatrix}
Args:
gamma: the probability of the interaction being dissipative.
p: the probability of the qubit and environment exchanging energy.
Raises:
ValueError: gamma or p is not a valid probability.
"""
return GeneralizedAmplitudeDampingChannel(p, gamma)
[docs]@value.value_equality
class AmplitudeDampingChannel(raw_types.Gate):
"""Dampen qubit amplitudes through dissipation.
This channel models the effect of energy dissipation to the
surrounding environment.
"""
[docs] def __init__(self, gamma) -> None:
r"""The amplitude damping channel.
Construct a channel that dissipates energy. The probability of
energy exchange occurring is given by gamma.
This channel evolves a density matrix as follows:
\rho -> M_0 \rho M_0^\dagger + M_1 \rho M_1^\dagger
With:
M_0 = \begin{bmatrix}
1 & 0 \\
0 & \sqrt{1 - \gamma}
\end{bmatrix}
M_1 = \begin{bmatrix}
0 & \sqrt{\gamma} \\
0 & 0
\end{bmatrix}
Args:
gamma: the probability of the interaction being dissipative.
Raises:
ValueError: is gamma is not a valid probability.
"""
def validate_probability(p, p_str):
if p < 0:
raise ValueError('{} was less than 0.'.format(p_str))
elif p > 1:
raise ValueError('{} was greater than 1.'.format(p_str))
return p
self._gamma = validate_probability(gamma, 'gamma')
self._delegate = GeneralizedAmplitudeDampingChannel(1.0, self._gamma)
def _channel_(self) -> Iterable[np.ndarray]:
# just return first two kraus ops, we don't care about
# the last two.
return list(self._delegate._channel_())[:2]
def _value_equality_values_(self):
return self._gamma
def __repr__(self) -> str:
return 'cirq.amplitude_damp(gamma={!r})'.format(self._gamma)
def __str__(self) -> str:
return 'amplitude_damp(gamma={!r})'.format(self._gamma)
def _circuit_diagram_info_(
self, args: protocols.CircuitDiagramInfoArgs
) -> str:
return 'AD({!r})'.format(self._gamma)
[docs]def amplitude_damp(gamma: float) -> AmplitudeDampingChannel:
r"""
Returns an AmplitudeDampingChannel with the given probability gamma.
This channel evolves a density matrix via:
\rho -> M_0 \rho M_0^\dagger + M_1 \rho M_1^\dagger
With:
M_0 = \begin{bmatrix}
1 & 0 \\
0 & \sqrt{1 - \gamma}
\end{bmatrix}
M_1 = \begin{bmatrix}
0 & \sqrt{\gamma} \\
0 & 0
\end{bmatrix}
Args:
gamma: the probability of the interaction being dissipative.
Raises:
ValueError: if gamma is not a valid probability.
"""
return AmplitudeDampingChannel(gamma)
[docs]@value.value_equality
class PhaseDampingChannel(raw_types.Gate):
"""Dampen qubit phase.
This channel models phase damping which is the loss of quantum
information without the loss of energy.
"""
[docs] def __init__(self, gamma) -> None:
r"""The phase damping channel.
Construct a channel that enacts a phase damping constant gamma.
This channel evolves a density matrix via:
\rho -> M_0 \rho M_0^\dagger + M_1 \rho M_1^\dagger
With:
M_0 = \begin{bmatrix}
1 & 0 \\
0 & \sqrt{1 - \gamma}
\end{bmatrix}
M_1 = \begin{bmatrix}
0 & 0 \\
0 & \sqrt{\gamma}
\end{bmatrix}
Args:
gamma: The damping constant.
Raises:
ValueError: if gamma is not a valid probability.
"""
def validate_probability(p, p_str):
if p < 0:
raise ValueError('{} was less than 0.'.format(p_str))
elif p > 1:
raise ValueError('{} was greater than 1.'.format(p_str))
return p
self._gamma = validate_probability(gamma, 'gamma')
def _channel_(self) -> Iterable[np.ndarray]:
return (
np.array([[1., 0.], [0., np.sqrt(1. - self._gamma)]]),
np.array([[0., 0.], [0., np.sqrt(self._gamma)]]),
)
def _value_equality_values_(self):
return self._gamma
def __repr__(self) -> str:
return 'cirq.phase_damp(gamma={!r})'.format(self._gamma)
def __str__(self) -> str:
return 'phase_damp(gamma={!r})'.format(self._gamma)
def _circuit_diagram_info_(
self, args: protocols.CircuitDiagramInfoArgs
) -> str:
return 'PD({!r})'.format(self._gamma)
[docs]def phase_damp(gamma: float) -> PhaseDampingChannel:
r"""
Creates a PhaseDampingChannel with damping constant gamma.
This channel evolves a density matrix via:
\rho -> M_0 \rho M_0^\dagger + M_1 \rho M_1^\dagger
With:
M_0 = \begin{bmatrix}
1 & 0 \\
0 & \sqrt{1 - \gamma}
\end{bmatrix}
M_1 = \begin{bmatrix}
0 & 0 \\
0 & \sqrt{\gamma}
\end{bmatrix}
Args:
gamma: The damping constant.
Raises:
ValueError: is gamma is not a valid probability.
"""
return PhaseDampingChannel(gamma)
[docs]@value.value_equality
class PhaseFlipChannel(raw_types.Gate):
"""Probabilistically flip the sign of the phase of a qubit."""
[docs] def __init__(self, p) -> None:
r"""The phase flip channel.
Construct a channel to flip the phase with probability p.
This channel evolves a density matrix via:
\rho -> M_0 \rho M_0^\dagger + M_1 \rho M_1^\dagger
With:
M_0 = \sqrt{p} \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
M_1 = \sqrt{1-p} \begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}
Args:
p: the probability of a phase flip.
Raises:
ValueError: if p is not a valid probability.
"""
def validate_probability(p, p_str):
if p < 0:
raise ValueError('{} was less than 0.'.format(p_str))
elif p > 1:
raise ValueError('{} was greater than 1.'.format(p_str))
return p
self._p = validate_probability(p, 'p')
self._delegate = AsymmetricDepolarizingChannel(0., 0., 1. - p)
def _channel_(self) -> Iterable[np.ndarray]:
kraus_ops = list(self._delegate._channel_())
# just return identity and z term
return (kraus_ops[0], kraus_ops[3])
def _value_equality_values_(self):
return self._p
def __repr__(self) -> str:
return 'cirq.phase_flip(p={!r})'.format(self._p)
def __str__(self) -> str:
return 'phase_flip(p={!r})'.format(self._p)
def _circuit_diagram_info_(
self, args: protocols.CircuitDiagramInfoArgs
) -> str:
return 'PF({!r})'.format(self._p)
def _phase_flip_Z() -> common_gates.ZPowGate:
"""
Returns a cirq.Z which corresponds to a guaranteed phase flip.
"""
return common_gates.ZPowGate()
def _phase_flip(p: float) -> PhaseFlipChannel:
r"""
Returns a PhaseFlipChannel that flips a qubit's phase with probability p.
This channel evolves a density matrix via:
\rho -> M_0 \rho M_0^\dagger + M_1 \rho M_1^\dagger
With:
M_0 = \sqrt{p} \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
M_1 = \sqrt{1-p} \begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}
Args:
p: the probability of a phase flip.
Raises:
ValueError: if p is not a valid probability.
"""
return PhaseFlipChannel(p)
[docs]def phase_flip(
p: Optional[float] = None
) -> Union[common_gates.ZPowGate, PhaseFlipChannel]:
r"""
Returns a PhaseFlipChannel that flips a qubit's phase with probability p
if p is None, return a guaranteed phase flip in the form of a Z operation.
This channel evolves a density matrix via:
\rho -> M_0 \rho M_0^\dagger + M_1 \rho M_1^\dagger
With:
M_0 = \sqrt{p} \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
M_1 = \sqrt{1-p} \begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}
Args:
p: the probability of a phase flip.
Raises:
ValueError: if p is not a valid probability.
"""
if p is None:
return _phase_flip_Z()
return _phase_flip(p)
[docs]@value.value_equality
class BitFlipChannel(raw_types.Gate):
r"""Probabilistically flip a qubit from 1 to 0 state or vice versa."""
[docs] def __init__(self, p) -> None:
r"""The bit flip channel.
Construct a channel that flips a qubit with probability p.
This channel evolves a density matrix via:
\rho -> M_0 \rho M_0^\dagger + M_1 \rho M_1^\dagger
With:
M_0 = \sqrt{p} \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
M_1 = \sqrt{1-p} \begin{bmatrix}
0 & 1 \\
1 & -0
\end{bmatrix}
Args:
p: the probability of a bit flip.
Raises:
ValueError: if p is not a valid probability.
"""
def validate_probability(p, p_str):
if p < 0:
raise ValueError('{} was less than 0.'.format(p_str))
elif p > 1:
raise ValueError('{} was greater than 1.'.format(p_str))
return p
self._p = validate_probability(p, 'p')
self._delegate = AsymmetricDepolarizingChannel(1. - p, 0., 0.)
def _channel_(self) -> Iterable[np.ndarray]:
# Return just the I and X pieces.
return list(self._delegate._channel_())[:2]
def _value_equality_values_(self):
return self._p
def __repr__(self) -> str:
return 'cirq.bit_flip(p={!r})'.format(self._p)
def __str__(self) -> str:
return 'bit_flip(p={!r})'.format(self._p)
def _circuit_diagram_info_(
self, args: protocols.CircuitDiagramInfoArgs
) -> str:
return 'BF({!r})'.format(self._p)
def _bit_flip(p: float) -> BitFlipChannel:
r"""
Construct a BitFlipChannel that flips a qubit state
with probability of a flip given by p.
This channel evolves a density matrix via:
\rho -> M_0 \rho M_0^\dagger + M_1 \rho M_1^\dagger
With:
M_0 = \sqrt{p} \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
M_1 = \sqrt{1-p} \begin{bmatrix}
0 & 1 \\
1 & -0
\end{bmatrix}
Args:
p: the probability of a bit flip.
Raises:
ValueError: if p is not a valid probability.
"""
return BitFlipChannel(p)
[docs]def bit_flip(
p: Optional[float] = None
) -> Union[common_gates.XPowGate, BitFlipChannel]:
r"""
Construct a BitFlipChannel that flips a qubit state
with probability of a flip given by p. If p is None, return
a guaranteed flip in the form of an X operation.
This channel evolves a density matrix via
\rho -> M_0 \rho M_0^\dagger + M_1 \rho M_1^\dagger
With
M_0 = \sqrt{p} \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
M_1 = \sqrt{1-p} \begin{bmatrix}
0 & 1 \\
1 & -0
\end{bmatrix}
Args:
p: the probability of a bit flip.
Raises:
ValueError: if p is not a valid probability.
"""
if p is None:
return common_gates.X
return _bit_flip(p)
[docs]@value.value_equality
class RotationErrorChannel(raw_types.Gate):
"""Channel to introduce rotation error in X, Y, Z."""
[docs] def __init__(self, eps_x, eps_y, eps_z) -> None:
r"""The rotation error channel.
This channel introduces rotation error by epsilon for
rotations in X, Y and Z that are constant in time.
This channel evolves a density matrix via
\rho ->
\exp{-i \epsilon_x \frac{X}{2}} \rho \exp{i \epsilon_x \frac{X}{2}}
+ \exp{-i \epsilon_y \frac{Y}{2}} \rho \exp{i \epsilon_y \frac{Y}{2}}
+ \exp{-i \epsilon_z \frac{Z}{2}} \rho \exp{i \epsilon_z \frac{Z}{2}}
Args:
eps_x: angle to over rotate in x.
eps_y: angle to over rotate in y.
eps_z: angle to over rotate in z.
"""
# Angles could be anything, so no validation nescessary ?
self._eps_x = eps_x
self._eps_y = eps_y
self._eps_z = eps_z
def _channel_(self) -> Iterable[np.ndarray]:
return (
np.exp(
0.5
* (0. - 1.0j)
* self._eps_x
* np.array([[0., 1.], [1., 0.]])
),
np.exp(
0.5
* (0. - 1.0j)
* self._eps_y
* np.array([[0., (0. - 1.0j)], [(0. + 1.0j), 0.]])
),
np.exp(
0.5
* (0. - 1.0j)
* self._eps_z
* np.array([[1., 0.], [0., -1.]])
),
)
def _value_equality_values_(self):
return self._eps_x, self._eps_y, self._eps_z
def __repr__(self) -> str:
return 'cirq.rotation_error(eps_x={!r},eps_y={!r},eps_z={!r})'.format(
self._eps_x, self._eps_y, self._eps_z
)
def __str__(self) -> str:
return 'rotation_error(eps_x={!r},eps_y={!r},eps_z={!r})'.format(
self._eps_x, self._eps_y, self._eps_z
)
def _circuit_diagram_info_(
self, args: protocols.CircuitDiagramInfoArgs
) -> str:
return 'RE({!r},{!r},{!r})'.format(
self._eps_x, self._eps_y, self._eps_z
)
[docs]def rotation_error(
eps_x: float, eps_y: float, eps_z: float
) -> RotationErrorChannel:
r"""
Constructs a RotationErrorChannel that can over/under rotate
a qubit in X, Y, Z by given error angles.
This channel evolves a density matrix via:
\rho ->
\exp{-i \epsilon_x \frac{X}{2}} \rho \exp{i \epsilon_x \frac{X}{2}}
+ \exp{-i \epsilon_y \frac{Y}{2}} \rho \exp{i \epsilon_y \frac{Y}{2}}
+ \exp{-i \epsilon_z \frac{Z}{2}} \rho \exp{i \epsilon_z \frac{Z}{2}}
Args:
eps_x: angle to over rotate in x.
eps_y: angle to over rotate in y.
eps_z: angle to over rotate in z.
"""
return RotationErrorChannel(eps_x, eps_y, eps_z)