Qutrit channel amplitude damping

doc/introduction/operations.rst

@@ -507,6 +507,7 @@ Qutrit noisy channels
  :nosignatures:
 
  ~pennylane.QutritDepolarizingChannel
+ ~pennylane.QutritAmplitudeDamping
 
 :html:`</div>`
 

doc/releases/changelog-dev.md

@@ -122,6 +122,9 @@
  returning a list of `QuantumTape`s and a post-processing function instead of simply the transformed circuit.
  [(#5693)](https://github.com/PennyLaneAI/pennylane/pull/5693)
 
+* `qml.QutritAmplitudeDamping` channel has been added, allowing for noise processes modelled by amplitude damping to be simulated on the `default.qutrit.mixed` device.
+ [(#5503)](https://github.com/PennyLaneAI/pennylane/pull/5503)
+
 <h3>Deprecations 👋</h3>
 
 * The `simplify` argument in `qml.Hamiltonian` and `qml.ops.LinearCombination` is deprecated. 

pennylane/ops/qutrit/__init__.py

@@ -26,7 +26,7 @@
 from .observables import *
 from .parametric_ops import *
 from .state_preparation import *
-from .channel import QutritDepolarizingChannel
+from .channel import *
 
 # TODO: Change `qml.Identity` for qutrit support or add `qml.TIdentity` for qutrits
 __ops__ = {
@@ -50,6 +50,7 @@
 }
 __channels__ = {
  "QutritDepolarizingChannel",
+ "QutritAmplitudeDamping",
 }
 
 __all__ = list(__ops__ | __obs__ | __channels__)

pennylane/ops/qutrit/channel.py

@@ -222,3 +222,94 @@ def compute_kraus_matrices(p): # pylint:disable=arguments-differ
  )
 
  return [identity] + Ks
+
+
+class QutritAmplitudeDamping(Channel):
+ r"""
+ Single-qutrit amplitude damping error channel.
+
+ Interaction with the environment can lead to changes in the state populations of a qutrit.
+ This can be modelled by the qutrit amplitude damping channel with the following Kraus matrices:
+
+ .. math::
+ K_0 = \begin{bmatrix}
+ 1 & 0 & 0\\
+ 0 & \sqrt{1-\gamma_1} & 0 \\
+ 0 & 0 & \sqrt{1-\gamma_2}
+ \end{bmatrix}, \quad
+ K_1 = \begin{bmatrix}
+ 0 & \sqrt{\gamma_1} & 0 \\
+ 0 & 0 & 0 \\
+ 0 & 0 & 0
+ \end{bmatrix}, \quad
+ K_2 = \begin{bmatrix}
+ 0 & 0 & \sqrt{\gamma_2} \\
+ 0 & 0 & 0 \\
+ 0 & 0 & 0
+ \end{bmatrix}
+
+ where :math:`\gamma_1 \in [0, 1]` and :math:`\gamma_2 \in [0, 1]` are the amplitude damping
+ probabilities for subspaces (0,1) and (0,2) respectively.
+
+ .. note::
+
+ The Kraus operators :math:`\{K_0, K_1, K_2\}` are adapted from [`1 <https://doi.org/10.48550/arXiv.1905.10481>`_] (Eq. 8).
+
+ **Details:**
+
+ * Number of wires: 1
+ * Number of parameters: 2
+
+ Args:
+ gamma_1 (float): :math:`|1 \rangle \rightarrow |0 \rangle` amplitude damping probability.
+ gamma_2 (float): :math:`|2 \rangle \rightarrow |0 \rangle` amplitude damping probability.
+ wires (Sequence[int] or int): the wire the channel acts on
+ id (str or None): String representing the operation (optional)
+ """
+
+ num_params = 2
+ num_wires = 1
+ grad_method = "F"
+
+ def __init__(self, gamma_1, gamma_2, wires, id=None):
+ # Verify gamma_1 and gamma_2
+ for gamma in (gamma_1, gamma_2):
+ if not (math.is_abstract(gamma_1) or math.is_abstract(gamma_2)):
+ if not 0.0 <= gamma <= 1.0:
+ raise ValueError("Each probability must be in the interval [0,1]")
+ super().__init__(gamma_1, gamma_2, wires=wires, id=id)
+
+ @staticmethod
+ def compute_kraus_matrices(gamma_1, gamma_2): # pylint:disable=arguments-differ
+ r"""Kraus matrices representing the ``QutritAmplitudeDamping`` channel.
+
+ Args:
+ gamma_1 (float): :math:`|1\rangle \rightarrow |0\rangle` amplitude damping probability.
+ gamma_2 (float): :math:`|2\rangle \rightarrow |0\rangle` amplitude damping probability.
+
+ Returns:
+ list(array): list of Kraus matrices
+
+ **Example**
+
+ >>> qml.QutritAmplitudeDamping.compute_kraus_matrices(0.5, 0.25)
+ [
+ array([ [1. , 0. , 0. ],
+ [0. , 0.70710678, 0. ],
+ [0. , 0. , 0.8660254 ]]),
+ array([ [0. , 0.70710678, 0. ],
+ [0. , 0. , 0. ],
+ [0. , 0. , 0. ]]),
+ array([ [0. , 0. , 0.5 ],
+ [0. , 0. , 0. ],
+ [0. , 0. , 0. ]])
+ ]
+ """
+ K0 = math.diag([1, math.sqrt(1 - gamma_1 + math.eps), math.sqrt(1 - gamma_2 + math.eps)])
+ K1 = math.sqrt(gamma_1 + math.eps) * math.convert_like(
+ math.cast_like(math.array([[0, 1, 0], [0, 0, 0], [0, 0, 0]]), gamma_1), gamma_1
+ )
+ K2 = math.sqrt(gamma_2 + math.eps) * math.convert_like(
+ math.cast_like(math.array([[0, 0, 1], [0, 0, 0], [0, 0, 0]]), gamma_2), gamma_2
+ )
+ return [K0, K1, K2]

tests/ops/qutrit/test_qutrit_channel_ops.py

@@ -19,6 +19,7 @@
 from numpy.linalg import matrix_power
 
 import pennylane as qml
+from pennylane import math
 from pennylane import numpy as pnp
 from pennylane.ops.qutrit import channel
 
@@ -74,7 +75,7 @@ def test_grad_depolarizing(self, angle):
  dev = qml.device("default.qutrit.mixed")
  prob = pnp.array(0.5, requires_grad=True)
 
- @qml.qnode(dev)
+ @qml.qnode(dev, diff_method="parameter-shift")
  def circuit(p):
  qml.TRX(angle, wires=0, subspace=(0, 1))
  qml.TRX(angle, wires=0, subspace=(1, 2))
@@ -113,28 +114,24 @@ def expected_jac_fn(p):
  @staticmethod
  def kraus_fn(p):
  """Gets a matrix of the Kraus matrices to be tested."""
- return qml.math.stack(channel.QutritDepolarizingChannel(p, wires=0).kraus_matrices())
+ return math.stack(channel.QutritDepolarizingChannel(p, wires=0).kraus_matrices())
 
  @staticmethod
  def kraus_fn_real(p):
  """Gets a matrix of the real part of the Kraus matrices to be tested."""
- return qml.math.real(
- qml.math.stack(channel.QutritDepolarizingChannel(p, wires=0).kraus_matrices())
- )
+ return math.real(math.stack(channel.QutritDepolarizingChannel(p, wires=0).kraus_matrices()))
 
  @staticmethod
  def kraus_fn_imag(p):
  """Gets a matrix of the imaginary part of the Kraus matrices to be tested."""
- return qml.math.imag(
- qml.math.stack(channel.QutritDepolarizingChannel(p, wires=0).kraus_matrices())
- )
+ return math.imag(math.stack(channel.QutritDepolarizingChannel(p, wires=0).kraus_matrices()))
 
  @pytest.mark.autograd
  def test_kraus_jac_autograd(self):
  """Tests Jacobian of Kraus matrices using autograd."""
  p = pnp.array(0.43, requires_grad=True)
  jac = qml.jacobian(self.kraus_fn_real)(p) + 1j * qml.jacobian(self.kraus_fn_imag)(p)
- assert qml.math.allclose(jac, self.expected_jac_fn(p))
+ assert math.allclose(jac, self.expected_jac_fn(p))
 
  @pytest.mark.torch
  def test_kraus_jac_torch(self):
@@ -144,7 +141,7 @@ def test_kraus_jac_torch(self):
  p = torch.tensor(0.43, requires_grad=True)
  jacobian = torch.autograd.functional.jacobian
  jac = jacobian(self.kraus_fn_real, p) + 1j * jacobian(self.kraus_fn_imag, p)
- assert qml.math.allclose(jac, self.expected_jac_fn(p.detach().numpy()))
+ assert math.allclose(jac, self.expected_jac_fn(p.detach().numpy()))
 
  @pytest.mark.tf
  def test_kraus_jac_tf(self):
@@ -157,10 +154,10 @@ def test_kraus_jac_tf(self):
  with tf.GradientTape() as imag_tape:
  imag_out = self.kraus_fn_imag(p)
 
- real_jac = qml.math.cast(real_tape.jacobian(real_out, p), complex)
- imag_jac = qml.math.cast(imag_tape.jacobian(imag_out, p), complex)
+ real_jac = math.cast(real_tape.jacobian(real_out, p), complex)
+ imag_jac = math.cast(imag_tape.jacobian(imag_out, p), complex)
  jac = real_jac + 1j * imag_jac
- assert qml.math.allclose(jac, self.expected_jac_fn(0.43))
+ assert math.allclose(jac, self.expected_jac_fn(0.43))
 
  @pytest.mark.jax
  def test_kraus_jac_jax(self):
@@ -171,4 +168,98 @@ def test_kraus_jac_jax(self):
 
  p = jax.numpy.array(0.43, dtype=jax.numpy.complex128)
  jac = jax.jacobian(self.kraus_fn, holomorphic=True)(p)
- assert qml.math.allclose(jac, self.expected_jac_fn(p))
+ assert math.allclose(jac, self.expected_jac_fn(p))
+
+
+class TestQutritAmplitudeDamping:
+ """Tests for the qutrit quantum channel QutritAmplitudeDamping"""
+
+ def test_gamma_zero(self, tol):
+ """Test gamma_1=gamma_2=0 gives correct Kraus matrices"""
+ kraus_mats = qml.QutritAmplitudeDamping(0, 0, wires=0).kraus_matrices()
+ assert np.allclose(kraus_mats[0], np.eye(3), atol=tol, rtol=0)
+ assert np.allclose(kraus_mats[1], np.zeros((3, 3)), atol=tol, rtol=0)
+ assert np.allclose(kraus_mats[2], np.zeros((3, 3)), atol=tol, rtol=0)
+
+ @pytest.mark.parametrize("gamma1,gamma2", ((0.1, 0.2), (0.75, 0.75)))
+ def test_gamma_arbitrary(self, gamma1, gamma2, tol):
+ """Test the correct Kraus matrices are returned, also ensures that the sum of gammas can be over 1."""
+ K_0 = np.diag((1, np.sqrt(1 - gamma1), np.sqrt(1 - gamma2)))
+
+ K_1 = np.zeros((3, 3))
+ K_1[0, 1] = np.sqrt(gamma1)
+
+ K_2 = np.zeros((3, 3))
+ K_2[0, 2] = np.sqrt(gamma2)
+
+ expected = [K_0, K_1, K_2]
+ damping_channel = qml.QutritAmplitudeDamping(gamma1, gamma2, wires=0)
+ assert np.allclose(damping_channel.kraus_matrices(), expected, atol=tol, rtol=0)
+
+ @pytest.mark.parametrize("gamma1,gamma2", ((1.5, 0.0), (0.0, 1.0 + math.eps)))
+ def test_gamma_invalid_parameter(self, gamma1, gamma2):
+ """Ensures that error is thrown when gamma_1 or gamma_2 are outside [0,1]"""
+ with pytest.raises(ValueError, match="Each probability must be in the interval"):
+ channel.QutritAmplitudeDamping(gamma1, gamma2, wires=0).kraus_matrices()
+
+ @staticmethod
+ def expected_jac_fn(gamma_1, gamma_2):
+ """Gets the expected Jacobian of Kraus matrices"""
+ partial_1 = [math.zeros((3, 3)) for _ in range(3)]
+ partial_1[0][1, 1] = -1 / (2 * math.sqrt(1 - gamma_1))
+ partial_1[1][0, 1] = 1 / (2 * math.sqrt(gamma_1))
+
+ partial_2 = [math.zeros((3, 3)) for _ in range(3)]
+ partial_2[0][2, 2] = -1 / (2 * math.sqrt(1 - gamma_2))
+ partial_2[2][0, 2] = 1 / (2 * math.sqrt(gamma_2))
+
+ return [partial_1, partial_2]
+
+ @staticmethod
+ def kraus_fn(gamma_1, gamma_2):
+ """Gets the Kraus matrices of QutritAmplitudeDamping channel, used for differentiation."""
+ damping_channel = qml.QutritAmplitudeDamping(gamma_1, gamma_2, wires=0)
+ return math.stack(damping_channel.kraus_matrices())
+
+ @pytest.mark.autograd
+ def test_kraus_jac_autograd(self):
+ """Tests Jacobian of Kraus matrices using autograd."""
+ gamma_1 = pnp.array(0.43, requires_grad=True)
+ gamma_2 = pnp.array(0.12, requires_grad=True)
+ jac = qml.jacobian(self.kraus_fn)(gamma_1, gamma_2)
+ assert math.allclose(jac, self.expected_jac_fn(gamma_1, gamma_2))
+
+ @pytest.mark.torch
+ def test_kraus_jac_torch(self):
+ """Tests Jacobian of Kraus matrices using PyTorch."""
+ import torch
+
+ gamma_1 = torch.tensor(0.43, requires_grad=True)
+ gamma_2 = torch.tensor(0.12, requires_grad=True)
+
+ jac = torch.autograd.functional.jacobian(self.kraus_fn, (gamma_1, gamma_2))
+ expected = self.expected_jac_fn(gamma_1.detach().numpy(), gamma_2.detach().numpy())
+ assert math.allclose(jac[0].detach().numpy(), expected[0])
+ assert math.allclose(jac[1].detach().numpy(), expected[1])
+
+ @pytest.mark.tf
+ def test_kraus_jac_tf(self):
+ """Tests Jacobian of Kraus matrices using TensorFlow."""
+ import tensorflow as tf
+
+ gamma_1 = tf.Variable(0.43)
+ gamma_2 = tf.Variable(0.12)
+ with tf.GradientTape() as tape:
+ out = self.kraus_fn(gamma_1, gamma_2)
+ jac = tape.jacobian(out, (gamma_1, gamma_2))
+ assert math.allclose(jac, self.expected_jac_fn(gamma_1, gamma_2))
+
+ @pytest.mark.jax
+ def test_kraus_jac_jax(self):
+ """Tests Jacobian of Kraus matrices using JAX."""
+ import jax
+
+ gamma_1 = jax.numpy.array(0.43)
+ gamma_2 = jax.numpy.array(0.12)
+ jac = jax.jacobian(self.kraus_fn, argnums=[0, 1])(gamma_1, gamma_2)
+ assert math.allclose(jac, self.expected_jac_fn(gamma_1, gamma_2))