def BasisEmbedding(features, wires):
r"""Encodes :math:`n` binary features into a basis state of :math:`n` qubits.
For example, for ``features=np.array([0, 1, 0])``, the quantum system will be
prepared in state :math:`|010 \rangle`.
.. warning::
``BasisEmbedding`` calls a circuit whose architecture depends on the binary features.
The ``features`` argument is therefore not differentiable when using the template, and
gradients with respect to the argument cannot be computed by PennyLane.
Args:
features (array): binary input array of shape ``(n, )``
wires (Sequence[int] or int): qubit indices that the template acts on
Raises:
ValueError: if inputs do not have the correct format
"""
#############
# Input checks
wires, n_wires = _check_wires(wires)
_check_shape(features, (n_wires,))
# basis_state is guaranteed to be a list
if any([b not in [0, 1] for b in features]):
raise ValueError("Basis state must only consist of 0s and 1s, got {}".format(features))
###############
features = np.array(features)
BasisState(features, wires=wires)
def AngleEmbedding(features, wires, rotation="X"):
r"""
Encodes :math:`N` features into the rotation angles of :math:`n` qubits, where :math:`N \leq n`.
The rotations can be chosen as either :class:`~pennylane.ops.RX`, :class:`~pennylane.ops.RY`
or :class:`~pennylane.ops.RZ` gates, as defined by the ``rotation`` parameter:
* ``rotation='X'`` uses the features as angles of RX rotations
* ``rotation='Y'`` uses the features as angles of RY rotations
* ``rotation='Z'`` uses the features as angles of RZ rotations
The length of ``features`` has to be smaller or equal to the number of qubits. If there are fewer entries in
``features`` than rotations, the circuit does not apply the remaining rotation gates.
Args:
features (array): input array of shape ``(N,)``, where N is the number of input features to embed,
with :math:`N\leq n`
wires (Sequence[int] or int): qubit indices that the template acts on
rotation (str): Type of rotations used
Raises:
ValueError: if inputs do not have the correct format
"""
#############
# Input checks
_check_no_variable(rotation, msg="'rotation' cannot be differentiable")
wires = _check_wires(wires)
_check_shape(
features,
(len(wires), ),
bound="max",
msg="'features' must be of shape {} or smaller; "
"got {}.".format((len(wires), ), _get_shape(features)),
)
_check_type(rotation, [str],
msg="'rotation' must be a string; got {}".format(rotation))
_check_is_in_options(
rotation,
["X", "Y", "Z"],
msg="did not recognize option {} for 'rotation'.".format(rotation),
)
###############
if rotation == "X":
for f, w in zip(features, wires):
RX(f, wires=w)
elif rotation == "Y":
for f, w in zip(features, wires):
RY(f, wires=w)
elif rotation == "Z":
for f, w in zip(features, wires):
RZ(f, wires=w)
def DisplacementEmbedding(features, wires, method="amplitude", c=0.1):
r"""Encodes :math:`N` features into the displacement amplitudes :math:`r` or phases :math:`\phi` of :math:`M` modes,
where :math:`N\leq M`.
The mathematical definition of the displacement gate is given by the operator
.. math::
D(\alpha) = \exp(r (e^{i\phi}\ad -e^{-i\phi}\a)),
where :math:`\a` and :math:`\ad` are the bosonic creation and annihilation operators.
``features`` has to be an array of at most ``len(wires)`` floats. If there are fewer entries in
``features`` than wires, the circuit does not apply the remaining displacement gates.
Args:
features (array): Array of features of size (N,)
wires (Sequence[int]): sequence of mode indices that the template acts on
method (str): ``'phase'`` encodes the input into the phase of single-mode displacement, while
``'amplitude'`` uses the amplitude
c (float): value of the phase of all displacement gates if ``execution='amplitude'``, or
the amplitude of all displacement gates if ``execution='phase'``
Raises:
ValueError: if inputs do not have the correct format
"""
#############
# Input checks
_check_no_variable(method, msg="'method' cannot be differentiable")
_check_no_variable(c, msg="'c' cannot be differentiable")
wires = _check_wires(wires)
expected_shape = (len(wires), )
_check_shape(
features,
expected_shape,
bound="max",
msg="'features' must be of shape {} or smaller; got {}."
"".format(expected_shape, _get_shape(features)),
)
_check_is_in_options(
method,
["amplitude", "phase"],
msg="did not recognize option {} for 'method'"
"".format(method),
)
#############
for idx, f in enumerate(features):
if method == "amplitude":
Displacement(f, c, wires=wires[idx])
elif method == "phase":
Displacement(c, f, wires=wires[idx])
def SqueezingEmbedding(features, wires, method='amplitude', c=0.1):
r"""Encodes :math:`N` features into the squeezing amplitudes :math:`r \geq 0` or phases :math:`\phi \in [0, 2\pi)`
of :math:`M` modes, where :math:`N\leq M`.
The mathematical definition of the squeezing gate is given by the operator
.. math::
S(z) = \exp\left(\frac{r}{2}\left(e^{-i\phi}\a^2 -e^{i\phi}{\ad}^{2} \right) \right),
where :math:`\a` and :math:`\ad` are the bosonic creation and annihilation operators.
``features`` has to be an iterable of at most ``len(wires)`` floats. If there are fewer entries in
``features`` than wires, the circuit does not apply the remaining squeezing gates.
Args:
features (array): Array of features of size (N,)
wires (Sequence[int]): sequence of mode indices that the template acts on
method (str): ``'phase'`` encodes the input into the phase of single-mode squeezing, while
``'amplitude'`` uses the amplitude
c (float): value of the phase of all squeezing gates if ``execution='amplitude'``, or the
amplitude of all squeezing gates if ``execution='phase'``
Raises:
ValueError: if inputs do not have the correct format
"""
#############
# Input checks
_check_no_variable(method, msg="'method' cannot be differentiable")
_check_no_variable(c, msg="'c' cannot be differentiable")
wires = _check_wires(wires)
expected_shape = (len(wires), )
_check_shape(features,
expected_shape,
bound='max',
msg="'features' must be of shape {} or smaller; got {}"
"".format(expected_shape, _get_shape(features)))
_check_is_in_options(
method, ['amplitude', 'phase'],
msg="did not recognize option {} for 'method'".format(method))
#############
for idx, f in enumerate(features):
if method == 'amplitude':
Squeezing(f, c, wires=wires[idx])
elif method == 'phase':
Squeezing(c, f, wires=wires[idx])
def BasisStatePreparation(basis_state, wires):
r"""
Prepares a basis state on the given wires using a sequence of Pauli X gates.
.. warning::
``basis_state`` influences the circuit architecture and is therefore incompatible with
gradient computations. Ensure that ``basis_state`` is not passed to the qnode by positional
arguments.
Args:
basis_state (array): Input array of shape ``(N,)``, where N is the number of wires
the state preparation acts on. ``N`` must be smaller or equal to the total
number of wires of the device.
wires (Sequence[int]): sequence of qubit indices that the template acts on
Raises:
ValueError: if inputs do not have the correct format
"""
######################
# Input checks
wires = _check_wires(wires)
expected_shape = (len(wires), )
_check_shape(
basis_state,
expected_shape,
msg=" 'basis_state' must be of shape {}; got {}."
"".format(expected_shape, _get_shape(basis_state)),
)
# basis_state cannot be trainable
_check_no_variable(
basis_state,
msg=
"'basis_state' cannot be differentiable; must be passed as a keyword argument "
"to the quantum node",
)
# basis_state is guaranteed to be a list of binary values
if any([b not in [0, 1] for b in basis_state]):
raise ValueError(
"'basis_state' must only contain values of 0 and 1; got {}".format(
basis_state))
######################
for wire, state in zip(wires, basis_state):
if state == 1:
qml.PauliX(wire)
def AngleEmbedding(features, wires, rotation='X'):
r"""
Encodes :math:`N` features into the rotation angles of :math:`n` qubits, where :math:`N \leq n`.
The rotations can be chosen as either :class:`~pennylane.ops.RX`, :class:`~pennylane.ops.RY`
or :class:`~pennylane.ops.RZ` gates, as defined by the ``rotation`` parameter:
* ``rotation='X'`` uses the features as angles of RX rotations
* ``rotation='Y'`` uses the features as angles of RY rotations
* ``rotation='Z'`` uses the features as angles of RZ rotations
The length of ``features`` has to be smaller or equal to the number of qubits. If there are fewer entries in
``features`` than rotations, the circuit does not apply the remaining rotation gates.
Args:
features (array): input array of shape ``(N,)``, where N is the number of input features to embed,
with :math:`N\leq n`
wires (Sequence[int] or int): qubit indices that the template acts on
rotation (str): Type of rotations used
Raises:
ValueError: if inputs do not have the correct format
"""
#############
# Input checks
_check_no_variable([rotation], ['rotation'])
wires, n_wires = _check_wires(wires)
msg = "AngleEmbedding cannot process more features than number of qubits {};" \
"got {}.".format(n_wires, len(features))
_check_shape(features, (n_wires,), bound='max', msg=msg)
_check_type(rotation, [str])
msg = "Rotation strategy {} not recognized.".format(rotation)
_check_hyperp_is_in_options(rotation, ['X', 'Y', 'Z'], msg=msg)
###############
if rotation == 'X':
for f, w in zip(features, wires):
RX(f, wires=w)
elif rotation == 'Y':
for f, w in zip(features, wires):
RY(f, wires=w)
elif rotation == 'Z':
for f, w in zip(features, wires):
RZ(f, wires=w)
def DisplacementEmbedding(features, wires, method='amplitude', c=0.1):
r"""Encodes :math:`N` features into the displacement amplitudes :math:`r` or phases :math:`\phi` of :math:`M` modes,
where :math:`N\leq M`.
The mathematical definition of the displacement gate is given by the operator
.. math::
D(\alpha) = \exp(r (e^{i\phi}\ad -e^{-i\phi}\a)),
where :math:`\a` and :math:`\ad` are the bosonic creation and annihilation operators.
``features`` has to be an array of at most ``len(wires)`` floats. If there are fewer entries in
``features`` than wires, the circuit does not apply the remaining displacement gates.
Args:
features (array): Array of features of size (N,)
wires (Sequence[int]): sequence of mode indices that the template acts on
method (str): ``'phase'`` encodes the input into the phase of single-mode displacement, while
``'amplitude'`` uses the amplitude
c (float): value of the phase of all displacement gates if ``execution='amplitude'``, or
the amplitude of all displacement gates if ``execution='phase'``
Raises:
ValueError: if inputs do not have the correct format
"""
#############
# Input checks
_check_no_variable([method, c], ['method', 'c'])
wires, n_wires = _check_wires(wires)
msg = "DisplacementEmbedding cannot process more features than number of wires {};" \
"got {}.".format(n_wires, len(features))
_check_shape(features, (n_wires,), bound='max', msg=msg)
msg = "Did not recognise parameter encoding method {}.".format(method)
_check_hyperp_is_in_options(method, ['amplitude', 'phase'], msg=msg)
#############
for idx, f in enumerate(features):
if method == 'amplitude':
Displacement(f, c, wires=wires[idx])
elif method == 'phase':
Displacement(c, f, wires=wires[idx])
def BasisStatePreparation(basis_state, wires):
r"""
Prepares a basis state on the given wires using a sequence of Pauli X gates.
.. warning::
``basis_state`` influences the circuit architecture and is therefore incompatible with
gradient computations. Ensure that ``basis_state`` is not passed to the qnode by positional
arguments.
Args:
basis_state (array): Input array of shape ``(N,)``, where N is the number of wires
the state preparation acts on. ``N`` must be smaller or equal to the total
number of wires of the device.
wires (Sequence[int]): sequence of qubit indices that the template acts on
Raises:
ValueError: if inputs do not have the correct format
"""
######################
# Input checks
wires, n_wires = _check_wires(wires)
msg = "The size of the basis state must match the number of qubits {}; got {}.".format(
n_wires, len(basis_state))
_check_shape(basis_state, (n_wires, ), msg=msg)
# basis_state cannot be trainable
msg = "Basis state influences circuit architecture and can therefore not be passed as a " \
"positional argument to the quantum node."
_check_no_variable([basis_state], ['basisstate'], msg=msg)
# basis_state is guaranteed to be a list
if any([b not in [0, 1] for b in basis_state]):
raise ValueError(
"Basis state must only consist of 0s and 1s, got {}".format(
basis_state))
######################
for wire, state in zip(wires, basis_state):
if state == 1:
qml.PauliX(wire)
def AmplitudeEmbedding(features, wires, pad=None, normalize=False):
r"""Encodes :math:`2^n` features into the amplitude vector of :math:`n` qubits.
By setting ``pad`` to a real or complex number, ``features`` is automatically padded to dimension
:math:`2^n` where :math:`n` is the number of qubits used in the embedding.
To represent a valid quantum state vector, the L2-norm of ``features`` must be one.
The argument ``normalize`` can be set to ``True`` to automatically normalize the features.
If both automatic padding and normalization are used, padding is executed *before* normalizing.
.. note::
On some devices, ``AmplitudeEmbedding`` must be the first operation of a quantum node.
.. note::
``AmplitudeEmbedding`` calls a circuit that involves non-trivial classical processing of the
features. The ``features`` argument is therefore **not differentiable** when using the template, and
gradients with respect to the features cannot be computed by PennyLane.
.. warning::
``AmplitudeEmbedding`` calls a circuit that involves non-trivial classical processing of the
features. The `features` argument is therefore not differentiable when using the template, and
gradients with respect to the argument cannot be computed by PennyLane.
Args:
features (array): input array of shape ``(2^n,)``
wires (Sequence[int] or int): :math:`n` qubit indices that the template acts on
pad (float or complex): if not None, the input is padded with this constant to size :math:`2^n`
normalize (Boolean): controls the activation of automatic normalization
Raises:
ValueError: if inputs do not have the correct format
.. UsageDetails::
Amplitude embedding encodes a normalized :math:`2^n`-dimensional feature vector into the state
of :math:`n` qubits:
.. code-block:: python
import pennylane as qml
from pennylane.templates import AmplitudeEmbedding
dev = qml.device('default.qubit', wires=2)
@qml.qnode(dev)
def circuit(f=None):
AmplitudeEmbedding(features=f, wires=range(2))
return qml.expval(qml.PauliZ(0))
circuit(f=[1/2, 1/2, 1/2, 1/2])
Checking the final state of the device, we find that it is equivalent to the input passed to the circuit:
>>> dev._state
[0.5+0.j 0.5+0.j 0.5+0.j 0.5+0.j]
**Passing features as positional arguments to a quantum node**
The ``features`` argument of ``AmplitudeEmbedding`` can in principle also be passed to the quantum node
as a positional argument:
.. code-block:: python
@qml.qnode(dev)
def circuit(f):
AmplitudeEmbedding(features=f, wires=range(2))
return qml.expval(qml.PauliZ(0))
However, due to non-trivial classical processing to construct the state preparation circuit,
the features argument is **not differentiable**.
>>> g = qml.grad(circuit, argnum=0)
>>> g([1,1,1,1])
ValueError: Cannot differentiate wrt parameter(s) {0, 1, 2, 3}.
**Normalization**
The template will raise an error if the feature input is not normalized.
One can set ``normalize=True`` to automatically normalize it:
.. code-block:: python
@qml.qnode(dev)
def circuit(f=None):
AmplitudeEmbedding(features=f, wires=range(2), normalize=True)
return qml.expval(qml.PauliZ(0))
circuit(f=[15, 15, 15, 15])
The re-normalized feature vector is encoded into the quantum state vector:
>>> dev._state
[0.5 + 0.j, 0.5 + 0.j, 0.5 + 0.j, 0.5 + 0.j]
**Padding**
If the dimension of the feature vector is smaller than the number of amplitudes,
one can automatically pad it with a constant for the missing dimensions using the ``pad`` option:
.. code-block:: python
from math import sqrt
@qml.qnode(dev)
def circuit(f=None):
AmplitudeEmbedding(features=f, wires=range(2), pad=0.)
return qml.expval(qml.PauliZ(0))
circuit(f=[1/sqrt(2), 1/sqrt(2)])
>>> dev._state
[0.70710678 + 0.j, 0.70710678 + 0.j, 0.0 + 0.j, 0.0 + 0.j]
**Operations before the embedding**
On some devices, ``AmplitudeEmbedding`` must be the first operation in the quantum node.
For example, ``'default.qubit'`` complains when running the following circuit:
.. code-block:: python
dev = qml.device('default.qubit', wires=2)
@qml.qnode(dev)
def circuit(f=None):
qml.Hadamard(wires=0)
AmplitudeEmbedding(features=f, wires=range(2))
return qml.expval(qml.PauliZ(0))
>>> circuit(f=[1/2, 1/2, 1/2, 1/2])
pennylane._device.DeviceError: Operation QubitStateVector cannot be used
after other Operations have already been applied on a default.qubit device.
"""
#############
# Input checks
_check_no_variable(pad, msg="'pad' cannot be differentiable")
_check_no_variable(normalize, msg="'normalize' cannot be differentiable")
wires = _check_wires(wires)
n_amplitudes = 2**len(wires)
expected_shape = (n_amplitudes,)
if pad is None:
shape = _check_shape(features, expected_shape, msg="'features' must be of shape {}; got {}. Use the 'pad' "
"argument for automated padding."
"".format(expected_shape, _get_shape(features)))
else:
shape = _check_shape(features, expected_shape, bound='max', msg="'features' must be of shape {} or smaller "
"to be padded; got {}"
"".format(expected_shape, _get_shape(features)))
_check_type(pad, [float, complex, type(None)], msg="'pad' must be a float or complex; got {}".format(pad))
_check_type(normalize, [bool], msg="'normalize' must be a boolean; got {}".format(normalize))
###############
#############
# Preprocessing
# pad
n_features = shape[0]
if pad is not None and n_amplitudes > n_features:
features = np.pad(features, (0, n_amplitudes-n_features), mode='constant', constant_values=pad)
# normalize
if isinstance(features[0], Variable):
feature_values = [s.val for s in features]
norm = np.sum(np.abs(feature_values)**2)
else:
norm = np.sum(np.abs(features)**2)
if not np.isclose(norm, 1.0, atol=TOLERANCE, rtol=0):
if normalize or pad:
features = features/np.sqrt(norm)
else:
raise ValueError("'features' must be a vector of length 1.0; got length {}."
"Use 'normalization=True' to automatically normalize.".format(norm))
###############
features = np.array(features)
QubitStateVector(features, wires=wires)
def RandomLayers(weights,
wires,
ratio_imprim=0.3,
imprimitive=CNOT,
rotations=None,
seed=42):
r"""Layers of randomly chosen single qubit rotations and 2-qubit entangling gates, acting
on randomly chosen qubits.
The argument ``weights`` contains the weights for each layer. The number of layers :math:`L` is therefore derived
from the first dimension of ``weights``.
The two-qubit gates of type ``imprimitive`` and the rotations are distributed randomly in the circuit.
The number of random rotations is derived from the second dimension of ``weights``. The number of
two-qubit gates is determined by ``ratio_imprim``. For example, a ratio of ``0.3`` with ``30`` rotations
will lead to the use of ``10`` two-qubit gates.
.. note::
If applied to one qubit only, this template will use no imprimitive gates.
This is an example of two 4-qubit random layers with four Pauli-Y/Pauli-Z rotations :math:`R_y, R_z`,
controlled-Z gates as imprimitives, as well as ``ratio_imprim=0.3``:
.. figure:: ../../_static/layer_rnd.png
:align: center
:width: 60%
:target: javascript:void(0);
.. note::
Using the default seed (or any other fixed integer seed) generates one and the same circuit in every
quantum node. To generate different circuit architectures, either use a different random seed, or use ``seed=None``
together with the ``cache=False`` option when creating a quantum node.
.. warning::
When using a random number generator anywhere inside the quantum function without the ``cache=False`` option,
a new random circuit architecture will be created every time the quantum node is evaluated.
Args:
weights (array[float]): array of weights of shape ``(L, k)``,
wires (Sequence[int]): sequence of qubit indices that the template acts on
ratio_imprim (float): value between 0 and 1 that determines the ratio of imprimitive to rotation gates
imprimitive (pennylane.ops.Operation): two-qubit gate to use, defaults to :class:`~pennylane.ops.CNOT`
rotations (list[pennylane.ops.Operation]): List of Pauli-X, Pauli-Y and/or Pauli-Z gates. The frequency
determines how often a particular rotation type is used. Defaults to the use of all three
rotations with equal frequency.
seed (int): seed to generate random architecture
Raises:
ValueError: if inputs do not have the correct format
"""
if seed is not None:
np.random.seed(seed)
if rotations is None:
rotations = [RX, RY, RZ]
#############
# Input checks
hyperparams = [ratio_imprim, imprimitive, rotations, seed]
hyperparam_names = ['ratio_imprim', 'imprimitive', 'rotations', 'seed']
_check_no_variable(hyperparams, hyperparam_names)
wires, _ = _check_wires(wires)
repeat = _check_number_of_layers([weights])
n_rots = _get_shape(weights)[1]
_check_shape(weights, (repeat, n_rots))
_check_type(ratio_imprim, [float, type(None)])
_check_type(n_rots, [int, type(None)])
_check_type(rotations, [list, type(None)])
_check_type(seed, [int, type(None)])
###############
for l in range(repeat):
_random_layer(weights=weights[l],
wires=wires,
ratio_imprim=ratio_imprim,
imprimitive=imprimitive,
rotations=rotations,
seed=seed)
def StronglyEntanglingLayers(weights, wires, ranges=None, imprimitive=CNOT):
r"""Layers consisting of single qubit rotations and entanglers, inspired by the circuit-centric classifier design
`arXiv:1804.00633 <https://arxiv.org/abs/1804.00633>`_.
The argument ``weights`` contains the weights for each layer. The number of layers :math:`L` is therefore derived
from the first dimension of ``weights``.
The 2-qubit gates, whose type is specified by the ``imprimitive`` argument,
act chronologically on the :math:`M` wires, :math:`i = 1,...,M`. The second qubit of each gate is given by
:math:`(i+r)\mod M`, where :math:`r` is a hyperparameter called the *range*, and :math:`0 < r < M`.
If applied to one qubit only, this template will use no imprimitive gates.
This is an example of two 4-qubit strongly entangling layers (ranges :math:`r=1` and :math:`r=2`, respectively) with
rotations :math:`R` and CNOTs as imprimitives:
.. figure:: ../../_static/layer_sec.png
:align: center
:width: 60%
:target: javascript:void(0);
Args:
weights (array[float]): array of weights of shape ``(:math:`L`, :math:`M`, 3)``
wires (Sequence[int] or int): qubit indices that the template acts on
ranges (Sequence[int]): sequence determining the range hyperparameter for each subsequent layer; if None
using :math:`r=l \mod M` for the :math:`l`th layer and :math:`M` wires.
imprimitive (pennylane.ops.Operation): two-qubit gate to use, defaults to :class:`~pennylane.ops.CNOT`
Raises:
ValueError: if inputs do not have the correct format
"""
#############
# Input checks
_check_no_variable([ranges, imprimitive], ['ranges', 'imprimitive'])
wires, n_wires = _check_wires(wires)
repeat = _check_number_of_layers([weights])
_check_shape(weights, (repeat, n_wires, 3))
_check_type(ranges, [list, type(None)])
if ranges is None:
# Tile ranges with iterations of range(1, n_wires)
ranges = [(l % (n_wires - 1)) + 1 for l in range(repeat)]
msg = "StronglyEntanglingLayers expects ``ranges`` to contain a range for each layer; " \
"got {}.".format(len(ranges))
_check_shape(ranges, (repeat, ), msg=msg)
msg = "StronglyEntanglingLayers expects ``ranges`` to be a list of integers; got {}.".format(
len(ranges))
_check_type(ranges[0], [int], msg=msg)
if any((r >= n_wires or r == 0) for r in ranges):
raise ValueError(
"The range hyperparameter for all layers needs to be smaller than the number of "
"qubits; got ranges {}.".format(ranges))
###############
for l in range(repeat):
_strongly_entangling_layer(weights=weights[l],
wires=wires,
r=ranges[l],
imprimitive=imprimitive)
def MottonenStatePreparation(state_vector, wires):
r"""
Prepares an arbitrary state on the given wires using a decomposition into gates developed
by Möttönen et al. (Quantum Info. Comput., 2005).
The state is prepared via a sequence
of "uniformly controlled rotations". A uniformly controlled rotation on a target qubit is
composed from all possible controlled rotations on said qubit and can be used to address individual
elements of the state vector. In the work of Mottonen et al., the inverse of their state preparation
is constructed by first equalizing the phases of the state vector via uniformly controlled Z rotations
and then rotating the now real state vector into the direction of the state :math:`|0\rangle` via
uniformly controlled Y rotations.
This code is adapted from code written by Carsten Blank for PennyLane-Qiskit.
Args:
state_vector (array): Input array of shape ``(2^N,)``, where N is the number of wires
the state preparation acts on. ``N`` must be smaller or equal to the total
number of wires.
wires (Sequence[int]): sequence of qubit indices that the template acts on
Raises:
ValueError: if inputs do not have the correct format
"""
###############
# Input checks
wires, n_wires = _check_wires(wires)
msg = "The state vector must be of size {}; got {}.".format(
2**n_wires, len(state_vector))
_check_shape(state_vector, (2**n_wires, ), msg=msg)
# check if state_vector is normalized
if isinstance(state_vector[0], Variable):
state_vector_values = [s.val for s in state_vector]
norm = np.sum(np.abs(state_vector_values)**2)
else:
norm = np.sum(np.abs(state_vector)**2)
if not np.isclose(norm, 1.0, atol=1e-3):
raise ValueError(
"State vector probabilities have to sum up to 1.0, got {}".format(
norm))
#######################
# Change ordering of indices, original code was for IBM machines
state_vector = np.array(state_vector).reshape(
[2] * n_wires).T.flatten()[:, np.newaxis]
state_vector = sparse.dok_matrix(state_vector)
wires = np.array(wires)
a = sparse.dok_matrix(state_vector.shape)
omega = sparse.dok_matrix(state_vector.shape)
for (i, j), v in state_vector.items():
if isinstance(v, Variable):
a[i, j] = np.absolute(v.val)
omega[i, j] = np.angle(v.val)
else:
a[i, j] = np.absolute(v)
omega[i, j] = np.angle(v)
# This code is directly applying the inverse of Carsten Blank's
# code to avoid inverting at the end
# Apply y rotations
for k in range(n_wires, 0, -1):
alpha_y_k = _get_alpha_y(a, n_wires, k) # type: sparse.dok_matrix
control = wires[k:]
target = wires[k - 1]
_uniform_rotation_y_dagger(alpha_y_k, control, target)
# Apply z rotations
for k in range(n_wires, 0, -1):
alpha_z_k = _get_alpha_z(omega, n_wires, k)
control = wires[k:]
target = wires[k - 1]
if len(alpha_z_k) > 0:
_uniform_rotation_z_dagger(alpha_z_k, control, target)
def test_check_shape_exception(self, inpt, target_shape, bound):
"""Tests that shape check fails for invalid arguments."""
with pytest.raises(ValueError, match="XXX"):
_check_shape(inpt, target_shape, bound=bound, msg="XXX")
def test_check_shape(self, inpt, target_shape, bound):
"""Tests that shape check succeeds for valid arguments."""
_check_shape(inpt, target_shape, bound=bound, msg="XXX")
def AmplitudeEmbedding(features, wires, pad=None, normalize=False):
r"""Encodes :math:`2^n` features into the amplitude vector of :math:`n` qubits.
If the total number of features to embed is less than the :math:`2^n` available amplitudes,
non-informative constants (zeros) can be padded to ``features``. To enable this, the argument
``pad`` should be set to ``True``.
The L2-norm of ``features`` must be one. By default, ``AmplitudeEmbedding`` expects a normalized
feature vector. The argument ``normalize`` can be set to ``True`` to automatically normalize it.
.. warning::
``AmplitudeEmbedding`` calls a circuit that involves non-trivial classical processing of the
features. The `features` argument is therefore not differentiable when using the template, and
gradients with respect to the argument cannot be computed by PennyLane.
Args:
features (array): input array of shape ``(2^n,)``
wires (Sequence[int] or int): qubit indices that the template acts on
pad (float or complex): if not None, the input is padded with this constant to size :math:`2^n`
normalize (Boolean): controls the activation of automatic normalization
Raises:
ValueError: if inputs do not have the correct format
"""
#############
# Input checks
_check_no_variable([pad, normalize], ['pad', 'normalize'])
wires, n_wires = _check_wires(wires)
n_ampl = 2**n_wires
if pad is None:
msg = "AmplitudeEmbedding must get a feature vector of size 2**len(wires), which is {}. Use 'pad' " \
"argument for automated padding.".format(n_ampl)
shp = _check_shape(features, (n_ampl,), msg=msg)
else:
msg = "AmplitudeEmbedding must get a feature vector of at least size 2**len(wires) = {}.".format(n_ampl)
shp = _check_shape(features, (n_ampl,), msg=msg, bound='max')
_check_type(pad, [float, complex, type(None)])
_check_type(normalize, [bool])
###############
# Pad
n_feats = shp[0]
if pad is not None and n_ampl > n_feats:
features = np.pad(features, (0, n_ampl-n_feats), mode='constant', constant_values=pad)
# Normalize
if isinstance(features[0], Variable):
feature_values = [s.val for s in features]
norm = np.sum(np.abs(feature_values)**2)
else:
norm = np.sum(np.abs(features)**2)
if not np.isclose(norm, 1.0, atol=TOLERANCE, rtol=0):
if normalize or pad:
features = features/np.sqrt(norm)
else:
raise ValueError("Vector of features has to be normalized to 1.0, got {}."
"Use 'normalization=True' to automatically normalize.".format(norm))
features = np.array(features)
QubitStateVector(features, wires=wires)
def QAOAEmbedding(features, weights, wires, local_field='Y'):
r"""
Encodes :math:`N` features into :math:`n>N` qubits, using a layered, trainable quantum
circuit that is inspired by the QAOA ansatz.
A single layer applies two circuits or "Hamiltonians": The first encodes the features, and the second is
a variational ansatz inspired by a 1-dimensional Ising model. The feature-encoding circuit associates features with
the angles of :class:`RX` rotations. The Ising ansatz consists of trainable two-qubit ZZ interactions
:math:`e^{-i \alpha \sigma_z \otimes \sigma_z}`,
and trainable local fields :math:`e^{-i \frac{\beta}{2} \sigma_{\mu}}`, where :math:`\sigma_{\mu}`
can be chosen to be :math:`\sigma_{x}`, :math:`\sigma_{y}` or :math:`\sigma_{z}`
(default choice is :math:`\sigma_{y}` or the ``RY`` gate), and :math:`\alpha, \beta` are adjustable gate parameters.
The number of features has to be smaller or equal to the number of qubits. If there are fewer features than
qubits, the feature-encoding rotation is replaced by a Hadamard gate.
The argument ``weights`` contains an array of the :math:`\alpha, \beta` parameters for each layer.
The number of layers :math:`L` is derived from the first dimension of ``weights``, which has the following
shape:
* :math:`(L, )`, if the embedding acts on a single wire,
* :math:`(L, 3)`, if the embedding acts on two wires,
* :math:`(L, 2n)` else.
After the :math:`L` th layer, another set of feature-encoding :class:`RX` gates is applied.
This is an example for the full embedding circuit using 2 layers, 3 features, 4 wires, and ``RY`` local fields:
|
.. figure:: ../../_static/qaoa_layers.png
:align: center
:width: 60%
:target: javascript:void(0);
|
.. note::
``QAOAEmbedding`` supports gradient computations with respect to both the ``features`` and the ``weights``
arguments. Note that trainable parameters need to be passed to the quantum node as positional arguments.
Args:
features (array): array of features to encode
weights (array): array of weights
wires (Sequence[int] or int): `n` qubit indices that the template acts on
local_field (str): type of local field used, one of ``'X'``, ``'Y'``, or ``'Z'``
Raises:
ValueError: if inputs do not have the correct format
.. UsageDetails::
The QAOA embedding encodes an :math:`n`-dimensional feature vector into at most :math:`n` qubits. The
embedding applies layers of a circuit, and each layer is defined by a set of weight parameters.
.. code-block:: python
import pennylane as qml
from pennylane.templates import QAOAEmbedding
dev = qml.device('default.qubit', wires=2)
@qml.qnode(dev)
def circuit(weights, f=None):
QAOAEmbedding(features=f, weights=weights, wires=range(2))
return qml.expval(qml.PauliZ(0))
features = [1., 2.]
layer1 = [0.1, -0.3, 1.5]
layer2 = [3.1, 0.2, -2.8]
weights = [layer1, layer2]
print(circuit(weights, f=features))
**Using parameter initialization functions**
The initial weight parameters can alternatively be generated by utility functions from the
``pennylane.init`` module, for example using the function :func:`~.qaoa_embedding_normal`:
.. code-block:: python
from pennylane.init import qaoa_embedding_normal
weights = qaoa_embedding_normal(n_layers=2, n_wires=2, mean=0, std=0.2)
**Training the embedding**
The embedding is typically trained with respect to a given cost. For example, one can train it to
minimize the PauliZ expectation of the first qubit:
.. code-block:: python
o = GradientDescentOptimizer()
for i in range(10):
weights = o.step(lambda w : circuit(w, f=features), weights)
print("Step ", i, " weights = ", weights)
**Training the features**
In principle, also the features are trainable, which means that gradients with respect to feature values
can be computed. To train both weights and features, they need to be passed to the qnode as
positional arguments. If the built-in optimizer is used, they have to be merged to one input:
.. code-block:: python
@qml.qnode(dev)
def circuit2(pars):
weights = pars[0]
features = pars[1]
QAOAEmbedding(features=features, weights=weights, wires=range(2))
return qml.expval(qml.PauliZ(0))
features = [1., 2.]
weights = [[0.1, -0.3, 1.5], [3.1, 0.2, -2.8]]
pars = [weights, features]
o = GradientDescentOptimizer()
for i in range(10):
pars = o.step(circuit2, pars)
print("Step ", i, " weights = ", pars[0], " features = ", pars[1])
**Local Fields**
While by default, ``RY`` gates are used as local fields, one may also choose ``local_field='Z'`` or
``local_field='X'`` as hyperparameters of the embedding.
.. code-block:: python
@qml.qnode(dev)
def circuit(weights, f=None):
QAOAEmbedding(features=f, weights=weights, wires=range(2), local_field='Z')
return qml.expval(qml.PauliZ(0))
Choosing ``'Z'`` fields implements a QAOAEmbedding where the second Hamiltonian is a
1-dimensional Ising model.
"""
#############
# Input checks
wires = _check_wires(wires)
expected_shape = (len(wires), )
_check_shape(features,
expected_shape,
bound='max',
msg="'features' must be of shape {} or smaller; got {}"
"".format((len(wires), ), _get_shape(features)))
_check_is_in_options(local_field, ['X', 'Y', 'Z'],
msg="did not recognize option {} for 'local_field'"
"".format(local_field))
repeat = _check_number_of_layers([weights])
if len(wires) == 1:
expected_shape = (repeat, 1)
_check_shape(weights,
expected_shape,
msg="'weights' must be of shape {}; got {}"
"".format(expected_shape, _get_shape(features)))
elif len(wires) == 2:
expected_shape = (repeat, 3)
_check_shape(weights,
expected_shape,
msg="'weights' must be of shape {}; got {}"
"".format(expected_shape, _get_shape(features)))
else:
expected_shape = (repeat, 2 * len(wires))
_check_shape(weights,
expected_shape,
msg="'weights' must be of shape {}; got {}"
"".format(expected_shape, _get_shape(features)))
#####################
n_features = _get_shape(features)[0]
if local_field == 'Z':
local_fields = RZ
elif local_field == 'X':
local_fields = RX
else:
local_fields = RY
for l in range(repeat):
# apply alternating Hamiltonians
qaoa_feature_encoding_hamiltonian(features, n_features, wires)
qaoa_ising_hamiltonian(weights, wires, local_fields, l)
# repeat the feature encoding once more at the end
qaoa_feature_encoding_hamiltonian(features, n_features, wires)
