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 (Iterable or Wires): Wires that the template acts on. Accepts an iterable of numbers or strings, or
a Wires object.
rotation (str): Type of rotations used
Raises:
ValueError: if inputs do not have the correct format
"""
#############
# Input checks
wires = 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":
broadcast(unitary=RX, pattern="single", wires=wires, parameters=features)
elif rotation == "Y":
broadcast(unitary=RY, pattern="single", wires=wires, parameters=features)
elif rotation == "Z":
broadcast(unitary=RZ, pattern="single", wires=wires, parameters=features)
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 (Iterable or Wires): Wires that the template acts on. Accepts an iterable of numbers or strings, or
a Wires object.
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
wires = 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),
)
#############
constants = [c] * len(features)
if method == "amplitude":
broadcast(
unitary=Displacement,
pattern="single",
wires=wires,
parameters=list(zip(features, constants)),
)
elif method == "phase":
broadcast(
unitary=Displacement,
pattern="single",
wires=wires,
parameters=list(zip(constants, features)),
)
def broadcast(unitary, wires, pattern, parameters=None, kwargs=None):
r"""Applies a unitary multiple times to a specific pattern of wires.
The unitary, defined by the argument ``unitary``, is either a quantum operation
(such as :meth:`~.pennylane.ops.RX`), or a
user-supplied template. Depending on the chosen pattern, ``unitary`` is applied to a wire or a subset of wires:
* ``pattern="single"`` applies a single-wire unitary to each one of the :math:`M` wires:
.. figure:: ../../_static/templates/broadcast_single.png
:align: center
:width: 20%
:target: javascript:void(0);
* ``pattern="double"`` applies a two-wire unitary to :math:`\lfloor \frac{M}{2} \rfloor`
subsequent pairs of wires:
.. figure:: ../../_static/templates/broadcast_double.png
:align: center
:width: 20%
:target: javascript:void(0);
* ``pattern="double_odd"`` applies a two-wire unitary to :math:`\lfloor \frac{M-1}{2} \rfloor`
subsequent pairs of wires, starting with the second wire:
.. figure:: ../../_static/templates/broadcast_double_odd.png
:align: center
:width: 20%
:target: javascript:void(0);
* ``pattern="chain"`` applies a two-wire unitary to all :math:`M-1` neighbouring pairs of wires:
.. figure:: ../../_static/templates/broadcast_chain.png
:align: center
:width: 20%
:target: javascript:void(0);
* ``pattern="ring"`` applies a two-wire unitary to all :math:`M` neighbouring pairs of wires,
where the last wire is considered to be a neighbour to the first one:
.. figure:: ../../_static/templates/broadcast_ring.png
:align: center
:width: 20%
:target: javascript:void(0);
.. note:: For 2 wires, the ring pattern is automatically replaced by ``pattern = 'chain'`` to avoid
a mere repetition of the unitary.
* ``pattern="pyramid"`` applies a two-wire unitary to wire pairs shaped in a pyramid declining to the right:
.. figure:: ../../_static/templates/broadcast_pyramid.png
:align: center
:width: 20%
:target: javascript:void(0);
* ``pattern="all_to_all"`` applies a two-wire unitary to wire pairs that connect all wires to each other:
.. figure:: ../../_static/templates/broadcast_alltoall.png
:align: center
:width: 20%
:target: javascript:void(0);
* A custom pattern can be passed by providing a list of wire lists to ``pattern``. The ``unitary`` is applied
to each set of wires specified in the list.
.. figure:: ../../_static/templates/broadcast_custom.png
:align: center
:width: 20%
:target: javascript:void(0);
Each ``unitary`` may depend on a different set of parameters. These are passed as a list by the ``parameters``
argument.
For more details, see *Usage Details* below.
Args:
unitary (func): quantum gate or template
pattern (str): specifies the wire pattern of the broadcast
parameters (list): sequence of parameters for each gate applied
wires (Iterable or Wires): Wires that the template acts on. Accepts an iterable of numbers or strings, or
a Wires object.
kwargs (dict): dictionary of auxilliary parameters for ``unitary``
Raises:
ValueError: if inputs do not have the correct format
.. UsageDetails::
**Broadcasting single gates**
In the simplest case the unitary is typically an :meth:`~.pennylane.operation.Operation` object
implementing a quantum gate.
.. code-block:: python
import pennylane as qml
from pennylane import broadcast
dev = qml.device('default.qubit', wires=3)
@qml.qnode(dev)
def circuit(pars):
broadcast(unitary=qml.RX, pattern="single", wires=[0,1,2], parameters=pars)
return qml.expval(qml.PauliZ(0))
circuit([1, 1, 2])
This is equivalent to the following circuit:
.. code-block:: python
@qml.qnode(dev)
def circuit(pars):
qml.RX(pars[0], wires=[0])
qml.RX(pars[1], wires=[1])
qml.RX(pars[2], wires=[2])
return qml.expval(qml.PauliZ(0))
circuit([1, 1, 2])
**Broadcasting templates**
Alternatively, one can broadcast a built-in or user-defined template:
.. code-block:: python
from pennylane.templates import template
@template
def mytemplate(pars, wires):
qml.Hadamard(wires=wires)
qml.RY(pars, wires=wires)
dev = qml.device('default.qubit', wires=3)
@qml.qnode(dev)
def circuit(pars):
broadcast(unitary=mytemplate, pattern="single", wires=[0,1,2], parameters=pars)
return qml.expval(qml.PauliZ(0))
print(circuit([1, 1, 0.1]))
**Constant unitaries**
If the ``unitary`` argument does not take parameters, no ``parameters`` argument is passed to
:func:`~.pennylane.broadcast`:
.. code-block:: python
dev = qml.device('default.qubit', wires=3)
@qml.qnode(dev)
def circuit():
broadcast(unitary=qml.Hadamard, pattern="single", wires=[0,1,2])
return qml.expval(qml.PauliZ(0))
circuit()
**Multiple parameters in unitary**
The unitary, whether it is a single gate or a user-defined template,
can take multiple parameters. For example:
.. code-block:: python
from pennylane.templates import template
@template
def mytemplate(pars1, pars2, wires):
qml.Hadamard(wires=wires)
qml.RY(pars1, wires=wires)
qml.RX(pars2, wires=wires)
@qml.qnode(dev)
def circuit(pars):
broadcast(unitary=mytemplate, pattern="single", wires=[0,1,2], parameters=pars)
return qml.expval(qml.PauliZ(0))
circuit([[1, 1], [2, 1], [0.1, 1]])
In general, the unitary takes D parameters and **must** have the following signature:
.. code-block:: python
unitary(parameter1, parameter2, ... parameterD, wires, **kwargs)
If ``unitary`` does not depend on parameters (:math:`D=0`), the signature is
.. code-block:: python
unitary(wires, **kwargs)
As a result, ``parameters`` must be a list or array of length-:math:`D` lists or arrays.
If :math:`D` becomes large, the signature can be simplified by wrapping each entry in ``parameters``:
.. code-block:: python
@template
def mytemplate(pars, wires):
qml.Hadamard(wires=wires)
qml.RY(pars[0], wires=wires)
qml.RX(pars[1], wires=wires)
@qml.qnode(dev)
def circuit(pars):
broadcast(unitary=mytemplate, pattern="single", wires=[0,1,2], parameters=pars)
return qml.expval(qml.PauliZ(0))
print(circuit([[[1, 1]], [[2, 1]], [[0.1, 1]]]))
If the number of parameters for each wire does not match the unitary, an error gets thrown:
.. code-block:: python
@template
def mytemplate(pars1, pars2, wires):
qml.Hadamard(wires=wires)
qml.RY(pars1, wires=wires)
qml.RX(pars2, wires=wires)
@qml.qnode(dev)
def circuit(pars):
broadcast(unitary=mytemplate, pattern="single", wires=[0, 1, 2], parameters=pars)
return qml.expval(qml.PauliZ(0))
>>> circuit([1, 2, 3]))
TypeError: mytemplate() missing 1 required positional argument: 'pars2'
**Keyword arguments**
The unitary can be a template that takes additional keyword arguments.
.. code-block:: python
@template
def mytemplate(wires, h=True):
if h:
qml.Hadamard(wires=wires)
qml.T(wires=wires)
@qml.qnode(dev)
def circuit(hadamard=None):
broadcast(unitary=mytemplate, pattern="single", wires=[0, 1, 2], kwargs={'h': hadamard})
return qml.expval(qml.PauliZ(0))
circuit(hadamard=False)
**Different patterns**
The basic usage of the different patterns works as follows:
* Double pattern
.. code-block:: python
dev = qml.device('default.qubit', wires=4)
@qml.qnode(dev)
def circuit(pars):
broadcast(unitary=qml.CRot, pattern='double',
wires=[0,1,2,3], parameters=pars)
return qml.expval(qml.PauliZ(0))
pars1 = [-1, 2.5, 3]
pars2 = [-1, 4, 2]
circuit([pars1, pars2])
* Double-odd pattern
.. code-block:: python
dev = qml.device('default.qubit', wires=4)
@qml.qnode(dev)
def circuit(pars):
broadcast(unitary=qml.CRot, pattern='double_odd',
wires=[0,1,2,3], parameters=pars)
return qml.expval(qml.PauliZ(0))
pars1 = [-5.3, 2.3, 3]
circuit([pars1])
* Chain pattern
.. code-block:: python
dev = qml.device('default.qubit', wires=4)
@qml.qnode(dev)
def circuit(pars):
broadcast(unitary=qml.CRot, pattern='chain',
wires=[0,1,2,3], parameters=pars)
return qml.expval(qml.PauliZ(0))
pars1 = [1.8, 2, 3]
pars2 = [-1, 3, 1]
pars3 = [2, -1.2, 4]
circuit([pars1, pars2, pars3])
* Ring pattern
In general, the number of parameter sequences has to match
the number of wires:
.. code-block:: python
dev = qml.device('default.qubit', wires=3)
@qml.qnode(dev)
def circuit(pars):
broadcast(unitary=qml.CRot, pattern='ring',
wires=[0,1,2], parameters=pars)
return qml.expval(qml.PauliZ(0))
pars1 = [1, -2.2, 3]
pars2 = [-1, 3, 1]
pars3 = [2.6, 1, 4]
circuit([pars1, pars2, pars3])
However, there is an exception for 2 wires, where only one set of parameters is needed.
This avoids repeating a gate over the
same wires twice:
.. code-block:: python
dev = qml.device('default.qubit', wires=2)
@qml.qnode(dev)
def circuit(pars):
broadcast(unitary=qml.CRot, pattern='ring',
wires=[0,1], parameters=pars)
return qml.expval(qml.PauliZ(0))
pars1 = [-3.2, 2, 1.2]
circuit([pars1])
* Pyramid pattern
.. code-block:: python
dev = qml.device('default.qubit', wires=4)
@qml.qnode(dev)
def circuit(pars):
broadcast(unitary=qml.CRot, pattern='pyramid',
wires=[0,1,2,3], parameters=pars)
return qml.expval(qml.PauliZ(0))
pars1 = [1.1, 2, 3]
pars2 = [-1, 3, 1]
pars3 = [2, 1, 4.2]
circuit([pars1, pars2, pars3])
* All-to-all pattern
.. code-block:: python
dev = qml.device('default.qubit', wires=4)
@qml.qnode(dev)
def circuit(pars):
broadcast(unitary=qml.CRot, pattern='ring',
wires=[0,1,2,3], parameters=pars)
return qml.expval(qml.PauliZ(0))
pars1 = [1, 2, 3]
pars2 = [-1, 3, 1]
pars3 = [2, 1, 4]
pars4 = [-1, -2, -3]
pars5 = [2, 1, 4]
pars6 = [3, -2, -3]
circuit([pars1, pars2, pars3, pars4, pars5, pars6])
* Custom pattern
For a custom pattern, the wire lists for each application of the unitary is
passed to ``pattern``:
.. code-block:: python
dev = qml.device('default.qubit', wires=5)
pattern = [[0, 1], [3, 4]]
@qml.qnode(dev)
def circuit():
broadcast(unitary=qml.CNOT, pattern=pattern,
wires=range(5))
return qml.expval(qml.PauliZ(0))
circuit()
When using a parametrized unitary, make sure that the number of wire lists in ``pattern`` corresponds to the
number of parameters in ``parameters``.
.. code-block:: python
pattern = [[0, 1], [3, 4]]
@qml.qnode(dev)
def circuit(pars):
broadcast(unitary=qml.CRot, pattern=pattern,
wires=range(5), parameters=pars)
return qml.expval(qml.PauliZ(0))
pars1 = [1, 2, 3]
pars2 = [-1, 3, 1]
pars = [pars1, pars2]
assert len(pars) == len(pattern)
circuit(pars)
"""
OPTIONS = [
"single", "double", "double_odd", "chain", "ring", "pyramid",
"all_to_all", "custom"
]
#########
# Input checks
wires = Wires(wires)
check_type(
parameters,
[Iterable, type(None)],
msg="'parameters' must be either of type None or "
"Iterable; got {}".format(type(parameters)),
)
if kwargs is None:
kwargs = {}
check_type(
kwargs,
[dict],
msg="'kwargs' must be a dictionary; got {}".format(type(kwargs)),
)
custom_pattern = None
if isinstance(pattern, str):
check_is_in_options(
pattern,
OPTIONS,
msg="did not recognize option {} for 'pattern'".format(pattern),
)
else:
# turn custom pattern into list of Wires objects
custom_pattern = [Wires(w) for w in pattern]
# set "pattern" to "custom", indicating that custom settings have to be used
pattern = "custom"
n_parameters = {
"single":
len(wires),
"double":
0 if len(wires) in [0, 1] else len(wires) // 2,
"double_odd":
0 if len(wires) in [0, 1] else (len(wires) - 1) // 2,
"chain":
0 if len(wires) in [0, 1] else len(wires) - 1,
"ring":
0 if len(wires) in [0, 1] else (1 if len(wires) == 2 else len(wires)),
"pyramid":
0 if len(wires) in [0, 1] else sum(i + 1
for i in range(len(wires) // 2)),
"all_to_all":
0 if len(wires) in [0, 1] else len(wires) * (len(wires) - 1) // 2,
"custom":
len(custom_pattern) if custom_pattern is not None else None,
}
# check that there are enough parameters for pattern
if parameters is not None:
shape = get_shape(parameters)
# specific error message for ring edge case of 2 wires
if (pattern == "ring") and (len(wires) == 2) and (shape[0] != 1):
raise ValueError(
"the ring pattern with 2 wires is an exception and only applies one unitary"
)
if shape[0] != n_parameters[pattern]:
raise ValueError(
"'parameters' must contain entries for {} unitaries; got {} entries"
.format(n_parameters[pattern], shape[0]))
# repackage for consistent unpacking
if len(shape) == 1:
parameters = [[p] for p in parameters]
else:
parameters = [[] for _ in range(n_parameters[pattern])]
#########
# define wire sequences for patterns
wire_sequence = {
"single": [wires[i] for i in range(len(wires))],
"double":
[wires.subset([i, i + 1]) for i in range(0,
len(wires) - 1, 2)],
"double_odd":
[wires.subset([i, i + 1]) for i in range(1,
len(wires) - 1, 2)],
"chain": [wires.subset([i, i + 1]) for i in range(len(wires) - 1)],
"ring":
wires_ring(wires),
"pyramid":
wires_pyramid(wires),
"all_to_all":
wires_all_to_all(wires),
"custom":
custom_pattern,
}
# broadcast the unitary
for wires, pars in zip(wire_sequence[pattern], parameters):
wires = wires.tolist(
) # TODO: Delete once operator takes Wires objects
unitary(*pars, wires=wires, **kwargs)
def Interferometer(theta,
phi,
varphi,
wires,
mesh="rectangular",
beamsplitter="pennylane"):
r"""General linear interferometer, an array of beamsplitters and phase shifters.
For :math:`M` wires, the general interferometer is specified by
providing :math:`M(M-1)/2` transmittivity angles :math:`\theta` and the same number of
phase angles :math:`\phi`, as well as :math:`M-1` additional rotation
parameters :math:`\varphi`.
By specifying the keyword argument ``mesh``, the scheme used to implement the interferometer
may be adjusted:
* ``mesh='rectangular'`` (default): uses the scheme described in
`Clements et al. <https://dx.doi.org/10.1364/OPTICA.3.001460>`__, resulting in a *rectangular* array of
:math:`M(M-1)/2` beamsplitters arranged in :math:`M` slices and ordered from left
to right and top to bottom in each slice. The first beamsplitter acts on
wires :math:`0` and :math:`1`:
.. figure:: ../../_static/clements.png
:align: center
:width: 30%
:target: javascript:void(0);
* ``mesh='triangular'``: uses the scheme described in `Reck et al. <https://dx.doi.org/10.1103/PhysRevLett.73.58>`__,
resulting in a *triangular* array of :math:`M(M-1)/2` beamsplitters arranged in
:math:`2M-3` slices and ordered from left to right and top to bottom. The
first and fourth beamsplitters act on wires :math:`M-1` and :math:`M`, the second
on :math:`M-2` and :math:`M-1`, and the third on :math:`M-3` and :math:`M-2`, and
so on.
.. figure:: ../../_static/reck.png
:align: center
:width: 30%
:target: javascript:void(0);
In both schemes, the network of :class:`~pennylane.ops.Beamsplitter` operations is followed by
:math:`M` local :class:`~pennylane.ops.Rotation` Operations.
The rectangular decomposition is generally advantageous, as it has a lower
circuit depth (:math:`M` vs :math:`2M-3`) and optical depth than the triangular
decomposition, resulting in reduced optical loss.
This is an example of a 4-mode interferometer with beamsplitters :math:`B` and rotations :math:`R`,
using ``mesh='rectangular'``:
.. figure:: ../../_static/layer_interferometer.png
:align: center
:width: 60%
:target: javascript:void(0);
.. note::
The decomposition as formulated in `Clements et al. <https://dx.doi.org/10.1364/OPTICA.3.001460>`__ uses a different
convention for a beamsplitter :math:`T(\theta, \phi)` than PennyLane, namely:
.. math:: T(\theta, \phi) = BS(\theta, 0) R(\phi)
For the universality of the decomposition, the used convention is irrelevant, but
for a given set of angles the resulting interferometers will be different.
If an interferometer consistent with the convention from `Clements et al. <https://dx.doi.org/10.1364/OPTICA.3.001460>`__
is needed, the optional keyword argument ``beamsplitter='clements'`` can be specified. This
will result in each :class:`~pennylane.ops.Beamsplitter` being preceded by a :class:`~pennylane.ops.Rotation` and
thus increase the number of elementary operations in the circuit.
Args:
theta (array): length :math:`M(M-1)/2` array of transmittivity angles :math:`\theta`
phi (array): length :math:`M(M-1)/2` array of phase angles :math:`\phi`
varphi (array): length :math:`M` array of rotation angles :math:`\varphi`
wires (Sequence[int]): wires the interferometer should act on
mesh (string): the type of mesh to use
beamsplitter (str): if ``clements``, the beamsplitter convention from
Clements et al. 2016 (https://dx.doi.org/10.1364/OPTICA.3.001460) is used; if ``pennylane``, the
beamsplitter is implemented via PennyLane's ``Beamsplitter`` operation.
Raises:
ValueError: if inputs do not have the correct format
"""
#############
# Input checks
check_no_variable(beamsplitter,
msg="'beamsplitter' cannot be differentiable")
check_no_variable(mesh, msg="'mesh' cannot be differentiable")
wires = check_wires(wires)
weights_list = [theta, phi, varphi]
n_wires = len(wires)
n_if = n_wires * (n_wires - 1) // 2
expected_shapes = [(n_if, ), (n_if, ), (n_wires, )]
check_shapes(weights_list,
expected_shapes,
msg="wrong shape of weight input(s) detected")
check_is_in_options(
beamsplitter,
["clements", "pennylane"],
msg="did not recognize option {} for 'beamsplitter'"
"".format(beamsplitter),
)
check_is_in_options(
mesh,
["triangular", "rectangular"],
msg="did not recognize option {} for 'mesh'"
"".format(mesh),
)
###############
M = len(wires)
if M == 1:
# the interferometer is a single rotation
Rotation(varphi[0], wires=wires[0])
return
n = 0 # keep track of free parameters
if mesh == "rectangular":
# Apply the Clements beamsplitter array
# The array depth is N
for l in range(M):
for k, (w1, w2) in enumerate(zip(wires[:-1], wires[1:])):
# skip even or odd pairs depending on layer
if (l + k) % 2 != 1:
if beamsplitter == "clements":
Rotation(phi[n], wires=[w1])
Beamsplitter(theta[n], 0, wires=[w1, w2])
else:
Beamsplitter(theta[n], phi[n], wires=[w1, w2])
n += 1
elif mesh == "triangular":
# apply the Reck beamsplitter array
# The array depth is 2*N-3
for l in range(2 * M - 3):
for k in range(abs(l + 1 - (M - 1)), M - 1, 2):
if beamsplitter == "clements":
Rotation(phi[n], wires=[wires[k]])
Beamsplitter(theta[n], 0, wires=[wires[k], wires[k + 1]])
else:
Beamsplitter(theta[n],
phi[n],
wires=[wires[k], wires[k + 1]])
n += 1
# apply the final local phase shifts to all modes
for i, p in enumerate(varphi):
Rotation(p, wires=[wires[i]])
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 (Iterable or Wires): Wires that the template acts on. Accepts an iterable of numbers or strings, or
a Wires object.
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
wires = Wires(wires)
check_no_variable(method, msg="'method' cannot be differentiable")
check_no_variable(c, msg="'c' cannot be differentiable")
check_type(c, [float, int], msg="'c' must be of type float or integer; got {}".format(type(c)))
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),
)
##############
constants = [c] * len(features)
if method == "amplitude":
broadcast(
unitary=Squeezing,
pattern="single",
wires=wires,
parameters=list(zip(features, constants)),
)
elif method == "phase":
broadcast(
unitary=Squeezing,
pattern="single",
wires=wires,
parameters=list(zip(constants, features)),
)
def _preprocess(features, wires, method, c):
"""Validate and pre-process inputs as follows:
* Check that the features tensor is one-dimensional.
* Check that the first dimension of the features tensor
has length :math:`n` or less, where :math:`n` is the number of qubits.
* Create a parameter tensor which combines the features with a tensor of constants.
Args:
features (tensor_like): input features to pre-process
wires (Wires): wires that template acts on
method (str): indicates whether amplitude or phase encoding is used
c (float): value of constant
Returns:
tensor_like: 2-dimensional tensor containing the features and constants
"""
if qml.tape_mode_active():
shape = qml.math.shape(features)
constants = [c] * shape[0]
if len(shape) != 1:
raise ValueError(
f"Features must be a one-dimensional tensor; got shape {shape}."
)
n_features = shape[0]
if n_features != len(wires):
raise ValueError(
f"Features must be of length {len(wires)}; got length {n_features}."
)
if method == "amplitude":
pars = qml.math.stack([features, constants], axis=1)
elif method == "phase":
pars = qml.math.stack([constants, features], axis=1)
else:
raise ValueError(f"did not recognize method {method}")
else:
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),
)
constants = [c] * len(features)
if method == "amplitude":
pars = list(zip(features, constants))
elif method == "phase":
pars = list(zip(constants, features))
return pars
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 \frac{\alpha}{2} \sigma_z \otimes \sigma_z}` (in PennyLane represented by the :class:`~.MultiRZ` gate),
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, 1)`, 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 (Iterable or Wires): Wires that the template acts on. Accepts an iterable of numbers or strings, or
a Wires object.
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 = 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)),
)
#####################
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, wires)
qaoa_ising_hamiltonian(weights[l], wires, local_fields)
# repeat the feature encoding once more at the end
qaoa_feature_encoding_hamiltonian(features, wires)
def test_check_options_exception(self, hp, opts):
"""Tests that option check throws error for invalid arguments."""
with pytest.raises(ValueError, match="XXX"):
check_is_in_options(hp, opts, msg="XXX")
def test_check_options(self, hp, opts):
"""Tests that option check succeeds for valid arguments."""
check_is_in_options(hp, opts, msg="XXX")
