Source code for pennylane.ops.qubit.matrix_ops
# Copyright 2018-2021 Xanadu Quantum Technologies Inc.
# 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
# http://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.
"""
This submodule contains the discrete-variable quantum operations that
accept a hermitian or an unitary matrix as a parameter.
"""
# pylint:disable=arguments-differ
import warnings
from itertools import product
import numpy as np
from pennylane.math import norm, cast, eye, zeros, transpose, conj, sqrt, sqrt_matrix
from pennylane import numpy as pnp
import pennylane as qml
from pennylane.operation import AnyWires, DecompositionUndefinedError, Operation
from pennylane.wires import Wires
_walsh_hadamard_matrix = np.array([[1, 1], [1, -1]]) / 2
def _walsh_hadamard_transform(D, n=None):
r"""Compute the Walsh–Hadamard Transform of a one-dimensional array.
Args:
D (tensor_like): The array or tensor to be transformed. Must have a length that
is a power of two.
Returns:
tensor_like: The transformed tensor with the same shape as the input ``D``.
Due to the execution of the transform as a sequence of tensor multiplications
with shapes ``(2, 2), (2, 2,... 2)->(2, 2,... 2)``, the theoretical scaling of this
method is the same as the one for the
`Fast Walsh-Hadamard transform <https://en.wikipedia.org/wiki/Fast_Walsh-Hadamard_transform>`__:
On ``n`` qubits, there are ``n`` calls to ``tensordot``, each multiplying a
``(2, 2)`` matrix to a ``(2,)*n`` vector, with a single axis being contracted. This means
that there are ``n`` operations with a FLOP count of ``4 * 2**(n-1)``, where ``4`` is the cost
of a single ``(2, 2) @ (2,)`` contraction and ``2**(n-1)`` is the number of copies due to the
non-contracted ``n-1`` axes.
Due to the large internal speedups of compiled matrix multiplication and compatibility
with autodifferentiation frameworks, the approach taken here is favourable over a manual
realization of the FWHT unless memory limitations restrict the creation of intermediate
arrays.
"""
orig_shape = qml.math.shape(D)
n = n or int(qml.math.log2(orig_shape[-1]))
# Reshape the array so that we may apply the Hadamard transform to each axis individually
if broadcasted := len(orig_shape) > 1:
new_shape = (orig_shape[0],) + (2,) * n
else:
new_shape = (2,) * n
D = qml.math.reshape(D, new_shape)
# Apply Hadamard transform to each axis, shifted by one for broadcasting
for i in range(broadcasted, n + broadcasted):
D = qml.math.tensordot(_walsh_hadamard_matrix, D, axes=[[1], [i]])
# The axes are in reverted order after all matrix multiplications, so we need to transpose;
# If D was broadcasted, this moves the broadcasting axis to first position as well.
# Finally, reshape to original shape
return qml.math.reshape(qml.math.transpose(D), orig_shape)
[docs]class QubitUnitary(Operation):
r"""QubitUnitary(U, wires)
Apply an arbitrary unitary matrix with a dimension that is a power of two.
**Details:**
* Number of wires: Any (the operation can act on any number of wires)
* Number of parameters: 1
* Number of dimensions per parameter: (2,)
* Gradient recipe: None
Args:
U (array[complex]): square unitary matrix
wires (Sequence[int] or int): the wire(s) the operation acts on
id (str): custom label given to an operator instance,
can be useful for some applications where the instance has to be identified
unitary_check (bool): check for unitarity of the given matrix
Raises:
ValueError: if the number of wires doesn't fit the dimensions of the matrix
**Example**
>>> dev = qml.device('default.qubit', wires=1)
>>> U = 1 / np.sqrt(2) * np.array([[1, 1], [1, -1]])
>>> @qml.qnode(dev)
... def example_circuit():
... qml.QubitUnitary(U, wires=0)
... return qml.expval(qml.PauliZ(0))
>>> print(example_circuit())
0.0
"""
num_wires = AnyWires
"""int: Number of wires that the operator acts on."""
num_params = 1
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = (2,)
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
grad_method = None
"""Gradient computation method."""
def __init__(
self, U, wires, id=None, unitary_check=False
): # pylint: disable=too-many-arguments
wires = Wires(wires)
U_shape = qml.math.shape(U)
dim = 2 ** len(wires)
# For pure QubitUnitary operations (not controlled), check that the number
# of wires fits the dimensions of the matrix
if len(U_shape) not in {2, 3} or U_shape[-2:] != (dim, dim):
raise ValueError(
f"Input unitary must be of shape {(dim, dim)} or (batch_size, {dim}, {dim}) "
f"to act on {len(wires)} wires."
)
# Check for unitarity; due to variable precision across the different ML frameworks,
# here we issue a warning to check the operation, instead of raising an error outright.
if unitary_check and not (
qml.math.is_abstract(U)
or qml.math.allclose(
qml.math.einsum("...ij,...kj->...ik", U, qml.math.conj(U)),
qml.math.eye(dim),
atol=1e-6,
)
):
warnings.warn(
f"Operator {U}\n may not be unitary. "
"Verify unitarity of operation, or use a datatype with increased precision.",
UserWarning,
)
super().__init__(U, wires=wires, id=id)
[docs] @staticmethod
def compute_matrix(U): # pylint: disable=arguments-differ
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.QubitUnitary.matrix`
Args:
U (tensor_like): unitary matrix
Returns:
tensor_like: canonical matrix
**Example**
>>> U = np.array([[0.98877108+0.j, 0.-0.14943813j], [0.-0.14943813j, 0.98877108+0.j]])
>>> qml.QubitUnitary.compute_matrix(U)
[[0.98877108+0.j, 0.-0.14943813j],
[0.-0.14943813j, 0.98877108+0.j]]
"""
return U
[docs] @staticmethod
def compute_decomposition(U, wires):
r"""Representation of the operator as a product of other operators (static method).
.. math:: O = O_1 O_2 \dots O_n.
A decomposition is only defined for matrices that act on either one or two wires. For more
than two wires, this method raises a ``DecompositionUndefined``.
See :func:`~.transforms.one_qubit_decomposition` and :func:`~.transforms.two_qubit_decomposition`
for more information on how the decompositions are computed.
.. seealso:: :meth:`~.QubitUnitary.decomposition`.
Args:
U (array[complex]): square unitary matrix
wires (Iterable[Any] or Wires): the wire(s) the operation acts on
Returns:
list[Operator]: decomposition of the operator
**Example:**
>>> U = 1 / np.sqrt(2) * np.array([[1, 1], [1, -1]])
>>> qml.QubitUnitary.compute_decomposition(U, 0)
[Rot(tensor(3.14159265, requires_grad=True), tensor(1.57079633, requires_grad=True), tensor(0., requires_grad=True), wires=[0])]
"""
# Decomposes arbitrary single-qubit unitaries as Rot gates (RZ - RY - RZ format),
# or a single RZ for diagonal matrices.
shape = qml.math.shape(U)
is_batched = len(shape) == 3
shape_without_batch_dim = shape[1:] if is_batched else shape
if shape_without_batch_dim == (2, 2):
return qml.transforms.decompositions.one_qubit_decomposition(U, Wires(wires)[0])
if shape_without_batch_dim == (4, 4):
# TODO[dwierichs]: Implement decomposition of broadcasted unitary
if is_batched:
raise DecompositionUndefinedError(
"The decomposition of a two-qubit QubitUnitary does not support broadcasting."
)
return qml.transforms.two_qubit_decomposition(U, Wires(wires))
return super(QubitUnitary, QubitUnitary).compute_decomposition(U, wires=wires)
# pylint: disable=arguments-renamed, invalid-overridden-method
@property
def has_decomposition(self):
return len(self.wires) < 3
[docs] def adjoint(self):
U = self.matrix()
return QubitUnitary(qml.math.moveaxis(qml.math.conj(U), -2, -1), wires=self.wires)
[docs] def pow(self, z):
if isinstance(z, int):
return [QubitUnitary(qml.math.linalg.matrix_power(self.matrix(), z), wires=self.wires)]
return super().pow(z)
def _controlled(self, wire):
return qml.ControlledQubitUnitary(*self.parameters, control_wires=wire, wires=self.wires)
[docs] def label(self, decimals=None, base_label=None, cache=None):
return super().label(decimals=decimals, base_label=base_label or "U", cache=cache)
[docs]class DiagonalQubitUnitary(Operation):
r"""DiagonalQubitUnitary(D, wires)
Apply an arbitrary diagonal unitary matrix with a dimension that is a power of two.
**Details:**
* Number of wires: Any (the operation can act on any number of wires)
* Number of parameters: 1
* Number of dimensions per parameter: (1,)
* Gradient recipe: None
Args:
D (array[complex]): diagonal of unitary matrix
wires (Sequence[int] or int): the wire(s) the operation acts on
"""
num_wires = AnyWires
"""int: Number of wires that the operator acts on."""
num_params = 1
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = (1,)
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
grad_method = None
"""Gradient computation method."""
[docs] @staticmethod
def compute_matrix(D): # pylint: disable=arguments-differ
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.DiagonalQubitUnitary.matrix`
Args:
D (tensor_like): diagonal of the matrix
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.DiagonalQubitUnitary.compute_matrix(torch.tensor([1, -1]))
tensor([[ 1, 0],
[ 0, -1]])
"""
D = qml.math.asarray(D)
if not qml.math.is_abstract(D) and not qml.math.allclose(
D * qml.math.conj(D), qml.math.ones_like(D)
):
raise ValueError("Operator must be unitary.")
# The diagonal is supposed to have one-dimension. If it is broadcasted, it has two
if qml.math.ndim(D) == 2:
return qml.math.stack([qml.math.diag(_D) for _D in D])
return qml.math.diag(D)
[docs] @staticmethod
def compute_eigvals(D): # pylint: disable=arguments-differ
r"""Eigenvalues of the operator in the computational basis (static method).
If :attr:`diagonalizing_gates` are specified and implement a unitary :math:`U^{\dagger}`,
the operator can be reconstructed as
.. math:: O = U \Sigma U^{\dagger},
where :math:`\Sigma` is the diagonal matrix containing the eigenvalues.
Otherwise, no particular order for the eigenvalues is guaranteed.
.. seealso:: :meth:`~.DiagonalQubitUnitary.eigvals`
Args:
D (tensor_like): diagonal of the matrix
Returns:
tensor_like: eigenvalues
**Example**
>>> qml.DiagonalQubitUnitary.compute_eigvals(torch.tensor([1, -1]))
tensor([ 1, -1])
"""
D = qml.math.asarray(D)
if not (
qml.math.is_abstract(D)
or qml.math.allclose(D * qml.math.conj(D), qml.math.ones_like(D))
):
raise ValueError("Operator must be unitary.")
return D
[docs] @staticmethod
def compute_decomposition(D, wires):
r"""Representation of the operator as a product of other operators (static method).
.. math:: O = O_1 O_2 \dots O_n.
``DiagonalQubitUnitary`` decomposes into :class:`~.QubitUnitary`, :class:`~.RZ`,
:class:`~.IsingZZ`, and/or :class:`~.MultiRZ` depending on the number of wires.
.. note::
The parameters of the decomposed operations are cast to the ``complex128`` dtype
as real dtypes can lead to ``NaN`` values in the decomposition.
.. seealso:: :meth:`~.DiagonalQubitUnitary.decomposition`.
Args:
D (tensor_like): diagonal of the matrix
wires (Iterable[Any] or Wires): the wire(s) the operation acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> diag = np.exp(1j * np.array([0.4, 2.1, 0.5, 1.8]))
>>> qml.DiagonalQubitUnitary.compute_decomposition(diag, wires=[0, 1])
[QubitUnitary(array([[0.36235775+0.93203909j, 0. +0.j ],
[0. +0.j , 0.36235775+0.93203909j]]), wires=[0]),
RZ(1.5000000000000002, wires=[1]),
RZ(-0.10000000000000003, wires=[0]),
IsingZZ(0.2, wires=[0, 1])]
"""
n = len(wires)
# Cast the diagonal into a complex dtype so that the logarithm works as expected
D_casted = qml.math.cast(D, "complex128")
phases = qml.math.real(qml.math.log(D_casted) * (-1j))
coeffs = _walsh_hadamard_transform(phases, n).T
global_phase = qml.math.exp(1j * coeffs[0])
# For all other gates, there is a prefactor -1/2 to be compensated.
coeffs = coeffs * (-2.0)
# TODO: Replace the following by a GlobalPhase gate.
ops = [QubitUnitary(qml.math.tensordot(global_phase, qml.math.eye(2), axes=0), wires[0])]
for wire0 in range(n):
# Single PauliZ generators correspond to the coeffs at powers of two
ops.append(qml.RZ(coeffs[1 << wire0], n - 1 - wire0))
# Double PauliZ generators correspond to the coeffs at the sum of two powers of two
ops.extend(
qml.IsingZZ(coeffs[(1 << wire0) + (1 << wire1)], [n - 1 - wire0, n - 1 - wire1])
for wire1 in range(wire0)
)
# Add all multi RZ gates that are not generated by single or double PauliZ generators
ops.extend(
qml.MultiRZ(c, [wires[k] for k in np.where(term)[0]])
for c, term in zip(coeffs, product((0, 1), repeat=n))
if sum(term) > 2
)
return ops
[docs] def adjoint(self):
return DiagonalQubitUnitary(qml.math.conj(self.parameters[0]), wires=self.wires)
[docs] def pow(self, z):
cast_data = qml.math.cast(self.data[0], np.complex128)
return [DiagonalQubitUnitary(cast_data**z, wires=self.wires)]
def _controlled(self, control):
return DiagonalQubitUnitary(
qml.math.hstack([np.ones_like(self.parameters[0]), self.parameters[0]]),
wires=control + self.wires,
)
[docs] def label(self, decimals=None, base_label=None, cache=None):
return super().label(decimals=decimals, base_label=base_label or "U", cache=cache)
[docs]class BlockEncode(Operation):
r"""BlockEncode(A, wires)
Construct a unitary :math:`U(A)` such that an arbitrary matrix :math:`A`
is encoded in the top-left block.
.. math::
\begin{align}
U(A) &=
\begin{bmatrix}
A & \sqrt{I-AA^\dagger} \\
\sqrt{I-A^\dagger A} & -A^\dagger
\end{bmatrix}.
\end{align}
**Details:**
* Number of wires: Any (the operation can act on any number of wires)
* Number of parameters: 1
* Number of dimensions per parameter: (2,)
* Gradient recipe: None
Args:
A (tensor_like): a general :math:`(n \times m)` matrix to be encoded
wires (Iterable[int, str], Wires): the wires the operation acts on
id (str or None): String representing the operation (optional)
Raises:
ValueError: if the number of wires doesn't fit the dimensions of the matrix
**Example**
We can define a matrix and a block-encoding circuit as follows:
>>> A = [[0.1,0.2],[0.3,0.4]]
>>> dev = qml.device('default.qubit', wires=2)
>>> @qml.qnode(dev)
... def example_circuit():
... qml.BlockEncode(A, wires=range(2))
... return qml.state()
We can see that :math:`A` has been block encoded in the matrix of the circuit:
>>> print(qml.matrix(example_circuit)())
[[ 0.1 0.2 0.97283788 -0.05988708]
[ 0.3 0.4 -0.05988708 0.86395228]
[ 0.94561648 -0.07621992 -0.1 -0.3 ]
[-0.07621992 0.89117368 -0.2 -0.4 ]]
We can also block-encode a non-square matrix and check the resulting unitary matrix:
>>> A = [[0.2, 0, 0.2],[-0.2, 0.2, 0]]
>>> op = qml.BlockEncode(A, wires=range(3))
>>> print(np.round(qml.matrix(op), 2))
[[ 0.2 0. 0.2 0.96 0.02 0. 0. 0. ]
[-0.2 0.2 0. 0.02 0.96 0. 0. 0. ]
[ 0.96 0.02 -0.02 -0.2 0.2 0. 0. 0. ]
[ 0.02 0.98 0. -0. -0.2 0. 0. 0. ]
[-0.02 0. 0.98 -0.2 -0. 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 1. 0. 0. ]
[ 0. 0. 0. 0. 0. 0. 1. 0. ]
[ 0. 0. 0. 0. 0. 0. 0. 1. ]]
.. note::
If the operator norm of :math:`A` is greater than 1, we normalize it to ensure
:math:`U(A)` is unitary. The normalization constant can be
accessed through :code:`op.hyperparameters["norm"]`.
Specifically, the norm is computed as the maximum of
:math:`\| AA^\dagger \|` and
:math:`\| A^\dagger A \|`.
"""
num_params = 1
"""int: Number of trainable parameters that the operator depends on."""
num_wires = AnyWires
"""int: Number of wires that the operator acts on."""
ndim_params = (2,)
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
grad_method = None
"""Gradient computation method."""
def __init__(self, A, wires, id=None):
shape_a = qml.math.shape(A)
if shape_a == () or all(x == 1 for x in shape_a):
A = qml.math.reshape(A, [1, 1])
wires = Wires(wires)
if pnp.sum(shape_a) <= 2:
normalization = A if A > 1 else 1
subspace = (1, 1, 2 ** len(wires))
else:
normalization = max(
norm(A @ qml.math.transpose(qml.math.conj(A)), ord=pnp.inf),
norm(qml.math.transpose(qml.math.conj(A)) @ A, ord=pnp.inf),
)
subspace = (*shape_a, 2 ** len(wires))
A = qml.math.array(A) / normalization if normalization > 1 else A
if subspace[2] < (subspace[0] + subspace[1]):
raise ValueError(
f"Block encoding a ({subspace[0]} x {subspace[1]}) matrix "
f"requires a Hilbert space of size at least "
f"({subspace[0] + subspace[1]} x {subspace[0] + subspace[1]})."
f" Cannot be embedded in a {len(wires)} qubit system."
)
super().__init__(A, wires=wires, id=id)
self.hyperparameters["norm"] = normalization
self.hyperparameters["subspace"] = subspace
[docs] @staticmethod
def compute_matrix(*params, **hyperparams):
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.BlockEncode.matrix`
Args:
*params (list): trainable parameters of the operator, as stored in the ``parameters`` attribute
**hyperparams (dict): non-trainable hyperparameters of the operator, as stored in the ``hyperparameters`` attribute
Returns:
tensor_like: canonical matrix
**Example**
>>> A = np.array([[0.1,0.2],[0.3,0.4]])
>>> A
tensor([[0.1, 0.2],
[0.3, 0.4]])
>>> qml.BlockEncode.compute_matrix(A, subspace=[2,2,4])
array([[ 0.1 , 0.2 , 0.97283788, -0.05988708],
[ 0.3 , 0.4 , -0.05988708, 0.86395228],
[ 0.94561648, -0.07621992, -0.1 , -0.3 ],
[-0.07621992, 0.89117368, -0.2 , -0.4 ]])
"""
A = params[0]
n, m, k = hyperparams["subspace"]
shape_a = qml.math.shape(A)
def _stack(lst, h=False, like=None):
if like == "tensorflow":
axis = 1 if h else 0
return qml.math.concat(lst, like=like, axis=axis)
return qml.math.hstack(lst) if h else qml.math.vstack(lst)
interface = qml.math.get_interface(A)
if qml.math.sum(shape_a) <= 2:
col1 = _stack([A, sqrt(1 - A * conj(A))], like=interface)
col2 = _stack([sqrt(1 - A * conj(A)), -conj(A)], like=interface)
u = _stack([col1, col2], h=True, like=interface)
else:
d1, d2 = shape_a
col1 = _stack(
[A, sqrt_matrix(cast(eye(d2, like=A), A.dtype) - qml.math.transpose(conj(A)) @ A)],
like=interface,
)
col2 = _stack(
[
sqrt_matrix(cast(eye(d1, like=A), A.dtype) - A @ transpose(conj(A))),
-transpose(conj(A)),
],
like=interface,
)
u = _stack([col1, col2], h=True, like=interface)
if n + m < k:
r = k - (n + m)
col1 = _stack([u, zeros((r, n + m), like=A)], like=interface)
col2 = _stack([zeros((n + m, r), like=A), eye(r, like=A)], like=interface)
u = _stack([col1, col2], h=True, like=interface)
return u
[docs] def adjoint(self):
A = self.parameters[0]
return BlockEncode(qml.math.transpose(qml.math.conj(A)), wires=self.wires)
[docs] def label(self, decimals=None, base_label=None, cache=None):
return super().label(decimals=decimals, base_label=base_label or "BlockEncode", cache=cache)
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