# Source code for pennylane.ops.op_math.prod

```
# Copyright 2018-2022 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 file contains the implementation of the Prod class which contains logic for
computing the product between operations.
"""
import itertools
from copy import copy
from functools import reduce
from itertools import combinations
from typing import List, Tuple, Union
import numpy as np
from scipy.sparse import kron as sparse_kron
import pennylane as qml
from pennylane import math
from pennylane.operation import Operator
from pennylane.ops.op_math.pow import Pow
from pennylane.ops.op_math.sprod import SProd
from pennylane.ops.op_math.sum import Sum
from pennylane.ops.qubit import Hamiltonian
from pennylane.ops.qubit.non_parametric_ops import PauliX, PauliY, PauliZ
from pennylane.queuing import QueuingManager
from pennylane.wires import Wires
from .composite import CompositeOp
MAX_NUM_WIRES_KRON_PRODUCT = 9
"""The maximum number of wires up to which using ``math.kron`` is faster than ``math.dot`` for
computing the sparse matrix representation."""
[docs]def prod(*ops, do_queue=True, id=None, lazy=True):
"""Construct an operator which represents the generalized product of the
operators provided.
The generalized product operation represents both the tensor product as
well as matrix composition. This can be resolved naturally from the wires
that the given operators act on.
Args:
ops (tuple[~.operation.Operator]): The operators we would like to multiply
Keyword Args:
do_queue (bool): determines if the product operator will be queued. Default is True.
id (str or None): id for the product operator. Default is None.
lazy=True (bool): If ``lazy=False``, a simplification will be performed such that when any of the operators is already a product operator, its operands will be used instead.
Returns:
~ops.op_math.Prod: the operator representing the product.
.. note::
This operator supports batched operands:
>>> op = qml.prod(qml.RX(np.array([1, 2, 3]), wires=0), qml.PauliX(1))
>>> op.matrix().shape
(3, 4, 4)
But it doesn't support batching of operators:
>>> op = qml.prod(np.array([qml.RX(0.5, 0), qml.RZ(0.3, 0)]), qml.PauliZ(0))
AttributeError: 'numpy.ndarray' object has no attribute 'wires'
.. seealso:: :class:`~.ops.op_math.Prod`
**Example**
>>> prod_op = prod(qml.PauliX(0), qml.PauliZ(0))
>>> prod_op
PauliX(wires=[0]) @ PauliZ(wires=[0])
>>> prod_op.matrix()
array([[ 0, -1],
[ 1, 0]])
"""
if lazy:
return Prod(*ops, do_queue=do_queue, id=id)
ops_simp = Prod(
*itertools.chain.from_iterable([op if isinstance(op, Prod) else [op] for op in ops]),
do_queue=do_queue,
id=id,
)
if do_queue:
for op in ops:
QueuingManager.remove(op)
return ops_simp
[docs]class Prod(CompositeOp):
r"""Symbolic operator representing the product of operators.
Args:
factors (tuple[~.operation.Operator]): a tuple of operators which will be multiplied
together.
Keyword Args:
do_queue (bool): determines if the product operator will be queued. Default is True.
id (str or None): id for the product operator. Default is None.
.. seealso:: :func:`~.ops.op_math.prod`
**Example**
>>> prop_op = Prod(qml.PauliX(wires=0), qml.PauliZ(wires=0))
>>> prop_op
PauliX(wires=[0]) @ PauliZ(wires=[0])
>>> qml.matrix(prop_op)
array([[ 0, -1],
[ 1, 0]])
>>> prop_op.terms()
([1.0], [PauliX(wires=[0]) @ PauliZ(wires=[0])])
.. note::
When a Prod operator is applied in a circuit, its factors are applied in the reverse order.
(i.e ``Prod(op1, op2)`` corresponds to :math:`\hat{op}_{1}\cdot\hat{op}_{2}` which indicates
first applying :math:`\hat{op}_{2}` then :math:`\hat{op}_{1}` in the circuit). We can see this
in the decomposition of the operator.
>>> op = Prod(qml.PauliX(wires=0), qml.PauliZ(wires=1))
>>> op.decomposition()
[PauliZ(wires=[1]), PauliX(wires=[0])]
.. details::
:title: Usage Details
The Prod operator represents both matrix composition and tensor products
between operators.
>>> prod_op = Prod(qml.RZ(1.23, wires=0), qml.PauliX(wires=0), qml.PauliZ(wires=1))
>>> prod_op.matrix()
array([[ 0. +0.j , 0. +0.j ,
0.81677345-0.57695852j, 0. +0.j ],
[ 0. +0.j , 0. +0.j ,
0. +0.j , -0.81677345+0.57695852j],
[ 0.81677345+0.57695852j, 0. +0.j ,
0. +0.j , 0. +0.j ],
[ 0. +0.j , -0.81677345-0.57695852j,
0. +0.j , 0. +0.j ]])
The Prod operation can be used inside a `qnode` as an operation which,
if parameterized, can be differentiated.
.. code-block:: python
dev = qml.device("default.qubit", wires=3)
@qml.qnode(dev)
def circuit(theta):
qml.prod(qml.PauliZ(0), qml.RX(theta, 1))
return qml.expval(qml.PauliZ(1))
>>> par = np.array(1.23, requires_grad=True)
>>> circuit(par)
tensor(0.33423773, requires_grad=True)
>>> qml.grad(circuit)(par)
tensor(-0.9424888, requires_grad=True)
The Prod operation can also be measured as an observable.
If the circuit is parameterized, then we can also differentiate through the
product observable.
.. code-block:: python
prod_op = Prod(qml.PauliZ(wires=0), qml.Hadamard(wires=1))
dev = qml.device("default.qubit", wires=2)
@qml.qnode(dev)
def circuit(weights):
qml.RX(weights[0], wires=0)
return qml.expval(prod_op)
>>> weights = np.array([0.1], requires_grad=True)
>>> qml.grad(circuit)(weights)
array([-0.07059289])
"""
_op_symbol = "@"
_math_op = math.prod
@property
def is_hermitian(self):
"""Check if the product operator is hermitian.
Note, this check is not exhaustive. There can be hermitian operators for which this check
yields false, which ARE hermitian. So a false result only implies a more explicit check
must be performed.
"""
for o1, o2 in combinations(self.operands, r=2):
if qml.wires.Wires.shared_wires([o1.wires, o2.wires]):
return False
return all(op.is_hermitian for op in self)
@property
def overlapping_ops(self) -> List[Tuple[Wires, List[Operator]]]:
"""Groups all operands of the composite operator that act on overlapping wires taking
into account operator commutivity.
Returns:
List[List[Operator]]: List of lists of operators that act on overlapping wires. All the
inner lists commute with each other.
"""
if self._overlapping_ops is None:
overlapping_ops = [] # [(wires, [ops])]
for op in self:
op_idx = False
ops = [op]
wires = op.wires
for idx, (old_wires, old_ops) in reversed(list(enumerate(overlapping_ops))):
if any(wire in old_wires for wire in wires):
ops = old_ops + ops
wires = old_wires + wires
op_idx = idx
old_wires, old_ops = overlapping_ops.pop(idx)
if op_idx is not False:
overlapping_ops.insert(op_idx, (wires, ops))
else:
overlapping_ops += [(wires, ops)]
self._overlapping_ops = [overlapping_op[1] for overlapping_op in overlapping_ops]
return self._overlapping_ops
# pylint: disable=arguments-renamed, invalid-overridden-method
@property
def has_decomposition(self):
return True
[docs] def decomposition(self):
r"""Decomposition of the product operator is given by each factor applied in succession.
Note that the decomposition is the list of factors returned in reversed order. This is
to support the intuition that when we write :math:`\hat{O} = \hat{A} \cdot \hat{B}` it is implied
that :math:`\hat{B}` is applied to the state before :math:`\hat{A}` in the quantum circuit.
"""
if qml.queuing.QueuingManager.recording():
return [qml.apply(op) for op in self[::-1]]
return list(self[::-1])
[docs] def matrix(self, wire_order=None):
"""Representation of the operator as a matrix in the computational basis."""
mats: List[np.ndarray] = [] # TODO: change type to `tensor_like` when available
batched: List[bool] = [] # batched[i] tells if mats[i] is batched or not
for ops in self.overlapping_ops:
gen = (
(qml.matrix(op) if isinstance(op, Hamiltonian) else op.matrix(), op.wires)
for op in ops
)
reduced_mat, _ = math.reduce_matrices(gen, reduce_func=math.matmul)
if self.batch_size is not None:
batched.append(any(op.batch_size is not None for op in ops))
else:
batched.append(False)
mats.append(reduced_mat)
if self.batch_size is None:
full_mat = reduce(math.kron, mats)
else:
full_mat = qml.math.stack(
[
reduce(math.kron, [m[i] if b else m for m, b in zip(mats, batched)])
for i in range(self.batch_size)
]
)
return math.expand_matrix(full_mat, self.wires, wire_order=wire_order)
[docs] def sparse_matrix(self, wire_order=None):
if self.has_overlapping_wires or self.num_wires > MAX_NUM_WIRES_KRON_PRODUCT:
gen = ((op.sparse_matrix(), op.wires) for op in self)
reduced_mat, prod_wires = math.reduce_matrices(gen, reduce_func=math.dot)
wire_order = wire_order or self.wires
return math.expand_matrix(reduced_mat, prod_wires, wire_order=wire_order)
mats = (op.sparse_matrix() for op in self)
full_mat = reduce(sparse_kron, mats)
return math.expand_matrix(full_mat, self.wires, wire_order=wire_order)
# pylint: disable=protected-access
@property
def _queue_category(self):
"""Used for sorting objects into their respective lists in `QuantumTape` objects.
This property is a temporary solution that should not exist long-term and should not be
used outside of ``QuantumTape._process_queue``.
Options are:
* `"_prep"`
* `"_ops"`
* `"_measurements"`
* `None`
Returns (str or None): "_ops" if the _queue_catagory of all factors is "_ops", else None.
"""
return "_ops" if all(op._queue_category == "_ops" for op in self) else None
# pylint: disable=arguments-renamed, invalid-overridden-method
@property
def has_adjoint(self):
return True
@property
def arithmetic_depth(self) -> int:
return 1 + max(factor.arithmetic_depth for factor in self)
def _build_pauli_rep(self):
"""PauliSentence representation of the Product of operations."""
if all(
operand_pauli_reps := [
op._pauli_rep for op in self.operands # pylint: disable=protected-access
]
):
return reduce(lambda a, b: a * b, operand_pauli_reps)
return None
def _simplify_factors(self, factors: Tuple[Operator]) -> Tuple[complex, Operator]:
"""Reduces the depth of nested factors and groups identical factors.
Returns:
Tuple[complex, List[~.operation.Operator]: tuple containing the global phase and a list
of the simplified factors
"""
new_factors = _ProductFactorsGrouping()
for factor in factors:
simplified_factor = factor.simplify()
new_factors.add(factor=simplified_factor)
new_factors.remove_factors(wires=self.wires)
return new_factors.global_phase, new_factors.factors
[docs] def simplify(self) -> Union["Prod", Sum]:
# try using pauli_rep:
if pr := self._pauli_rep:
pr.simplify()
return pr.operation(wire_order=self.wires)
global_phase, factors = self._simplify_factors(factors=self.operands)
factors = list(itertools.product(*factors))
if len(factors) == 1:
factor = factors[0]
if len(factor) == 0:
op = qml.Identity(self.wires)
else:
op = factor[0] if len(factor) == 1 else Prod(*factor)
return op if global_phase == 1 else qml.s_prod(global_phase, op)
factors = [Prod(*factor).simplify() if len(factor) > 1 else factor[0] for factor in factors]
op = Sum(*factors).simplify()
return op if global_phase == 1 else qml.s_prod(global_phase, op).simplify()
@classmethod
def _sort(cls, op_list, wire_map: dict = None) -> List[Operator]:
"""Insertion sort algorithm that sorts a list of product factors by their wire indices, taking
into account the operator commutivity.
Args:
op_list (List[.Operator]): list of operators to be sorted
wire_map (dict): Dictionary containing the wire values as keys and its indexes as values.
Defaults to None.
Returns:
List[.Operator]: sorted list of operators
"""
if isinstance(op_list, tuple):
op_list = list(op_list)
for i in range(1, len(op_list)):
key_op = op_list[i]
j = i - 1
while j >= 0 and _swappable_ops(op1=op_list[j], op2=key_op, wire_map=wire_map):
op_list[j + 1] = op_list[j]
j -= 1
op_list[j + 1] = key_op
return op_list
def _swappable_ops(op1, op2, wire_map: dict = None) -> bool:
"""Boolean expression that indicates if op1 and op2 don't have intersecting wires and if they
should be swapped when sorting them by wire values.
Args:
op1 (.Operator): First operator.
op2 (.Operator): Second operator.
wire_map (dict): Dictionary containing the wire values as keys and its indexes as values.
Defaults to None.
Returns:
bool: True if operators should be swapped, False otherwise.
"""
wires1 = op1.wires
wires2 = op2.wires
if wire_map is not None:
wires1 = wires1.map(wire_map)
wires2 = wires2.map(wire_map)
wires1 = set(wires1)
wires2 = set(wires2)
# compare strings of wire labels so that we can compare arbitrary wire labels like 0 and "a"
return False if wires1 & wires2 else str(wires1.pop()) > str(wires2.pop())
class _ProductFactorsGrouping:
"""Utils class used for grouping identical product factors."""
_identity_map = {
"Identity": (1.0, "Identity"),
"PauliX": (1.0, "PauliX"),
"PauliY": (1.0, "PauliY"),
"PauliZ": (1.0, "PauliZ"),
}
_x_map = {
"Identity": (1.0, "PauliX"),
"PauliX": (1.0, "Identity"),
"PauliY": (1.0j, "PauliZ"),
"PauliZ": (-1.0j, "PauliY"),
}
_y_map = {
"Identity": (1.0, "PauliY"),
"PauliX": (-1.0j, "PauliZ"),
"PauliY": (1.0, "Identity"),
"PauliZ": (1.0j, "PauliX"),
}
_z_map = {
"Identity": (1.0, "PauliZ"),
"PauliX": (1.0j, "PauliY"),
"PauliY": (-1.0j, "PauliX"),
"PauliZ": (1.0, "Identity"),
}
_pauli_mult = {"Identity": _identity_map, "PauliX": _x_map, "PauliY": _y_map, "PauliZ": _z_map}
_paulis = {"PauliX": PauliX, "PauliY": PauliY, "PauliZ": PauliZ}
def __init__(self):
self._pauli_factors = {} # {wire: (pauli_coeff, pauli_word)}
self._non_pauli_factors = {} # {wires: [hash, exponent, operator]}
self._factors = []
self.global_phase = 1
def add(self, factor: Operator):
"""Add factor.
Args:
factor (Operator): Factor to add.
"""
wires = factor.wires
if isinstance(factor, Prod):
for prod_factor in factor:
self.add(prod_factor)
elif isinstance(factor, Sum):
self._remove_pauli_factors(wires=wires)
self._remove_non_pauli_factors(wires=wires)
self._factors += (factor.operands,)
elif not isinstance(factor, qml.Identity):
if isinstance(factor, SProd):
self.global_phase *= factor.scalar
factor = factor.base
if isinstance(factor, (qml.Identity, qml.PauliX, qml.PauliY, qml.PauliZ)):
self._add_pauli_factor(factor=factor, wires=wires)
self._remove_non_pauli_factors(wires=wires)
else:
self._add_non_pauli_factor(factor=factor, wires=wires)
self._remove_pauli_factors(wires=wires)
def _add_pauli_factor(self, factor: Operator, wires: List[int]):
"""Adds the given Pauli operator to the temporary ``self._pauli_factors`` dictionary. If
there was another Pauli operator acting on the same wire, the two operators are grouped
together using the ``self._pauli_mult`` dictionary.
Args:
factor (Operator): Factor to be added.
wires (List[int]): Factor wires. This argument is added to avoid calling
``factor.wires`` several times.
"""
wire = wires[0]
op2_name = factor.name
old_coeff, old_word = self._pauli_factors.get(wire, (1, "Identity"))
coeff, new_word = self._pauli_mult[old_word][op2_name]
self._pauli_factors[wire] = old_coeff * coeff, new_word
def _add_non_pauli_factor(self, factor: Operator, wires: List[int]):
"""Adds the given non-Pauli factor to the temporary ``self._non_pauli_factors`` dictionary.
If there alerady exists an identical operator in the dictionary, the two are grouped
together.
If there isn't an identical operator in the dictionary, all non Pauli factors that act on
the same wires are removed and added to the ``self._factors`` tuple.
Args:
factor (Operator): Factor to be added.
wires (List[int]): Factor wires. This argument is added to avoid calling
``factor.wires`` several times.
"""
if isinstance(factor, Pow):
exponent = factor.z
factor = factor.base
else:
exponent = 1
op_hash = factor.hash
old_hash, old_exponent, old_op = self._non_pauli_factors.get(wires, [None, None, None])
if isinstance(old_op, (qml.RX, qml.RY, qml.RZ)) and factor.name == old_op.name:
self._non_pauli_factors[wires] = [
op_hash,
old_exponent,
factor.__class__(factor.data[0] + old_op.data[0], wires).simplify(),
]
elif op_hash == old_hash:
self._non_pauli_factors[wires][1] += exponent
else:
self._remove_non_pauli_factors(wires=wires)
self._non_pauli_factors[wires] = [op_hash, copy(exponent), factor]
def _remove_non_pauli_factors(self, wires: List[int]):
"""Remove all factors from the ``self._non_pauli_factors`` dictionary that act on the given
wires and add them to the ``self._factors`` tuple.
Args:
wires (List[int]): Wires of the operators to be removed.
"""
if not self._non_pauli_factors:
return
for wire in wires:
for key, (_, exponent, op) in list(self._non_pauli_factors.items()):
if wire in key:
self._non_pauli_factors.pop(key)
if exponent == 0:
continue
if exponent != 1:
op = Pow(base=op, z=exponent).simplify()
if not isinstance(op, qml.Identity):
self._factors += ((op,),)
def _remove_pauli_factors(self, wires: List[int]):
"""Remove all Pauli factors from the ``self._pauli_factors`` dictionary that act on the
given wires and add them to the ``self._factors`` tuple.
Args:
wires (List[int]): Wires of the operators to be removed.
"""
if not self._pauli_factors:
return
for wire in wires:
pauli_coeff, pauli_word = self._pauli_factors.pop(wire, (1, "Identity"))
if pauli_word != "Identity":
pauli_op = self._paulis[pauli_word](wire)
self._factors += ((pauli_op,),)
self.global_phase *= pauli_coeff
def remove_factors(self, wires: List[int]):
"""Remove all factors from the ``self._pauli_factors`` and ``self._non_pauli_factors``
dictionaries that act on the given wires and add them to the ``self._factors`` tuple.
Args:
wires (List[int]): Wires of the operators to be removed.
"""
self._remove_pauli_factors(wires=wires)
self._remove_non_pauli_factors(wires=wires)
@property
def factors(self):
"""Grouped factors tuple.
Returns:
tuple: Tuple of grouped factors.
"""
return tuple(self._factors)
```

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