# Source code for pennylane.qaoa.mixers

```
# 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.
r"""
Methods for constructing QAOA mixer Hamiltonians.
"""
# pylint: disable=unnecessary-lambda-assignment
import itertools
import functools
from typing import Iterable, Union
import networkx as nx
import rustworkx as rx
import pennylane as qml
from pennylane.wires import Wires
[docs]def x_mixer(wires: Union[Iterable, Wires]):
r"""Creates a basic Pauli-X mixer Hamiltonian.
This Hamiltonian is defined as:
.. math:: H_M \ = \ \displaystyle\sum_{i} X_{i},
where :math:`i` ranges over all wires, and :math:`X_i`
denotes the Pauli-X operator on the :math:`i`-th wire.
This is mixer is used in *A Quantum Approximate Optimization Algorithm*
by Edward Farhi, Jeffrey Goldstone, Sam Gutmann [`arXiv:1411.4028 <https://arxiv.org/abs/1411.4028>`__].
Args:
wires (Iterable or Wires): The wires on which the Hamiltonian is applied
Returns:
Hamiltonian: Mixer Hamiltonian
**Example**
The mixer Hamiltonian can be called as follows:
>>> from pennylane import qaoa
>>> wires = range(3)
>>> mixer_h = qaoa.x_mixer(wires)
>>> print(mixer_h)
(1) [X0]
+ (1) [X1]
+ (1) [X2]
"""
wires = Wires(wires)
coeffs = [1 for w in wires]
obs = [qml.PauliX(w) for w in wires]
H = qml.Hamiltonian(coeffs, obs)
# store the valuable information that all observables are in one commuting group
H.grouping_indices = [list(range(len(H.ops)))]
return H
[docs]def xy_mixer(graph: Union[nx.Graph, rx.PyGraph]):
r"""Creates a generalized SWAP/XY mixer Hamiltonian.
This mixer Hamiltonian is defined as:
.. math:: H_M \ = \ \frac{1}{2} \displaystyle\sum_{(i, j) \in E(G)} X_i X_j \ + \ Y_i Y_j,
for some graph :math:`G`. :math:`X_i` and :math:`Y_i` denote the Pauli-X and Pauli-Y operators on the :math:`i`-th
wire respectively.
This mixer was introduced in *From the Quantum Approximate Optimization Algorithm
to a Quantum Alternating Operator Ansatz* by Stuart Hadfield, Zhihui Wang, Bryan O'Gorman,
Eleanor G. Rieffel, Davide Venturelli, and Rupak Biswas `Algorithms 12.2 (2019) <https://doi.org/10.3390/a12020034>`__.
Args:
graph (nx.Graph or rx.PyGraph): A graph defining the collections of wires on which the Hamiltonian acts.
Returns:
Hamiltonian: Mixer Hamiltonian
**Example**
The mixer Hamiltonian can be called as follows:
>>> from pennylane import qaoa
>>> from networkx import Graph
>>> graph = Graph([(0, 1), (1, 2)])
>>> mixer_h = qaoa.xy_mixer(graph)
>>> print(mixer_h)
(0.5) [X0 X1]
+ (0.5) [Y0 Y1]
+ (0.5) [X1 X2]
+ (0.5) [Y1 Y2]
>>> import rustworkx as rx
>>> graph = rx.PyGraph()
>>> graph.add_nodes_from([0, 1, 2])
>>> graph.add_edges_from([(0, 1, ""), (1, 2, "")])
>>> mixer_h = xy_mixer(graph)
>>> print(mixer_h)
(0.5) [X0 X1]
+ (0.5) [Y0 Y1]
+ (0.5) [X1 X2]
+ (0.5) [Y1 Y2]
"""
if not isinstance(graph, (nx.Graph, rx.PyGraph)):
raise ValueError(
f"Input graph must be a nx.Graph or rx.PyGraph object, got {type(graph).__name__}"
)
is_rx = isinstance(graph, rx.PyGraph)
edges = graph.edge_list() if is_rx else graph.edges
# In RX each node is assigned to an integer index starting from 0;
# thus, we use the following lambda function to get node-values.
get_nvalue = lambda i: graph.nodes()[i] if is_rx else i
coeffs = 2 * [0.5 for e in edges]
obs = []
for node1, node2 in edges:
obs.append(qml.PauliX(get_nvalue(node1)) @ qml.PauliX(get_nvalue(node2)))
obs.append(qml.PauliY(get_nvalue(node1)) @ qml.PauliY(get_nvalue(node2)))
return qml.Hamiltonian(coeffs, obs)
[docs]def bit_flip_mixer(graph: Union[nx.Graph, rx.PyGraph], b: int):
r"""Creates a bit-flip mixer Hamiltonian.
This mixer is defined as:
.. math:: H_M \ = \ \displaystyle\sum_{v \in V(G)} \frac{1}{2^{d(v)}} X_{v}
\displaystyle\prod_{w \in N(v)} (\mathbb{I} \ + \ (-1)^b Z_w)
where :math:`V(G)` is the set of vertices of some graph :math:`G`, :math:`d(v)` is the
`degree <https://en.wikipedia.org/wiki/Degree_(graph_theory)>`__ of vertex :math:`v`, and
:math:`N(v)` is the `neighbourhood <https://en.wikipedia.org/wiki/Neighbourhood_(graph_theory)>`__
of vertex :math:`v`. In addition, :math:`Z_v` and :math:`X_v`
are the Pauli-Z and Pauli-X operators on vertex :math:`v`, respectively,
and :math:`\mathbb{I}` is the identity operator.
This mixer was introduced in `Hadfield et al. (2019) <https://doi.org/10.3390/a12020034>`__.
Args:
graph (nx.Graph or rx.PyGraph): A graph defining the collections of wires on which the Hamiltonian acts.
b (int): Either :math:`0` or :math:`1`. When :math:`b=0`, a bit flip is performed on
vertex :math:`v` only when all neighbouring nodes are in state :math:`|0\rangle`.
Alternatively, for :math:`b=1`, a bit flip is performed only when all the neighbours of
:math:`v` are in the state :math:`|1\rangle`.
Returns:
Hamiltonian: Mixer Hamiltonian
**Example**
The mixer Hamiltonian can be called as follows:
>>> from pennylane import qaoa
>>> from networkx import Graph
>>> graph = Graph([(0, 1), (1, 2)])
>>> mixer_h = qaoa.bit_flip_mixer(graph, 0)
>>> print(mixer_h)
(0.25) [X1]
+ (0.5) [X0]
+ (0.5) [X2]
+ (0.25) [X1 Z2]
+ (0.25) [X1 Z0]
+ (0.5) [X0 Z1]
+ (0.5) [X2 Z1]
+ (0.25) [X1 Z0 Z2]
>>> import rustworkx as rx
>>> graph = rx.PyGraph()
>>> graph.add_nodes_from([0, 1, 2])
>>> graph.add_edges_from([(0, 1, ""), (1, 2, "")])
>>> mixer_h = qaoa.bit_flip_mixer(graph, 0)
>>> print(mixer_h)
(0.25) [X1]
+ (0.5) [X0]
+ (0.5) [X2]
+ (0.25) [X1 Z0]
+ (0.25) [X1 Z2]
+ (0.5) [X0 Z1]
+ (0.5) [X2 Z1]
+ (0.25) [X1 Z2 Z0]
"""
if not isinstance(graph, (nx.Graph, rx.PyGraph)):
raise ValueError(
f"Input graph must be a nx.Graph or rx.PyGraph object, got {type(graph).__name__}"
)
if b not in [0, 1]:
raise ValueError(f"'b' must be either 0 or 1, got {b}")
sign = 1 if b == 0 else -1
coeffs = []
terms = []
is_rx = isinstance(graph, rx.PyGraph)
graph_nodes = graph.node_indexes() if is_rx else graph.nodes
# In RX each node is assigned to an integer index starting from 0;
# thus, we use the following lambda function to get node-values.
get_nvalue = lambda i: graph.nodes()[i] if is_rx else i
for i in graph_nodes:
neighbours = sorted(graph.neighbors(i)) if is_rx else list(graph.neighbors(i))
degree = len(neighbours)
n_terms = [[qml.PauliX(get_nvalue(i))]] + [
[qml.Identity(get_nvalue(n)), qml.PauliZ(get_nvalue(n))] for n in neighbours
]
n_coeffs = [[1, sign] for n in neighbours]
final_terms = [qml.operation.Tensor(*list(m)).prune() for m in itertools.product(*n_terms)]
final_coeffs = [
(0.5**degree) * functools.reduce(lambda x, y: x * y, list(m), 1)
for m in itertools.product(*n_coeffs)
]
coeffs.extend(final_coeffs)
terms.extend(final_terms)
return qml.Hamiltonian(coeffs, terms)
```

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