# qml.qaoa.mixers.bit_flip_mixer¶

bit_flip_mixer(graph, b)[source]

Creates a bit-flip mixer Hamiltonian.

This mixer is defined as:

$H_M \ = \ \displaystyle\sum_{v \in V(G)} \frac{1}{2^{d(v)}} X_{v} \displaystyle\prod_{w \in N(v)} (\mathbb{I} \ + \ (-1)^b Z_w)$

where $$V(G)$$ is the set of vertices of some graph $$G$$, $$d(v)$$ is the degree of vertex $$v$$, and $$N(v)$$ is the neighbourhood of vertex $$v$$. In addition, $$Z_v$$ and $$X_v$$ are the Pauli-Z and Pauli-X operators on vertex $$v$$, respectively, and $$\mathbb{I}$$ is the identity operator.

This mixer was introduced in [arXiv:1709.03489].

Parameters
• graph (nx.Graph) – A graph defining the collections of wires on which the Hamiltonian acts.

• b (int) – Either $$0$$ or $$1$$. When $$b=0$$, a bit flip is performed on vertex $$v$$ only when all neighbouring nodes are in state $$|0\rangle$$. Alternatively, for $$b=1$$, a bit flip is performed only when all the neighbours of $$v$$ are in the state $$|1\rangle$$.

Returns

Mixer Hamiltonian

Return type

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]