qml.qaoa.mixers.bit_flip_mixer

bit_flip_mixer(graph, b)

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 Hadfield et al. (2019).

Parameters

• graph (nx.Graph or rx.PyGraph) – 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)
>>> mixer_h
(
    0.5 * X(0)
  + 0.5 * (X(0) @ Z(1))
  + 0.25 * X(1)
  + 0.25 * (X(1) @ Z(2))
  + 0.25 * (X(1) @ Z(0))
  + 0.25 * (X(1) @ Z(0) @ Z(2))
  + 0.5 * X(2)
  + 0.5 * (X(2) @ Z(1))
)

>>> 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)
(
    0.5 * X(0)
  + 0.5 * (X(0) @ Z(1))
  + 0.25 * X(1)
  + 0.25 * (X(1) @ Z(2))
  + 0.25 * (X(1) @ Z(0))
  + 0.25 * (X(1) @ Z(0) @ Z(2))
  + 0.5 * X(2)
  + 0.5 * (X(2) @ Z(1))
)

code/api/pennylane.qaoa.mixers.bit_flip_mixer

 

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