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 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)) )