qml.qaoa.cost.max_weight_cycle¶

max_weight_cycle
(graph, constrained=True)[source]¶ Returns the QAOA cost Hamiltonian and the recommended mixer corresponding to the maximumweighted cycle problem, for a given graph.
The maximumweighted cycle problem is defined in the following way (see here for more details). The product of weights of a subset of edges in a graph is given by
\[P = \prod_{(i, j) \in E} [(c_{ij}  1)x_{ij} + 1]\]where \(E\) are the edges of the graph, \(x_{ij}\) is a binary number that selects whether to include the edge \((i, j)\) and \(c_{ij}\) is the corresponding edge weight. Our objective is to maximimize \(P\), subject to selecting the \(x_{ij}\) so that our subset of edges composes a cycle.
 Parameters
graph (nx.Graph) – the directed graph on which the Hamiltonians are defined
constrained (bool) – specifies the variant of QAOA that is performed (constrained or unconstrained)
 Returns
The cost and mixer Hamiltonians, as well as a dictionary mapping from wires to the graph’s edges
 Return type
(Hamiltonian, Hamiltonian, dict)
Usage Details
There are two variations of QAOA for this problem, constrained and unconstrained:
Constrained
Note
This method of constrained QAOA was introduced by Hadfield, Wang, Gorman, Rieffel, Venturelli, and Biswas in arXiv:1709.03489.
The maximum weighted cycle cost Hamiltonian for unconstrained QAOA is
\[H_C = H_{\rm loss}.\]Here, \(H_{\rm loss}\) is a loss Hamiltonian:
\[H_{\rm loss} = \sum_{(i, j) \in E} Z_{ij}\log c_{ij}\]where \(E\) are the edges of the graph and \(Z_{ij}\) is a qubit PauliZ matrix acting upon the wire specified by the edge \((i, j)\) (see
loss_hamiltonian()
for more details).The returned mixer Hamiltonian is
cycle_mixer()
given by\[H_M = \frac{1}{4}\sum_{(i, j)\in E} \left(\sum_{k \in V, k\neq i, k\neq j, (i, k) \in E, (k, j) \in E} \left[X_{ij}X_{ik}X_{kj} +Y_{ij}Y_{ik}X_{kj} + Y_{ij}X_{ik}Y_{kj}  X_{ij}Y_{ik}Y_{kj}\right] \right).\]This mixer provides transitions between collections of cycles, i.e., any subset of edges in \(E\) such that all the graph’s nodes \(V\) have zero net flow (see the
net_flow_constraint()
function).Note
 Recommended initialization circuit:
Your circuit must prepare a state that corresponds to a cycle (or a superposition of cycles). Follow the example code below to see how this is done.
Unconstrained
The maximum weighted cycle cost Hamiltonian for constrained QAOA is defined as:
\[H_C \ = H_{\rm loss} + 3 H_{\rm netflow} + 3 H_{\rm outflow}.\]The netflow constraint Hamiltonian
net_flow_constraint()
is given by\[H_{\rm netflow} = \sum_{i \in V} \left((d_{i}^{\rm out}  d_{i}^{\rm in})\mathbb{I}  \sum_{j, (i, j) \in E} Z_{ij} + \sum_{j, (j, i) \in E} Z_{ji} \right)^{2},\]where \(d_{i}^{\rm out}\) and \(d_{i}^{\rm in}\) are the outdegree and indegree, respectively, of node \(i\). It is minimized whenever a subset of edges in \(E\) results in zero net flow from each node in \(V\).
The outflow constraint Hamiltonian
out_flow_constraint()
is given by\[H_{\rm outflow} = \sum_{i\in V}\left(d_{i}^{out}(d_{i}^{out}  2)\mathbb{I}  2(d_{i}^{out}1)\sum_{j,(i,j)\in E}\hat{Z}_{ij} + \left( \sum_{j,(i,j)\in E}\hat{Z}_{ij} \right)^{2}\right).\]It is minimized whenever a subset of edges in \(E\) results in an outflow of at most one from each node in \(V\).
The returned mixer Hamiltonian is
x_mixer()
applied to all wires.Note
 Recommended initialization circuit:
Even superposition over all basis states.
Example
First set up a simple graph:
import pennylane as qml import numpy as np import networkx as nx a = np.random.random((4, 4)) np.fill_diagonal(a, 0) g = nx.DiGraph(a)
The cost and mixer Hamiltonian as well as the mapping from wires to edges can be loaded using:
>>> cost, mixer, mapping = qml.qaoa.max_weight_cycle(g, constrained=True)
Since we are using
constrained=True
, we must ensure that the input state to the QAOA algorithm corresponds to a cycle. Consider the mapping:>>> mapping {0: (0, 1), 1: (0, 2), 2: (0, 3), 3: (1, 0), 4: (1, 2), 5: (1, 3), 6: (2, 0), 7: (2, 1), 8: (2, 3), 9: (3, 0), 10: (3, 1), 11: (3, 2)}
A simple cycle is given by the edges
(0, 1)
and(1, 0)
and corresponding wires0
and3
. Hence, the state \(100100000000\rangle\) corresponds to a cycle and can be prepared usingBasisState
or simplePauliX
rotations on the0
and3
wires.
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