def _inner_net_flow_constraint_hamiltonian(graph: nx.DiGraph,
node) -> Hamiltonian:
r"""Calculates the squared inner portion of the Hamiltonian in :func:`net_flow_constraint`.
For a given :math:`i`, this function returns:
.. math::
\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}.
Args:
graph (nx.DiGraph): the directed graph specifying possible edges
node: a fixed node
Returns:
qml.Hamiltonian: The inner part of the net-flow constraint Hamiltonian.
"""
edges_to_qubits = edges_to_wires(graph)
coeffs = []
ops = []
out_edges = graph.out_edges(node)
in_edges = graph.in_edges(node)
coeffs.append(len(out_edges) - len(in_edges))
ops.append(qml.Identity(0))
for edge in out_edges:
wires = (edges_to_qubits[edge], )
coeffs.append(-1)
ops.append(qml.PauliZ(wires))
for edge in in_edges:
wires = (edges_to_qubits[edge], )
coeffs.append(1)
ops.append(qml.PauliZ(wires))
coeffs, ops = _square_hamiltonian_terms(coeffs, ops)
H = Hamiltonian(coeffs, ops)
H.simplify()
# store the valuable information that all observables are in one commuting group
H.grouping_indices = [list(range(len(H.ops)))]
return H
def _inner_out_flow_constraint_hamiltonian(graph: nx.DiGraph,
node) -> Hamiltonian:
r"""Calculates the inner portion of the Hamiltonian in :func:`out_flow_constraint`.
For a given :math:`i`, this function returns:
.. math::
d_{i}^{out}(d_{i}^{out} - 2)\mathbb{I}
- 2(d_{i}^{out}-1)\sum_{j,(i,j)\in E}\hat{Z}_{ij} +
( \sum_{j,(i,j)\in E}\hat{Z}_{ij} )^{2}
Args:
graph (nx.DiGraph): the directed graph specifying possible edges
node: a fixed node
Returns:
qml.Hamiltonian: The inner part of the out-flow constraint Hamiltonian.
"""
coeffs = []
ops = []
edges_to_qubits = edges_to_wires(graph)
out_edges = graph.out_edges(node)
d = len(out_edges)
for edge in out_edges:
wire = (edges_to_qubits[edge], )
coeffs.append(1)
ops.append(qml.PauliZ(wire))
coeffs, ops = _square_hamiltonian_terms(coeffs, ops)
for edge in out_edges:
wire = (edges_to_qubits[edge], )
coeffs.append(-2 * (d - 1))
ops.append(qml.PauliZ(wire))
coeffs.append(d * (d - 2))
ops.append(qml.Identity(0))
H = Hamiltonian(coeffs, ops)
H.simplify()
# store the valuable information that all observables are in one commuting group
H.grouping_indices = [list(range(len(H.ops)))]
return H
def loss_hamiltonian(graph: nx.Graph) -> Hamiltonian:
r"""Calculates the loss Hamiltonian for the maximum-weighted cycle problem.
We consider the problem of selecting a cycle from a graph that has the greatest product of edge
weights, as outlined `here <https://1qbit.com/whitepaper/arbitrage/>`__. The product of weights
of a subset of edges in a graph is given by
.. math:: P = \prod_{(i, j) \in E} [(c_{ij} - 1)x_{ij} + 1]
where :math:`E` are the edges of the graph, :math:`x_{ij}` is a binary number that selects
whether to include the edge :math:`(i, j)` and :math:`c_{ij}` is the corresponding edge weight.
Our objective is to maximimize :math:`P`, subject to selecting the :math:`x_{ij}` so that
our subset of edges composes a cycle.
The product of edge weights is maximized by equivalently considering
.. math:: \sum_{(i, j) \in E} x_{ij}\log c_{ij},
assuming :math:`c_{ij} > 0`.
This can be restated as a minimization of the expectation value of the following qubit
Hamiltonian:
.. math::
H = \sum_{(i, j) \in E} Z_{ij}\log c_{ij}.
where :math:`Z_{ij}` is a qubit Pauli-Z matrix acting upon the wire specified by the edge
:math:`(i, j)`. Mapping from edges to wires can be achieved using :func:`~.edges_to_wires`.
.. note::
The expectation value of the returned Hamiltonian :math:`H` is not equal to :math:`P`, but
minimizing the expectation value of :math:`H` is equivalent to maximizing :math:`P`.
Also note that the returned Hamiltonian does not impose that the selected set of edges is
a cycle. This constraint can be enforced using a penalty term or by selecting a QAOA
mixer Hamiltonian that only transitions between states that correspond to cycles.
**Example**
>>> import networkx as nx
>>> g = nx.complete_graph(3).to_directed()
>>> edge_weight_data = {edge: (i + 1) * 0.5 for i, edge in enumerate(g.edges)}
>>> for k, v in edge_weight_data.items():
g[k[0]][k[1]]["weight"] = v
>>> h = loss_hamiltonian(g)
>>> print(h)
(-0.6931471805599453) [Z0]
+ (0.0) [Z1]
+ (0.4054651081081644) [Z2]
+ (0.6931471805599453) [Z3]
+ (0.9162907318741551) [Z4]
+ (1.0986122886681098) [Z5]
Args:
graph (nx.Graph): the graph specifying possible edges
Returns:
qml.Hamiltonian: the loss Hamiltonian
Raises:
ValueError: if the graph contains self-loops
KeyError: if one or more edges do not contain weight data
"""
edges_to_qubits = edges_to_wires(graph)
coeffs = []
ops = []
edges_data = graph.edges(data=True)
for edge_data in edges_data:
edge = edge_data[:2]
if edge[0] == edge[1]:
raise ValueError("Graph contains self-loops")
try:
weight = edge_data[2]["weight"]
except KeyError as e:
raise KeyError(f"Edge {edge} does not contain weight data") from e
coeffs.append(np.log(weight))
ops.append(qml.PauliZ(wires=edges_to_qubits[edge]))
H = Hamiltonian(coeffs, ops)
# store the valuable information that all observables are in one commuting group
H.grouping_indices = [list(range(len(H.ops)))]
return H
def _inner_net_flow_constraint_hamiltonian(graph: Union[nx.DiGraph,
rx.PyDiGraph],
node: int) -> Hamiltonian:
r"""Calculates the squared inner portion of the Hamiltonian in :func:`net_flow_constraint`.
For a given :math:`i`, this function returns:
.. math::
\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}.
Args:
graph (nx.DiGraph or rx.PyDiGraph): the directed graph specifying possible edges
node: a fixed node
Returns:
qml.Hamiltonian: The inner part of the net-flow constraint Hamiltonian.
"""
if not isinstance(graph, (nx.DiGraph, rx.PyDiGraph)):
raise ValueError(
f"Input graph must be a nx.DiGraph or rx.PyDiGraph, got {type(graph).__name__}"
)
edges_to_qubits = edges_to_wires(graph)
coeffs = []
ops = []
is_rx = isinstance(graph, rx.PyDiGraph)
out_edges = graph.out_edges(node)
in_edges = graph.in_edges(node)
# To ensure out_edges and in_edges methods in both RX and NX return
# the lists of edges in the same order, we sort results.
if is_rx:
out_edges = sorted(out_edges)
in_edges = sorted(in_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_nvalues = lambda T: (graph.nodes().index(T[0]), graph.nodes().index(T[
1])) if is_rx else T
coeffs.append(len(out_edges) - len(in_edges))
ops.append(qml.Identity(0))
for edge in out_edges:
if len(edge) > 2:
edge = tuple(edge[:2])
wires = (edges_to_qubits[get_nvalues(edge)], )
coeffs.append(-1)
ops.append(qml.PauliZ(wires))
for edge in in_edges:
if len(edge) > 2:
edge = tuple(edge[:2])
wires = (edges_to_qubits[get_nvalues(edge)], )
coeffs.append(1)
ops.append(qml.PauliZ(wires))
coeffs, ops = _square_hamiltonian_terms(coeffs, ops)
H = Hamiltonian(coeffs, ops)
H.simplify()
# store the valuable information that all observables are in one commuting group
H.grouping_indices = [list(range(len(H.ops)))]
return H