unitary_infidelity

Graph.unitary_infidelity(unitary_operator, target, *, name=None)

Calculate the infidelity between a target operation and the actual implemented unitary.

Both operators must be square and have shapes broadcastable to each other.

Parameters

unitary_operator (np.ndarray or Tensor) – The actual unitary operator, \(U\), with shape (..., D, D). Its last two dimensions must be equal and the same as target, and its batch dimensions, if any, must be broadcastable with target.
target (np.ndarray or Tensor) – The target operation with respect to which the infidelity will be calculated, \(V\), with shape (..., D, D). Its last two dimensions must be equal and the same as unitary_operator, and its batch dimensions, if any, must be broadcastable with unitary_operator.
name (str, optional) – The name of the node.

Returns

The infidelity between the two operators, with shape (...).

Return type

Tensor

See also

density_matrix_infidelity: Infidelity between two density matrices.
infidelity_pwc: Total infidelity of a system with a piecewise-constant Hamiltonian.
infidelity_stf: Total infidelity of a system with a sampleable Hamiltonian.
state_infidelity: Infidelity between two quantum states.

Notes

The operational infidelity between the actual unitary and target operators is defined as

\[\mathcal{I} = 1 - \left| \frac{\mathrm{Tr} (V^\dagger U)}{\mathrm{Tr} (V^\dagger V)} \right|^2 .\]

Examples

Calculate the infidelity of a unitary with respect to a \(\sigma_x\) gate.

>>> theta = 0.5
>>> sigma_x = np.array([[0, 1], [1, 0]])
>>> unitary = np.array([[np.cos(theta), np.sin(theta)], [np.sin(theta), -np.cos(theta)]])
>>> graph.unitary_infidelity(unitary_operator=unitary, target=sigma_x, name="infidelity")
<Tensor: name="infidelity", operation_name="unitary_infidelity", shape=()>
>>> result = qctrl.functions.calculate_graph(graph=graph, output_node_names=["infidelity"])
>>> result.output["infidelity"]["value"]
0.7701511529340699

Calculate the time-dependent infidelity of the identity gate for a noiseless single qubit.

>>> sigma_x = np.array([[0, 1], [1, 0]])
>>> hamiltonian = sigma_x * graph.pwc_signal(
...     duration=1, values=np.pi * np.array([0.25, 1, 0.25])
... )
>>> unitaries = graph.time_evolution_operators_pwc(
...     hamiltonian=hamiltonian, sample_times=np.linspace(0, 1, 10)
... )
>>> graph.unitary_infidelity(unitary_operator=unitaries, target=np.eye(2), name="infidelities")
<Tensor: name="infidelities", operation_name="unitary_infidelity", shape=(10,)>
>>> result = qctrl.functions.calculate_graph(
...     graph=graph, output_node_names=["infidelities"]
... )
>>> result.output["infidelities"]["value"]
array([0.        , 0.00759612, 0.03015369, 0.0669873 , 0.32898993,
       0.67101007, 0.9330127 , 0.96984631, 0.99240388, 1.        ])