Graph.time_evolution_operators_stf(hamiltonian, sample_times, evolution_times=None, *, name=None)

Calculate the time-evolution operators for a system defined by an STF Hamiltonian by using a 4th order Runge–Kutta method.


  • hamiltonian (Stf) – The control Hamiltonian, or batch of control Hamiltonians.
  • sample_times (np.ndarray) – The N times at which you want to sample the unitaries. Must be ordered and contain at least one element. If you don’t provide evolution_times, sample_times must start with 0.
  • evolution_times (np.ndarray or None , optional) – The times at which the Hamiltonian should be sampled for the Runge–Kutta integration. If you provide it, must start with 0 and be ordered. If you don’t provide it, the sample_times are used for the integration.
  • name (str or None , optional) – The name of the node.


Tensor of shape (..., N, D, D), representing the unitary time evolution. The n-th element (along the -3 dimension) represents the unitary (or batch of unitaries) from t = 0 to sample_times[n].

Return type



Graph.time_evolution_operators_pwc : Corresponding operation for Pwc Hamiltonians.


For more information on Stf nodes see the Working with time-dependent functions in Boulder Opal topic.


Simulate the dynamics of a qubit, where a simple Gaussian drive rotate the qubit along the x-axis.

>>> duration = np.pi
>>> initial_state = np.array([1, 0])
>>> sigma_x = np.array([[0, 1], [1, 0]])
>>> time = graph.identity_stf()
>>> gaussian_drive = graph.exp(-(time ** 2))
>>> hamiltonian = gaussian_drive * sigma_x * np.sqrt(np.pi) / 2
>>> graph.time_evolution_operators_stf(
...     hamiltonian=hamiltonian,
...     sample_times=[duration],
...     evolution_times=np.linspace(0, duration, 200),
...     name="unitaries",
... )
<Tensor: name="unitaries", operation_name="time_evolution_operators_stf", shape=(1, 2, 2)>
>>> result = bo.execute_graph(graph=graph, output_node_names="unitaries")
>>> result["output"]["unitaries"]["value"].dot(initial_state)
array([[0.70711169 + 0.0j, 0.0 - 0.70710187j]])

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