Graph.estimated_krylov_subspace_dimension_lanczos(spectral_range, duration, maximum_segment_duration, error_tolerance=1e-06, *, name=None)

Calculate an appropriate Krylov subspace dimension (kk

Note that you can provide your own estimation of the Hamiltonian spectral range or use the spectral_range operation to perform that calculation.


  • spectral_range (float or Tensor) – Estimated order of magnitude of Hamiltonian spectral range (difference between largest and smallest eigenvalues).
  • duration (float) – The total evolution time.
  • maximum_segment_duration (float) – The maximum duration of the piecewise-constant Hamiltonian segments.
  • error_tolerance (float , optional) – Tolerance for the error in the integration, defined as the Frobenius norm of the vectorial difference between the exact state and the estimated state. Defaults to 1e-6.
  • name (str or None , optional) – The name of the node.


Recommended value of kk

Return type



Graph.spectral_range : Range of the eigenvalues of a Hermitian operator.

Graph.state_evolution_pwc : Evolve a quantum state.


To provide the recommended kk1 2 as an estimate for the error. For a single time step this gives

error12exp((wτ)216k)(ewτ4k)k \mathrm{error} \leq 12 \exp \left( - \frac{(w\tau)^2}{16 k} \right) \left (\frac{e w \tau}{ 4 k} \right )^{k}

where τ\tau is the time step and ww

As this bound overestimates the error, the actual resulting errors with the recommended parameter are expected to be (a few orders of magnitude) smaller than the requested tolerance.


[1] N. Del Buono and L. Lopez, Lect. Notes Comput. Sci. 2658, 111 (2003).

[2] M. Hochbruck and C. Lubich, SIAM J. Numer. Anal. 34, 1911 (1997).


>>> graph.estimated_krylov_subspace_dimension_lanczos(
...     spectral_range=30.0,
...     duration=5e-7,
...     maximum_segment_duration=2.5e-8,
...     error_tolerance=1e-5,
...     name="dim",
... )
<Tensor: name="dim", operation_name="estimated_krylov_subspace_dimension_lanczos", shape=()>
>>> result = bo.execute_graph(graph=graph, output_node_names="dim")
>>> result["output"]["dim"]["value"]

See more examples in the How to optimize controls on large sparse Hamiltonians user guide.

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