estimated_krylov_subspace_dimension_lanczos

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

Calculate an appropriate Krylov subspace dimension (\(k\)) to use in the Lanczos integrator while keeping the total error in the evolution below a given error tolerance.

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

Parameters

  • spectral_range (Tensor or float) – 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, optional) – The name of the node.

Returns

Recommended value of \(k\) to use in a Lanczos integration with a Hamiltonian with a similar spectral range to the one passed.

Return type

Tensor

See also

spectral_range

Range of the eigenvalues of a Hermitian operator.

state_evolution_pwc

Evolve a quantum state.

Notes

To provide the recommended \(k\) parameter, this function uses the bound in the error for the Lanczos algorithm 1 2 as an estimate for the error. For a single time step this gives

\[\mathrm{error} \leq 12 \exp \left( - \frac{(w\tau)^2}{16 k} \right) \left (\frac{e w \tau}{ 4 k} \right )^{k}\]

where \(\tau\) is the time step and \(w\) is the spectral range of the Hamiltonian.

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.

References

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).

Examples

>>> 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 = qctrl.functions.calculate_graph(graph=graph, output_node_names=["dim"])
>>> result.output["dim"]["value"]
2

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