spectral_range

Graph.spectral_range(operator, iteration_count=3000, seed=None, *, name=None)

Obtain the range of the eigenvalues of a Hermitian operator.

This function provides an estimate of the difference between the highest and the lowest eigenvalues of the operator. You can adjust its precision by modifying its default parameters.

Parameters:

operator (np.ndarray or scipy.sparse.spmatrix or Tensor) – The Hermitian operator \(M\) whose range of eigenvalues you want to determine.
iteration_count (int, optional) – The number of iterations \(N\) in the calculation. Defaults to 3000. Choose a higher number to improve the precision, or a smaller number to make the estimation run faster.
seed (int or None, optional) – The random seed that the function uses to choose the initial random vector \(\left| r \right\rangle\). Defaults to None, which means that the function uses a different seed in each run.
name (str or None, optional) – The name of the node.

Returns:

The difference between the largest and the smallest eigenvalues of the operator.

Return type:

Tensor (scalar, real)

Warning

This calculation can be expensive, so we recommend that you run it before the optimization, if possible. You can do this by using a representative or a worst-case operator.

Notes

This function repeatedly multiplies the operator \(M\) with a random vector \(\left| r \right\rangle\). In terms of the operator’s eigenvalues \(\{ v_i \}\) and eigenvectors \(\{\left|v_i \right\rangle\}\), the result of \(N\) matrix multiplications is:

\[M^N \left|r\right\rangle = \sum_i v_i^N \left|v_i\right\rangle \left\langle v_i \right. \left| r \right\rangle.\]

For large \(N\), the term corresponding to the eigenvalue with largest absolute value \(V\) will dominate the sum, as long as \(\left|r\right\rangle\) has a non-zero overlap with its eigenvector. The function then retrieves the eigenvalue \(V\) via:

\[V \approx \frac{\left\langle r \right| M^{2N+1} \left| r \right\rangle}{\left\| M^N \left| r \right\rangle \right\|^2}.\]

The same procedure applied to the matrix \(M-V\) allows the function to find the eigenvalue at the opposite end of the spectral range.

Examples

>>> operator = np.diag([10, 40])
>>> graph.spectral_range(operator, name="spectral_range")
<Tensor: name="spectral_range", operation_name="spectral_range", shape=()>
>>> result = qctrl.functions.calculate_graph(
...     graph=graph, output_node_names=["spectral_range"]
... )
>>> result.output["spectral_range"]["value"]
30.0

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