hiperwalk.Coined.uniform_state#

Coined.uniform_state(vertices=None, arcs=None)[source]#

Create a uniform state.

The uniform state is a unit vector with entries that have the same real amplitudes. If both vertices is None and arcs is None, create the uniform superposition of all arcs. Otherwise, create the uniform superposition of all arcs in arcs and all the arcs with tail in vertices.

Parameters:

vertices: list of vertices, default=None: If vertices is not None, every arc with tail in vertices is added to the uniform superposition.
arcs: list of arcs, default=None: If arcs is not None, every arc in arcs is added to the uniform superposition.

Returns:

numpy.ndarray

Notes

An example of the uniform state is

\[\ket{d} = \frac{1}{\sqrt{N}} \sum_{i = 0}^{N - 1} \ket{i}\]

where \(N\) represents the dimension of the Hilbert space, and \(i\) is a label within the graph. In the continuous-time quantum walk model, \(i\) corresponds to the label of a vertex, while in the coined quantum walk model, \(i\) is the label of an arc.