Problem definition in terms of Fractal distribution and Complexity Connection to Graph and Hypergraph description

Phase space dynamics and control of the quantum particles associated to hypergraph states

Overview

• Problem definition in terms of Fractal distribution and Complexity

• Connection to Graph and Hypergraph description

• Fractional covering number as entropy parameter

• Fractional entropy descriptor of a hypergraph

• Topological order relation to nonlocality and quantum states

• Multilevel hypergraph partitioning algorithms

• Conclusion

Complexity measures

• Algorithmic complexity (length of the shortest code).

• Fractional dimension.

• Shannon information (entropy).

• Correlation dimension (topologic dimension of an attractor). Topological entropy.

• Functional clustering.

Fractals as complex systems

What about describing the shape of nature via Graph or… Hypergraph

V={1,2,3,4,5}

• V={1,2,3,4,5}
• E={{1,2},{2,3},{3,4},{4,5},{5,1},{1,4},{3,5}}

H=(X,E), X={1,2,3,4}, E={{1,2,3},{2,3,4},{1,4}}={E1,E2,E3,}

• H=(X,E), X={1,2,3,4}, E={{1,2,3},{2,3,4},{1,4}}={E1,E2,E3,}

A hypergraph can be defined by its incidence matrix A with columns representing the edges El ,E2.,. .,Em and rows representing the vertices x1,x2,. ..,xn.

Both graphs and hypergraphs may be partitioned to optimize some objective.

• A hypergraph is used to represent the connectivity information from the circuit specification. Each vertex in the hypergraph represents a cell in the circuit and each hyperedge represents a net from the circuit’s netlist.

Fractional entropy descriptor of a hypergraph

Using function, from topological Rényi’s entropy of order a we obtain an integral entropic measure integrating a fractional parameter.

• Using function, from topological Rényi’s entropy of order a we obtain an integral entropic measure integrating a fractional parameter.

• Moreover, let’s note that at the limit a = 1, we obtain

• the Ahmad-Lin estimator of Shannon’s entropy

Example for function:

• Example for function:

• Fractional covering number of H:

Nonlocality signifies that the statistical behaviour of a system cannot be described by a local realistic theory.

• Nonlocality signifies that the statistical behaviour of a system cannot be described by a local realistic theory.

• For nonlocality it is essential that the correlation probabilities of such theories obey so-called Bell inequalities, which are violated for certain quantum states.

• The computational mapping f projects the 4-level (S, T0, T+/

• - system to a two-level subspace. In the case of a two- ½ spin qubit, f sends two degrees of freedom to zero computational meaning. Over the remaining subspace it is a linear mapping that maps the two basis states to computational states:

Conclusions

• We proposed an original fractional entropy measure inspired from Rényi’s topological order making possible description of the complex systems and strong variations of the shapes of the non parametrically estimated related PDF.

• The main motivation was to overcome the limitations of Shannon’s entropy which appeared not adapted to partition problem.

• Method is proposed for hypergraph structures which reflect nonlocal characteristics of correlations between separate objects and can be used for description of entanglement resources.

Thank you for attention!