TSS Graphs for Hadamard Matrices: Real vs Complex

Abstract:We have observed that Hadamard matrices whether real or complex lead to a dense output of identical probabilities for any given single input state. Complex Hadamard matrices have unusual orders and introduce phase which prove useful when controlled phase transformation gates are used. By taking a superposed combinations of input states, we discovered that the Hadamards generate non-uniform probabilities which is practically significant towards amplitude amplification without the need for manual parameterization like Grover's operator~\cite{grover1996}. A variety of graph theoretic properties are applied to TSS graphs and their trends were explored. Additionally, graphs turn out to be nearly isomorphic for the same set of input states which has potential applications for developing Quantum Algorithms. Keywords: Quantum Algorithms, Hadamard Matrices, Unitary Matrix Approach, Topological Structure of Superpositions (TSS), Graph Theory

arXiv:2609.16813 [quant-ph]

(or arXiv:2609.16813v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.16813 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.