TSS Graphs for Hadamard Matrices: Real vs Complex

@misc{lewis2026graphs,
  title = {{TSS Graphs for Hadamard Matrices: Real vs Complex}},
  author = {Wesley Lewis and Darsh Pareek and Ravi Janjam},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.16813},
  eprint = {2609.16813},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16813},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - TSS Graphs for Hadamard Matrices: Real vs Complex
AU  - Wesley Lewis
AU  - Darsh Pareek
AU  - Ravi Janjam
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.16813
UR  - https://arxiv.org/abs/2609.16813
AB  - 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
ER  -