Imperfect Graphs from Unitary Matrices -- I
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915824150773760 |
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| author | Lewis, Wesley Pareek, Darsh Kumar, Umesh Janjam, Ravi |
| author_facet | Lewis, Wesley Pareek, Darsh Kumar, Umesh Janjam, Ravi |
| contents | Matrix representations of quantum operators are computationally complete but often obscure the structural topology of information flow within a quantum circuit \cite{nielsen2000}. In this paper, we introduce a generalized graph-theoretic framework for analyzing quantum operators by mapping unitary matrices to directed graphs; we term these structures \emph{Imperfect Graphs} or more formally as \emph{Topological Structure of Superpositions}(TSS) as a tool to devise better Quantum Algorithms. In this framework, we represent computational basis states as vertices. A directed edge exists between two vertices if and only if there is a non-zero amplitude transition between them, effectively mapping the support of the unitary operator. In this paper we deliberately discard probability amplitudes and phase information to isolate the connectivity and reachability properties of the operator. We demonstrate how TSS intuitively helps describe gates such as the Hadamard, Pauli-(X,Y,Z) gates, etc \cite{nielsen2000}. This framework provides a novel perspective for viewing quantum circuits as discrete dynamical systems \cite{childs2009,aharonov2001}
Keywords: Quantum Algorithms, Unitary Matrix Approach, Topological Structure of Superpositions (TSS), Graph Theory |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21808 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Imperfect Graphs from Unitary Matrices -- I Lewis, Wesley Pareek, Darsh Kumar, Umesh Janjam, Ravi Quantum Physics Mathematical Physics Computational Physics Matrix representations of quantum operators are computationally complete but often obscure the structural topology of information flow within a quantum circuit \cite{nielsen2000}. In this paper, we introduce a generalized graph-theoretic framework for analyzing quantum operators by mapping unitary matrices to directed graphs; we term these structures \emph{Imperfect Graphs} or more formally as \emph{Topological Structure of Superpositions}(TSS) as a tool to devise better Quantum Algorithms. In this framework, we represent computational basis states as vertices. A directed edge exists between two vertices if and only if there is a non-zero amplitude transition between them, effectively mapping the support of the unitary operator. In this paper we deliberately discard probability amplitudes and phase information to isolate the connectivity and reachability properties of the operator. We demonstrate how TSS intuitively helps describe gates such as the Hadamard, Pauli-(X,Y,Z) gates, etc \cite{nielsen2000}. This framework provides a novel perspective for viewing quantum circuits as discrete dynamical systems \cite{childs2009,aharonov2001} Keywords: Quantum Algorithms, Unitary Matrix Approach, Topological Structure of Superpositions (TSS), Graph Theory |
| title | Imperfect Graphs from Unitary Matrices -- I |
| topic | Quantum Physics Mathematical Physics Computational Physics |
| url | https://arxiv.org/abs/2602.21808 |