CATEGORICAL TENSOR-GRAPH SEMANTICS FOR QUANTUM ALGORITHMS

This paper introduces a categorical approach to quantum computing, moving away from traditional Hilbert-space formalisms—which often obscure algorithmic structure behind complex matrix algebra—toward a topological framework. By utilizing tensor-graph semantics within the category FHilb, the authors map quantum algorithms to graphical skeletons. This method allows for a more intuitive understanding of how quantum gates, oracles, and entanglement interact to produce computational advantages.

The authors provide a comprehensive topological reinterpretation of several foundational quantum algorithms, including the Bernstein-Vazirani and Simon algorithms. By applying topological transformations, they distill these algorithms into minimal graphical representations, clarifying the role of auxiliary qubits and the mechanism of phase kickback. Furthermore, the paper extends these concepts beyond the standard qubit paradigm by formalizing the qutrit-adapted generalized Deutsch-Jozsa algorithm and the generalized single-shot Grover algorithm. These graphical representations offer a scalable, composable toolkit for designing and optimizing quantum circuits.

A significant portion of the work focuses on the diagrammatic genesis of quantum entanglement. By implementing CNOT gates via complementary Frobenius structures, the authors trace the construction of Bell and GHZ states and provide a simplified diagrammatic protocol for W-state preparation. This approach not only aids in theoretical understanding but also provides a foundation for automated circuit optimization, which is essential for managing the constraints of the current Noisy Intermediate-Scale Quantum (NISQ) era.