Categorical Tensor-Graph Semantics for Quantum Algorithms

Abstract

This paper systematically investigates quantum computing protocols from the intuitive perspective of categorical tensor-graph semantics within the category FHilb. While conventional Hilbert-space formalisms conceal the structural nature of quantum algorithms behind high-dimensional matrix operations, the topological framework developed herein directly encodes algorithmic functionalities to their graphical skeletons. We provide a comprehensive topological reinterpretation of the Bernstein-Vazirani and the Simon algorithms, demonstrating how topological transformations distill their core mathematical essence and clarify the operational mechanism of oracles. Going beyond standard qubit models, we formalize the qutrit-adapted generalized Deutsch-Jozsa algorithm as well as the generalized single-shot Grover algorithm, explicitly laying out their graphical representations. We further implement CNOT gates via complementary Frobenius structures, trace the diagrammatic genesis of quantum entanglement including Bell and GHZ states, and present a diagrammatic simplification for W-state preparation protocol. By bridging tensor category theory with practical quantum algorithmic design, this work furnishes composable, scalable diagrammatic toolkit essential for automated circuit optimization across the evolving quantum hardware ecosystem.