High-Level Fault-Tolerant Abstractions for Quantum-Gate Circuit Design and Synthesis: PQC and Topological Anyon Architectures (TQC) for Categorical Computations in SU(2)_3 TQFT and D-brane Stability

Abstract

We propose a dual-architecture quantum simulation framework for modeling morphisms and stability conditions in the bounded derived category $\mathbf{D}^b(\mathrm{Coh}(X))$, with applications to D-brane physics on K\"ahler and non-K\"ahler manifolds. Two physically executable quantum realizations are constructed: parameterized quantum circuits (PQCs) implemented on conventional gate-based qubit platforms, and a topological quantum computing (TQC) realization using braiding and fusion of Fibonacci anyons modeled via SU(2)$_3$ modular tensor categories. In the PQC model, we encode slope functionals S(F) and stability constraints as variational observables, mapping derived morphisms to unitaries that evolve over parameterized angles. The output expectation values simulate quantum-corrected Chern class inequalities with deformation terms $\delta$, capturing quantum corrections to classical geometric stability. In the TQC model, we engineer braid group representations to implement functorial transformations such as spherical twists and autoequivalences as sequences of fault-tolerant braid operations. This bifurcated approach provides a robust engineering pipeline for simulating categorical stability and homological algebra on quantum hardware, bridging abstract derived category theory with executable quantum architectures.