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

TL;DR

The paper develops a dual-architecture quantum framework to study stability and morphisms in the bounded derived category $D^b(\mathrm{Coh}(X))$, targeting D-brane configurations on non-Kähler manifolds. It introduces a novel stability function $S(\mathcal{F})$ based on the slope $\mu(\mathcal{F})$, plus higher-Chern-class invariants, and proves consistency within the derived-category setting. A PQC realization and a TQC realization using Fibonacci anyons are constructed to compute quantum-corrected stability measures, with explicit circuits (Circuit-1 to Circuit-5) providing correction factors $\delta$ that modify classical stability predicates. Topological quantum corrections are further modeled via braid-group representations and $R$-symbols to simulate autoequivalences (e.g., spherical twists) on brane configurations, yielding a fault-tolerant, categorically faithful framework. Collectively, the work bridges abstract homological algebra and noncommutative geometry with executable quantum architectures, enabling concrete simulations of D-brane dynamics and stability landscapes under quantum deformation and topological protection.

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 Kähler and non-Kähler 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 $δ$, 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.

Figures (25)

• Figure 1: Higher Chern Classes: The graph shows the first four Chern classes as functions of the parameter $x$. Each class is calculated using the combinatorial formula described above.
• Figure 2: Stability Region Visualization: The contour plot displays the stability condition across a parameter space defined by $x$ and $y$. The color gradient indicates regions of stability, helping identify stable configurations of D-branes.
• Figure 3: Parametric Study of Stability Conditions: This plot depicts how the stability condition varies with the parameter $x$, showcasing the impact of parameter changes on the stability of D-branes.
• Figure 4: Complex Torus Representation: This 3D plot shows the complex torus, highlighting its geometric structure, which is crucial for understanding the relationships between D-branes and coherent sheaves.
• Figure 5: Quantum Circuit-3: 2-qubits