Weighted Projective Line ZX Calculus: Quantized Orbifold Geometry for Quantum Compilation

We develop a unified geometric framework for quantum circuit compilation based on quantized orbifold phases and their diagrammatic semantics. Physical qubit platforms impose heterogeneous phase resolutions, anisotropic Bloch-ball contractions, and hardware-dependent $2\pi$ winding behavior. We show that these effects admit a natural description on the weighted projective line $\mathbb{P}(a,b)$, whose orbifold points encode discrete phase grids and whose monodromy captures winding accumulation under realistic noise channels. Building on this geometry, we introduce the WPL–ZX calculus, an extension of the standard ZX formalism in which each spider carries a weight–phase–winding triple $(a,\alpha ,k)$. We prove soundness of LCM-based fusion and normalization rules, derive curvature predictors for phase-grid compatibility, and present the Weighted ZX Circuit Compression (WZCC) algorithm, which performs geometry-aware optimization on heterogeneous phase lattices. To connect circuit-level structure with fault-tolerant architectures, we introduce Monodromy-Aware Surface-Code Decoding (MASD), a winding-regularized modification of minimum-weight matching on syndrome graphs. MASD incorporates orbifold-weighted edge costs, producing monotone decoder-risk metrics and improved robustness across phase-quantized noise models. All results are validated through symbolic and numerical simulations, demonstrating that quantized orbifold geometry provides a coherent and hardware-relevant extension of diagrammatic quantum compilation.