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

• URL: http://arxiv.org/abs/2512.00682v1
• Date: Sun, 30 Nov 2025 00:56:39 GMT
• Title: Weighted Projective Line ZX Calculus: Quantized Orbifold Geometry for Quantum Compilation
• Abstract summary: We develop a unified framework for quantum circuit compilation based on quantized orbifold phases and their diagrammatic semantics.<n>We show that these effects admit a natural description on the weighted projective line $mathbbP(a,b)$, whose orbifold points encode discrete phase grids.<n>We introduce the WPL--ZX calculus, an extension of the standard ZX formalism in which each spider carries a weight--phase--winding triple $(a,,k)$.
• Score: 0.764671395172401
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $2π$ 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 $(a,α,k)$. 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.

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