Weighted Projective Line ZX Calculus: Quantized Orbifold Geometry for Quantum Compilation
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| Format: | Preprint |
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2025
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| _version_ | 1866918224620158976 |
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| author | Cho, Gunhee Cheng, Jason Li, Evelyn |
| author_facet | Cho, Gunhee Cheng, Jason Li, Evelyn |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_00682 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weighted Projective Line ZX Calculus: Quantized Orbifold Geometry for Quantum Compilation Cho, Gunhee Cheng, Jason Li, Evelyn Quantum Physics 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. |
| title | Weighted Projective Line ZX Calculus: Quantized Orbifold Geometry for Quantum Compilation |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2512.00682 |