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

TL;DR

This paper introduces a geometric framework based on weighted projective lines for quantum circuit compilation, capturing hardware-specific phase effects and enabling optimized, noise-aware quantum circuit design.

Contribution

It develops the WPL--ZX calculus, a novel extension of ZX calculus incorporating orbifold phases, and proposes geometry-aware algorithms for circuit compression and fault-tolerant decoding.

Findings

WPL--ZX calculus is sound and captures hardware phase effects.

The WZCC algorithm improves circuit compression by geometry-aware optimization.

MASD enhances surface-code decoding robustness under phase-quantized noise.

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 $2\pi$ 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.