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

TL;DR

The paper presents a geometry-aware extension of ZX-calculus to hardware with quantized and heterogeneous phase grids by introducing weighted projective line geometry. It defines the WPL–ZX calculus, where spiders carry a triple $(a,\alpha,k)$ encoding local phase resolution, base phase, and winding, with LCM-based fusion and a total-angle semantics that respects orbifold monodromy. Two key contributions are WZCC, a quantization-aware circuit normalization that yields canonical forms while preserving semantics, and MASD, a winding-aware surface-code decoder that penalizes phase-winding mismatches during decoding. Extensive symbolic and numerical experiments, including IBM Q hardware noise models, demonstrate strong circuit-size compression (CSC) and high fidelity preservation (FP) while achieving phase-grid alignment (PQVR) and compatibility with standard transpiler optimizations. The work connects diagrammatic reasoning, orbifold geometry, and hardware-level noise into a unified pipeline, offering geometry-guided strategies for compiling and decoding in the NISQ era with potential extensions to industrial stacks and learning-guided normalization.

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.

Figures (18)

• Figure 1: Single-qubit example of ZX-based simplification (circuit view)
• Figure 2: ZX diagrams: before vs. after simplification (spider view)
• Figure 3: Illustration of spider properties: fusion, identity, bialgebra, and Hadamard color change.
• Figure 4: WPL--ZX example illustrating how weighted spiders track grids $a$, phases $\alpha$, and windings $k$. (a) The initial diagram lives on heterogeneous grids $a_Z$ and $a_X$. (b) Z--Z fusion preserves the dominant grid $a_Z$ while adding phases and winding numbers. (c) A mixed-color interaction is mediated by an $H$ gate and then lifted to the common LCM grid $\mathrm{lcm}(a_Z,a_X)$, which is the natural domain for WZCC normalisation.
• Figure 5: Scaling of PQVR (phase-quantization variance ratio), CSC (circuit-size compression), and FP (fidelity preservation) as functions of the number of qubits and the number of spiders for random WPLZX diagrams in D1. Each point is averaged over $20$ random instances; error bars indicate one standard deviation across instances.

Theorems & Definitions (146)

• Definition 2.1: Quantized phase grid
• Remark 2.2
• Example 2.3: Phase resolution
• Definition 2.4: Mixed-weight fusion grid
• Remark 2.5
• Lemma 2.6: Discrete phase closure under fusion
• proof
• Remark 2.7
• Definition 2.8: Accumulated phase and winding number
• Remark 2.9