Graph Rewriting Language as a Platform for Quantum Diagrammatic Calculi

TL;DR

This paper introduces a novel platform using the general-purpose graph rewriting language LMNtal for quantum circuit optimization, enabling direct diagram manipulation, pattern matching, and interactive visualization, thus aiding the discovery of optimization strategies.

Contribution

It presents a new approach leveraging LMNtal for quantum diagrammatic calculus, expanding beyond existing domain-specific tools with a flexible, model-checking enabled framework.

Findings

Demonstrated manipulation of ZX-diagrams with native graph transformation rules

Enabled simplified rule specification through quantified pattern matching

Provided interactive visualization for exploring optimization paths

Abstract

Systematic discovery of optimization paths in quantum circuit simplification remains a challenge. Today, ZX-calculus, a computing model for quantum circuit transformation, is attracting attention for its highly abstract graph-based approach. Whereas existing tools such as PyZX and Quantomatic offer domain-specific support for quantum circuit optimization, visualization and theorem-proving, we present a complementary approach using LMNtal, a general-purpose hierarchical graph rewriting language, to establish a diagrammatic transformation and verification platform with model checking. Our methodology shows three advantages: (1) manipulation of ZX-diagrams through native graph transformation rules, enabling direct implementation of basic rules; (2) quantified pattern matching via QLMNtal extensions, greatly simplifying rule specification; and (3) interactive visualization and validation of optimization paths through state space exploration. Through case studies, we demonstrate how our framework helps understand optimization paths and design new algorithms and strategies. This suggests that the declarative language LMNtal and its toolchain could serve as a new platform to investigate quantum circuit transformation from a different perspective.