Optimizing quantum circuits is a key challenge for quantum computing. The PyZX compiler broke new ground by optimizing circuits via the ZX calculus, a powerful graphical alternative to the quantum circuit model. Still, it carries no guarantee of its correctness. To address this, we developed VyZX, a verified ZX-calculus in the Coq proof assistant. VyZX provides two distinct representations of ZX diagrams for ease of programming and proof: A graph-based representation for writing high-level functions on diagrams and a block-based representation for proving ZX diagrams equivalent. Through these two different views, VyZX provides the tools necessary to verify properties and transformations of ZX diagrams. This paper explores the proofs and design choices underlying VyZX and its application and the challenges of verifying a graphical programming language.
Optimizing quantum circuits is a key challenge for quantum computing. The PyZX compiler broke new ground by optimizing circuits via the ZX calculus, a powerful graphical alternative to the quantum circuit model. Still, it carries no guarantee of its correctness. To address this, we developed VyZX, a verified ZX-calculus in the Coq proof assistant. VyZX provides two distinct representations of ZX diagrams for ease of programming and proof: A graph-based representation for writing high-level functions on diagrams and a block-based representation for proving ZX diagrams equivalent. Through these two different views, VyZX provides the tools necessary to verify properties and transformations of ZX diagrams. This paper explores the proofs and design choices underlying VyZX and its application and the challenges of verifying a graphical programming language.
Graphical languages are a convenient shorthand to represent computation, with rewrite rules relating one graph to another. In contrast, proof assistants rely heavily on inductive datatypes, particularly when giving semantics to embedded languages. This creates obstacles to formally reasoning about graphical languages, since imposing an inductive structure obfuscates the diagrammatic nature of graphical languages, along with their corresponding equational theories. To address this gap, we present VyZX, a verified library for reasoning about inductively defined graphical languages. These inductive constructs arise naturally from category-theoretic definitions. We developed VyZX to Verify the ZX-calculus, a graphical langauge for reasoning about quantum computation. The ZX-calculus comes with a collection of diagrammatic rewrite rules that preserve the graph's semantic interpretation. We show how inductive graphs in VyZX are used to prove the soundness of the ZX-calculus rewrite rules and apply them in practice using standard proof assistant techniques. We also provide an IDE-integrated visualizer for proof engineers to directly reason about diagrams in graphical form.
Graphical languages are a convenient shorthand to represent computation, with rewrite rules relating one graph to another. In contrast, proof assistants rely heavily on inductive datatypes, particularly when giving semantics to embedded languages. This creates obstacles to formally reasoning about graphical languages, since imposing an inductive structure obfuscates the diagrammatic nature of graphical languages, along with their corresponding equational theories. To address this gap, we present Vy ZX, a verified library for reasoning about inductively defined graphical languages. These inductive constructs arise naturally from category-theoretic definitions. We developed Vy ZX to V erify the ZX -calculus, a graphical language for reasoning about quantum computation. The ZX-calculus comes with a collection of diagrammatic rewrite rules that preserve the graph’s semantic interpretation. We show how inductive graphs in Vy ZX are used to prove the soundness of the ZX-calculus rewrite rules and apply them in practice using standard proof assistant techniques. We also provide an IDE-integrated visualizer for proof engineers to directly reason about diagrams in graphical form.
This dissertation is an exploration of quantum formal verification through the development of VyZX, a new formalization of the ZX-calculus. VyZX uses the categorical structure of the ZX-calculus to reason about ZX diagrams, providing a framework to establish completeness, encode common rules, and investigate applications like peephole optimization. However, working within a symmetric monoidal category introduces challenges, as the same ZX-diagram can have many equivalent representations in VyZX. This results in significant manual proof effort, as there is no single "best"; normal form, which is both tedious and difficult to fully automate. To address this, use E-Graphs, a data structure for managing equivalence classes, to streamline reasoning about equivalence. I build a system that can automatically use VyZX lemmas to prove the equivalence of two semantically equivalent diagrams if they are equal up to structure. The system performs well, solving complex associativity problems in less than a second. In benchmarks, the system is able to handle much larger problems than practical. In addition, I explore the solving of more complex equalities using custom rewrite databases. The system is able to replace and outperform Rocq's autorewrite tactic for a common problem class in VyZX, showing its general applicability.
Quantum computing is an emerging computational paradigm with the potential to outperform classical computers in solving a variety of problems. To achieve this, quantum programs are typically represented as quantum circuits, which must be optimized and adapted for target hardware through quantum circuit compilation. We introduce ZX-DB, a data-driven system that performs quantum circuit simplification and rewriting inside a graph database using ZX-calculus, a complete graphical formalism for quantum mechanics. ZX-DB encodes ZX-calculus rewrite rules as standard openCypher queries and executes them on an example graph database engine, Memgraph, enabling efficient, database-native transformations of large-scale quantum circuits. ZX-DB integrates correctness validation via tensor and graph equivalence checks and is evaluated against the state-of-the-art PyZX framework. Experimental results show that ZX-DB achieves up to an order-of-magnitude speedup for independent rewrites, while exposing pattern-matching bottlenecks in current graph database engines. By uniting quantum compilation and graph data management, ZX-DB opens a new systems direction toward scalable, database-supported quantum computing pipelines.
As state-of-the-art quantum computers are capable of running increasingly complex algorithms, the need for automated methods to design and test potential applications rises. Equivalence checking of quantum circuits is an important, yet hardly automated, task in the development of the quantum software stack. Recently, new methods have been proposed that tackle this problem from widely different perspectives. One of them is based on the ZX-calculus, a graphical rewriting system for quantum computing. However, the power and capability of this equivalence checking method has barely been explored. The aim of this work is to evaluate the ZX-calculus as a tool for equivalence checking of quantum circuits. To this end, it is demonstrated how the ZX-calculus based approach for equivalence checking can be expanded in order to verify the results of compilation flows and optimizations on quantum circuits. It is also shown that the ZX-calculus based method is not complete—especially for quantum circuits with ancillary qubits. In order to properly evaluate the proposed method, we conduct a detailed case study by comparing it to two other state-of-the-art methods for equivalence checking: one based on path-sums and another based on decision diagrams. The proposed methods have been integrated into the publicly available QCEC tool (https://github.com/cda-tum/qcec) which is part of the Munich Quantum Toolkit (MQT).
The ZX-calculus is a graphical language for reasoning about ZX-diagrams, a type of tensor networks that can represent arbitrary linear maps between qubits. Using the ZX-calculus, we can intuitively reason about quantum theory, and optimise and validate quantum circuits. In this paper we introduce PyZX, an open source library for automated reasoning with large ZX-diagrams. We give a brief introduction to the ZX-calculus, then show how PyZX implements methods for circuit optimisation, equality validation, and visualisation and how it can be used in tandem with other software. We end with a set of challenges that when solved would enhance the utility of automated diagrammatic reasoning.