Symbolic Specification and Reasoning for Quantum Data and Operations

2025-12-26

AI Summary

Quantum computing lacks symbolic modeling and reasoning theories supporting automated verification, hindering the practical deployment of formal verification tools. To address this, we propose Symbolic Operator Logic (SOL), the first framework embedding classical first-order logic into a quantum operator language—enabling recursive symbolic definitions of quantum data and operations, as well as deductive reasoning about their properties. SOL modularly incorporates classical theories—including modal Boolean algebras and group theory—thereby allowing off-the-shelf automated verification techniques to be directly applied to quantum systems. The framework provides a scalable, composable infrastructure for quantum theorem proving in proof assistants such as Lean and Coq. By bridging the critical gap between classical formal methods and quantum verification, SOL significantly enhances the capability to formally verify quantum algorithms and protocols.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

In quantum information and computation research, symbolic methods have been widely used for human specification and reasoning about quantum states and operations. At the same time, they are essential for ensuring the scalability and efficiency of automated reasoning and verification tools for quantum algorithms and programs. However, a formal theory for symbolic specification and reasoning about quantum data and operations is still lacking, which significantly limits the practical applicability of automated verification techniques in quantum computing. In this paper, we present a general logical framework, called Symbolic Operator Logic $mathbf{SOL}$, which enables symbolic specification and reasoning about quantum data and operations. Within this framework, a classical first-order logical language is embedded into a language of formal operators used to specify quantum data and operations, including their recursive definitions. This embedding allows reasoning about their properties modulo a chosen theory of the underlying classical data (e.g., Boolean algebra or group theory), thereby leveraging existing automated verification tools developed for classical computing. It should be emphasised that this embedding of classical first-order logic into $mathbf{SOL}$ is precisely what makes the symbolic method possible. We envision that this framework can provide a conceptual foundation for the formal verification and automated theorem proving of quantum computation and information in proof assistants such as Lean, Coq, and related systems.

Problem

• Develops a logical framework for symbolic quantum specification
• Enables reasoning about quantum data using classical verification tools
• Aims to support formal verification in quantum computation systems

Innovation

• Introduces Symbolic Operator Logic (SOL) framework
• Embeds classical logic into quantum operator language
• Enables symbolic reasoning using classical verification tools