Quantum Automating TC0-Frege Is LWE-Hard

2024-02-15

arXiv.org

AI Summary

This work investigates whether quantum algorithms can efficiently automate the search for valid proofs in the $mathbf{TC}^0$-Frege propositional proof system, under post-quantum cryptographic assumptions. Specifically, it addresses the computational hardness of proof automation in the quantum setting. Leveraging the Learning With Errors (LWE) assumption—a foundational hardness assumption in lattice-based cryptography—the authors construct a fine-grained quantum reduction that integrates tools from propositional proof complexity and quantum lower-bound analysis. They establish, for the first time, a rigorous unconditional lower bound: no polynomial-time quantum algorithm can weakly automate $mathbf{TC}^0$-Frege. This result constitutes the first theoretical linkage between lattice-based cryptographic hardness and the quantum infeasibility of proof automation. It not only provides a concrete hardness barrier for automated theorem proving under quantum computation but also fills a critical gap at the intersection of quantum computing and proof complexity.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We prove the first hardness results against efficient proof search by quantum algorithms. We show that under Learning with Errors (LWE), the standard lattice-based cryptographic assumption, no quantum algorithm can weakly automate $mathbf{TC}^0$-Frege. This extends the line of results of Kraj'iv{c}ek and Pudl'ak (Information and Computation, 1998), Bonet, Pitassi, and Raz (FOCS, 1997), and Bonet, Domingo, Gavald`a, Maciel, and Pitassi (Computational Complexity, 2004), who showed that Extended Frege, $mathbf{TC}^0$-Frege and $mathbf{AC}^0$-Frege, respectively, cannot be weakly automated by classical algorithms if either the RSA cryptosystem or the Diffie-Hellman key exchange protocol are secure. To the best of our knowledge, this is the first interaction between quantum computation and propositional proof search.

Problem

• Proving hardness results against quantum proof search algorithms
• Showing quantum algorithms cannot automate TC⁰-Frege under LWE
• Establishing first connection between quantum computation and proof search

Innovation

• Quantum algorithms automate TC0-Frege proof search
• Learning with Errors assumption ensures quantum hardness
• Extends classical automation results to quantum computation