Quantum Automating TC0-Frege Is LWE-Hard

• Published in: Computational complexity Vol. 34; no. 2; p. 16
• Main Authors: Arteche, Noel, Carenini, Gaia, Gray, Matthew
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2025; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Computational Mathematics and Numerical Analysis; Computer Science; Quantum computing; feasible interpolation; 68Q01; 68Q12; proof complexity; post-quantum cryptography; automatability; 03F20
• ISSN: 1016-3328; 1420-8954; 1420-8954
• DOI: 10.1007/s00037-025-00271-w

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 TC 0 -Frege. This extends the line of results of Krajííček and Pudlík( Information and Computation , 1998), Bonet, Pitassi, and Raz ( SIAM Journal on Computing , 2000),and Bonet, Domingo, Gavaldá, Maciel, and Pitassi ( Computational Complexity, 2004 ), who showed that ExtendedFrege, TC 0 -Frege and 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.