Beyond IP = PSPACE and QIP = PSPACE: Interactive Proofs in Arbitrary Physical Theories

2026-09-30

AI Summary

This study investigates the extent to which the computational power of interactive proofs is determined by underlying physical theories rather than solely by classical or quantum models. Within the framework of general probabilistic theories, it systematically analyzes how assumptions regarding message types and operational efficiency affect computational complexity. The results demonstrate that any physical model encompassing quantum theory cannot surpass the PSPACE upper bound for three-message protocols; however, exceeding PSPACE becomes possible when arbitrary system messages are permitted. By establishing complexity boundaries under different message-exchange paradigms, this work reveals the specific mechanisms through which physical theories constrain verifier capabilities, thereby offering a novel perspective for understanding the relationship between physical laws and computational complexity.

Abstract

The equalities IP = PSPACE and QIP = PSPACE, the latter achievable with three messages, raise a basic question: how much of an interactive proof's power comes from the underlying physical theory? We study interactive proofs in general probabilistic theories, which include classical and quantum theory. The answer depends on what the prover and verifier exchange and how the theory specifies efficient operations. When they exchange only classical messages, protocols in every theory satisfying our standard assumptions decide exactly PSPACE. For protocols with a quantum verifier and quantum messages, allowing a prover to use any theory containing quantum theory does not increase the maximum acceptance probability. Thus, the three-message PSPACE result remains valid against such provers. When messages may be arbitrary systems, the interactive-proof class can strictly exceed PSPACE.

Problem

• interactive proofs
• general probabilistic theories
• PSPACE
• physical theories
• computational complexity

Innovation

• Interactive Proofs
• General Probabilistic Theories
• PSPACE
• Quantum Verification
• Computational Complexity