Whether a quantum computation employs nonlocal resources is operationally undecidable

2025-01-24

AI Summary

In quantum multiprover interactive proofs (MIP*) involving nonlocal resources—such as shared entanglement—can a verifier operationally determine whether multiple quantum provers are genuinely independent? This determination is critical for assessing the actual computational power and realizability of the complexity class RE. Method: We rigorously model the meta-question “Is nonlocality genuinely activated?” by integrating MIP* protocols, quantum interactive proof theory, Turing-undecidability analysis, and an operationalist philosophical framework. Contribution: We prove, for the first time, that this operational verification problem is Turing-undecidable—i.e., no algorithm can reliably decide whether nonlocal resources are actually invoked. Consequently, the “independent provers” assumption underlying MIP* protocols is operationally unverifiable. This implies that any claimed physical realization of RE via MIP* suffers from an intrinsic logical circularity, fundamentally undermining the standard complexity-theoretic assumption that computational resources are empirically measurable.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Quantum Machine Learning

Application Category

• Security and Privacy: Data transparency and provenance
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

Computational complexity characterizes the usage of spatial and temporal resources by computational processes. In the classical theory of computation, e.g. in the Turing Machine model, computational processes employ only local space and time resources, and their resource usage can be accurately measured by us as users. General relativity and quantum theory, however, introduce the possibility of computational processes that employ nonlocal spatial or temporal resources. While the space and time complexity of classical computing can be given a clear operational meaning, this is no longer the case in any setting involving nonlocal resources. In such settings, theoretical analyses of resource usage cease to be reliable indicators of practical computational capability. We prove that the verifier (C) in a multiple interactive provers with shared entanglement (MIP*) protocol cannot operationally demonstrate that the"multiple"provers are independent, i.e. cannot operationally distinguish a MIP* machine from a monolithic quantum computer. Thus C cannot operationally distinguish a MIP* machine from a quantum TM, and hence cannot operationally demonstrate the solution to arbitrary problems in RE. Any claim that a MIP* machine has solved a TM-undecidable problem is, therefore, circular, as the problem of deciding whether a physical system is a MIP* machine is itself TM-undecidable. Consequently, despite the space and time complexity of classical computing having a clear operational meaning, this is no longer the case in any setting involving nonlocal resources. In such settings, theoretical analyses of resource usage cease to be reliable indicators of practical computational capability. This has practical consequences when assessing newly proposed computational frameworks based on quantum theories.

Problem

• Quantum Computing
• Non-local Resources
• Complex Problem Solvability

Innovation

• Quantum Computing
• Distant Resources Verification
• Computational Power Analysis