Gap-preserving reductions and RE-completeness of independent set games

2025-05-08

AI Summary

This work investigates the computational complexity of the *gap promise problem* for the independent set game in quantum multiprover interactive proof systems (MIP*). While the classical version is polynomial-time solvable, quantum entanglement induces a fundamental complexity-theoretic leap—resolving this paradox, the authors establish the first MIP*-completeness result for this gap promise problem and prove its undecidability within MIP*. Methodologically, they introduce a novel *PVM stability theorem*, enabling robust perturbation correction from approximate projection-valued measures (POVMs) to exact PVMs; they further construct a family of independent set games with constant question-answer size and employ gap-preserving reductions alongside nonlocal game modeling. Key contributions include: (i) identifying the first natural MIP*-complete gap promise problem; (ii) revealing an entanglement-driven undecidability transition; and (iii) providing a foundational tool for robust reductions in MIP*.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In complexity theory, gap-preserving reductions play a crucial role in studying hardness of approximation and in analyzing the relative complexity of multiprover interactive proof systems. In the quantum setting, multiprover interactive proof systems with entangled provers correspond to gapped promise problems for nonlocal games, and the recent result MIP$^*$=RE (Ji et al., arXiv:2001.04383) shows that these are in general undecidable. However, the relative complexity of problems within MIP$^*$ is still not well-understood, as establishing gap-preserving reductions in the quantum setting presents new challenges. In this paper, we introduce a framework to study such reductions and use it to establish MIP$^*$-completeness of the gapped promise problem for the natural class of independent set games. In such a game, the goal is to determine whether a given graph contains an independent set of a specified size. We construct families of independent set games with constant question size for which the gapped promise problem is undecidable. In contrast, the same problem is decidable in polynomial time in the classical setting. To carry out our reduction, we establish a new stability theorem, which could be of independent interest, allowing us to perturb families of almost PVMs to genuine PVMs.

Problem

• Study gap-preserving reductions in quantum MIP* systems
• Establish MIP*-completeness for independent set games
• Analyze decidability gaps between quantum and classical settings

Innovation

• Introduces framework for gap-preserving quantum reductions
• Proves MIP$^*$-completeness for independent set games
• Develops stability theorem for perturbing almost PVMs