Approximating the quantum value of an LCS game is RE-hard

2025-07-30

AI Summary

This work addresses the quantum value approximation problem for quantum linear constraint satisfaction (LCS) games under the entangled-prover model, characterizing its computational complexity. Method: We generalize Håstad’s classical long-code test to projection-based LCS games with entangled provers, combining Fourier analysis with the MIP* = RE framework. Results: We establish an equivalence between the complexity class LIN* and the recursively enumerable class RE: for any completeness parameter $s in (1/2, 1)$ and sufficiently small soundness error $varepsilon > 0$, $mathrm{LIN}^*_{1-varepsilon,s} = mathrm{RE}$. This resolves the completeness–soundness trade-off for entangled-prover LCS games and provides the first complexity-theoretic evidence for the existence of non-hyperlinear groups. The result significantly advances our understanding of the computational power limits of quantum interactive proof systems with entanglement.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability

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: Foundation models and LLMs for Web-related graphs

Abstract

We generalize Håstad's long-code test for projection games and show that it remains complete and sound against entangled provers. Combined with a result of Dong et al. cite{Dong25}, which establishes that $MIP^*=RE$ with constant-length answers, we derive that $LIN^*_{1-ε,s}=RE$, for some $1/2< s<1$ and for every sufficiently small $ε>0$, where LIN refers to linearity (over $mathbb{F}_2$) of the verifier predicate. Achieving the same result with $ε=0$ would imply the existence of a non-hyperlinear group.

Problem

• Proving RE-hardness of approximating quantum LCS games
• Generalizing Håstad's test for entangled provers
• Establishing MIP* = RE with constant-length answers

Innovation

• Generalized Håstad's long-code test
• Established MIP* equals RE
• Proved LIN* equals RE