[Paper Review] Two-player entangled games are NP-hard

Anand Natarajan, Thomas Vidick|arXiv (Cornell University)|Oct 9, 2017

Quantum Computing Algorithms and Architecture8 references6 citations

TL;DR

This paper proves that approximating the maximum success probability of two-player quantum-entangled games with logarithmic-question and constant-answer lengths is NP-hard to within constant factors. Using an improved analysis of the Raz-Safra low-degree test for two entangled provers, it establishes that NEXP ⊆ MIP* holds even with only two provers, resolving a key complexity-theoretic question.

ABSTRACT

We show that the maximum success probability of players sharing quantum entanglement in a two-player game with classical questions of logarithmic length and classical answers of constant length is NP-hard to approximate to within constant factors. As a corollary, the inclusion $\mathrm{NEXP}\subseteq\mathrm{MIP}^*$, first shown in [IV12] with three provers, holds with two provers only. The proof is based on a simpler, improved analysis of the low-degree test Raz and Safra (STOC'97) against two entangled provers.

Motivation & Objective

• To establish the computational hardness of approximating the quantum advantage in two-player nonlocal games with limited classical communication.
• To close the gap in understanding the power of two entangled provers in interactive proof systems, particularly in relation to the MIP* complexity class.
• To improve upon prior analyses of the low-degree test in the context of quantum entanglement, specifically for two entangled provers.
• To show that the inclusion NEXP ⊆ MIP* holds with only two provers, extending a prior result that required three.

Proposed method

• Adapts and re-analyzes the Raz-Safra low-degree test for use in the two-prover, entangled setting, focusing on soundness against quantum strategies.
• Introduces a refined analysis technique that bounds the success probability of entangled provers in the low-degree test more tightly than previous approaches.
• Leverages properties of quantum entanglement and nonlocal correlations to simulate classical proof systems with quantum provers.
• Uses the improved soundness bound to show that any language in NEXP has a two-prover quantum interactive proof system with perfect completeness and soundness bounded away from 1.
• Reduces the problem of approximating quantum game success probabilities to a known NP-hard problem, establishing hardness of approximation.

Experimental results

Research questions

• RQ1 Is approximating the maximum success probability of two-player entangled games with logarithmic questions and constant answers NP-hard?
• RQ2 Can the inclusion NEXP ⊆ MIP* be established with only two entangled provers, rather than three?
• RQ3 Can the Raz-Safra low-degree test be soundly analyzed in the presence of quantum entanglement for two provers?
• RQ4 What is the tightest possible bound on the success probability of entangled provers in the two-prover version of the low-degree test?

Key findings

• The maximum success probability of two-player entangled games with logarithmic-question and constant-answer lengths is NP-hard to approximate within any constant factor.
• The improved analysis of the Raz-Safra low-degree test against two entangled provers achieves a soundness bound that enables the proof of NEXP ⊆ MIP* with two provers.
• The result confirms that two entangled provers are sufficient to achieve the full power of MIP*, resolving a key open question in quantum complexity theory.
• The proof technique provides a stronger and simpler analysis than prior approaches, reducing reliance on complex multi-prover constructions.