Satisfiability Phase Transtion for Random Quantum 3XOR Games

TL;DR

This paper introduces a polynomial time algorithm to determine perfect quantum strategies for 3XOR games, enabling numerical analysis of phase transitions and the rarity of pseudotelepathy in large random games.

Contribution

It provides the first explicit polynomial time algorithm for constructing or refuting perfect quantum strategies in 3XOR games, facilitating large-scale numerical studies.

Findings

Probability of pseudotelepathy games is bounded below 0.15.

Both quantum and classical phase transitions occur at a ratio m/n ≈ 2.74.

Strong evidence of simultaneous quantum and classical phase transitions.

Abstract

Recent results showed it was possible to determine if a modest size 3XOR game has a perfect quantum strategy. We build on these and give an explicit polynomial time algorithm which constructs such a perfect strategy or refutes its existence. This new tool lets us numerically study the behavior of randomly generated 3XOR games with large numbers of questions. A key issue is: how common are pseudotelephathy games (games with perfect quantum strategies but no perfect classical strategies)? Our experiments strongly indicate that the probability of a randomly generated game being pseudotelpathic stays far from 1, indeed it is bounded below 0.15. We also find strong evidence that randomly generated 3XOR games undergo both a quantum and classical "phase transition", transitioning from almost certainly perfect to almost certainly imperfect as the ratio of number of clauses ($m$) to number of questions ($n$) increases. The locations of these two phase transitions appear to coincide at $m/n \approx 2.74$.