Quantum Optimality in the Odd-Cycle game: the topological odd-blocker, marked connected components of the giant, consistency of pearls, vanishing homotopy

TL;DR

This paper characterizes the optimal quantum strategies for the Odd-Cycle game, linking topological and geometric properties to quantum winning probabilities, and introduces new objects like the topological odd-blocker.

Contribution

It introduces novel topological and geometric tools, such as the topological odd-blocker, to analyze quantum strategies in the Odd-Cycle game, advancing understanding of quantum game optimality.

Findings

Quantifies the relation between the giant connected component and quantum winning probability.

Introduces the topological odd-blocker and pearls as new analytical tools.

Connects the foam problem to quantum strategy optimality in the Odd-Cycle game.

Abstract

We characterize optimality of Quantum strategies for the Odd-Cycle game. Separate from other game-theoretic settings, parallel repetition for the Odd-Cycle game is related to the foam problem, which can be formulated through a minimization of the surface area. In comparison to previous works on minimizing the surface area, we quantify how properties of the marked giant connected component can be related to the maximum winning probability using Quantum strategies. Objects that we introduce to formulate such connections include the topological odd-blocker, previous examples of error bounds for other Quantum games that have been formulated by the author, pearls, consistent regions, and the cycle elimination problem.