arXiv:2611.15839 · v1

k-partite quantum homomorphisms

首次提交 2026-11-12 02:50

版本更新 2026-11-12 02:50

雷达观测 2026-11-13 14:16

摘要

$k$-partite XOR games and linear binary constraint system (BCS) games arise from systems of linear equations over $\mathbb{Z}_2$, providing distinct quantum relaxations of the same classical constraint satisfaction problem (CSP). Linear BCS games have been generalised to synchronous homomorphism games on arbitrary relational structures, with compositional semantics provided by the quantum monad. We introduce $k$-partite homomorphism games, extending $k$-XOR games from linear systems to arbitrary relational structures. This yields a framework for studying multipartite quantum CSPs and quantum homomorphisms. We characterise perfect strategies in both the tensor-product and commuting-operator models as Kleisli morphisms of suitable state-dependent graded quantum monads on a category of $k$-partite relational structures. We then study quantum advantage in the pseudotelepathic sense: the existence of a perfect quantum strategy without a perfect classical strategy. We show that synchronous and $k$-partite quantum advantage are incomparable at the level of CSP instances. The GHZ instance admits three-player quantum advantage but no synchronous commuting-operator advantage, whereas the magic-square instance admits synchronous quantum advantage but no $k$-partite commuting-operator advantage for any number of players or proper colouring. We also exhibit an explicit six-element template whose CSP is NP-complete and admits two-player quantum advantage. Finally, we prove that no bounded-width CSP template admits $k$-partite quantum advantage, even in the commuting-operator model.