The largest known gap between quantum and classical chromatic number of graphs, obtained via quantum protocols for colouring Hadamard graphs based on the Deutsch--Jozsa algorithm and the quantum Fourier transform, is exponential. We put forth a quantum pseudo-telepathy version of Khot's $d$-to-$1$ Games Conjecture and prove that, conditional to its validity, the gap is unbounded: There exist graphs whose quantum chromatic number is $3$ and whose classical chromatic number is arbitrarily large. Furthermore, we show that the existence of a certain form of pseudo-telepathic XOR games would imply the conjecture and, thus, the unboundedness of the quantum chromatic gap. As two technical steps of our proof that might be of independent interest, we establish a quantum adjunction theorem for Pultr functors between categories of relational structures, and we prove that the Dinur--Khot--Kindler--Minzer--Safra reduction, recently used for proving the $2$-to-$2$ Games Theorem, is quantum complete.

翻译：基于Deutsch--Jozsa算法和量子傅里叶变换的Hadamard图着色量子协议，所获得的图量子色数与经典色数之间的最大已知间隙是指数级的。我们提出Khot的$d$-对-$1$博弈猜想的一个量子心灵感应版本，并证明在其成立条件下，该间隙是无界的：存在量子色数为$3$而经典色数任意大的图。此外，我们证明特定形式的伪心灵感应XOR博弈的存在性将蕴含该猜想，从而蕴含量子色数间隙的无界性。作为证明中两个可能具有独立价值的技术步骤，我们建立了关系结构范畴间Pultr函子的量子伴随定理，并证明了近期用于证明$2$-对-$2$博弈定理的Dinur--Khot--Kindler--Minzer--Safra归约是量子完备的。