We consider the problem of zero-error function computation with side information. Alice has a source $X$ and Bob has correlated source $Y$ and they can communicate via either classical or a quantum channel. Bob wants to calculate $f(X,Y)$ with zero error. We aim to characterize the minimum amount of information that Alice needs to send to Bob for this to happen with zero-error. In the classical setting, this quantity depends on the asymptotic growth of $\chi(G^{(m)})$, the chromatic number of an appropriately defined $m$-instance "confusion graph". In this work we present structural characterizations of $G^{(m)}$ and demonstrate two function computation scenarios that have the same single-instance confusion graph. However, in one case there a strict advantage in using quantum transmission as against classical transmission, whereas there is no such advantage in the other case.

翻译：我们研究带边信息的零误差函数计算问题。Alice拥有信源$X$，Bob拥有相关信源$Y$，双方可通过经典或量子信道进行通信。Bob需零误差地计算$f(X,Y)$。我们的目标是刻画实现该零误差计算时Alice需向Bob发送的最小信息量。在经典情形下，该量取决于适当定义的$m$实例“混淆图”的色数$\chi(G^{(m)})$的渐近增长。本文给出$G^{(m)}$的结构化特征，并展示两种具有相同单实例混淆图的函数计算场景。然而，其中一种场景中量子传输相对于经典传输具有严格优势，而另一种场景则不存在这种优势。