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Duan and Winter studied the one-shot zero-error classical capacity of aquantum channel assisted by quantum non-signalling correlations, and formulatedthis problem as a semidefinite program depending only on the Kraus operatorspace of the channel. For the class of classical-quantum channels, they showedthat the asymptotic zero-error classical capacity assisted by quantumnon-signalling correlations, minimized over all classical-quantum channels witha confusability graph $G$, is exactly $\log \vartheta(G)$, where $\vartheta(G)$is the celebrated Lov\'{a}sz theta function. In this paper, we show that theone-shot capacity for a classical-quantum channel, induced from a circulantgraph $G$ defined by equal-sized cyclotomic cosets, is $\log \lfloor\vartheta(G) \rfloor$, which further implies that its asymptotic capacity is$\log \vartheta(G)$. This type of graphs include the cycle graphs of oddlength, the Paley graphs of prime vertices, and the cubit residue graphs ofprime vertices. Examples of other graphs are also discussed. This endows theLov\'{a}sz $\theta$ function with a more straightforward operational meaning.

On the One-Shot Zero-Error Classical Capacity of Classical-Quantum Channels Assisted by Quantum Non-signalling Correlations