Abstract
From a given topological hypermap H, we define two related hypermaps H and H^∇ as complements of the ordinary dual hypermap H^* along with the concepts of their edge hypermap quantum codes C and C^∇. We then show that, when the sets of special darts are naturally corresponded, the duality between the ordinary hypermap quantum code C from H and the one C^* from H^* can be greatly simplified to the duality between C and C^∇.